ALMOST ETALE EXTENSIONS

Gerd Faltings

0. Introduction

The theory of almost étale coverings goes back to Tate for the case of discrete valuation rings. The higher dimensional theory has been invented by the author, and used to prove conjectures about Hodge-Tate and crystalline structures on the étale cohomology of varieties over p-adic fields. The main purpose of these notes is to extend the theory to cover toroidal singularities, especially all schemes with semistable reduction. This is possible because we have found a new method of proof for the main technical result, the purity theorem. This new method makes heavy use of Frobenius.

At the end we explain how to derive comparison maps to crystalline cohomology. However this is only sketched as all these ideas have already been presented in a number of papers ([Fa 3], [Fa 5], [Fa 6], [Fa 7], so that everybody interested in the matter can learn it there. We also add some (a little bit) incomplete results about cohomology of isocrystals, which we could not find in the literature.

All in all this suffices to prove the C_{st} -conjecture of Fontaine-Jansen. This had been done previously by T. Tsuji ([T]). However we can also treat the case of non-constant coefficients where some new phenomena occur. Anyway the author profited form his participation in the p-adic year at Institut Poincaré in Paris, first by giving lectures on a preliminary version and then by learning about T. Tsuji’s technique.

1. Almost-Mathematics

Let V̄ denote a (commutative with unit) ring together with a sequence of principal ideals 𝔪_{α} ⊆ V̄ parametrized by positive elements α ∈ Λ^{+}, where Y ⊆ Λ ⊆ ℚ denotes a subgroup done in ℝ. Denote by π a generator of 𝔪_{1}, and π^{α} a generator of 𝔪_{α}. Assume furthermore that π^{α}π^{β} = unit ·π^{α+β}.

Examples.

a) V̄ = ℤ_{p} = integral closure of ℤ_{p} in ℚ̄_{p}, 𝔪_{α} = elements of valuation ≥ α, π =

p (Λ = ℚ)

b) Consider R = \underleftarrow{lim}(ℤ̄_{p}/pℤ̄_{p}, Frobenius), V̄ = W (R), π = [\underline{p}, 0, …, 0] ∈

W (R), \underline{p} = \underleftarrow{lim} p^{1/p^{n}} ∈ R, Λ = ℤ[\frac{1}{p}], π^{α} = the obvious element.

If 𝔪 = ⋃_{α>0}𝔪_{α}, then a V̄ -module is almost zero if it is annihilated by 𝔪. Denote

this by M ≈ 0. Assume R is a V̄ -algebra and M an R-module.

1. Definition. M is almost projective if Ext_{R}^{i}(M, N) ≈ 0, all R-modules N and all i > 0 (or only i = 1)

Typeset by AMS-TEX

M is almost flat if T or_{i}^{R}(M, N) ≈ 0, all R-modules N and all i > 0 (or only i = 1) M is almost finitely generated if for each α > 0, α ∈ Λ, there exists a finitely generated R-module N and a π^{α} -isomorphism ψ_{α} : N → M (i.e., there exists ϕ_{α} : M → N with ϕ_{α} ∘ ψ_{α} = π^{α} · id_{N}, ψ_{α} ∘ ϕ_{α} = π^{α_{0}} ∘ id_{M}). M is almost finitely presented: dito with N finitely presented.

2. Remarks.

a) P almost projective ⇔ for all surjections f : M → N and maps g : P → N, π^{α} ·g factors through f(α > 0, α ∈ Λ) ⇔ all α > 0, α ∈ Λ, there exists a free R-module L and maps L ⟶^{g} P ⟶^{f} L with g ∘ f = π^{α} · id_{p}.

b) If P is almost projective and almost finitely generated, then P is almost finitely presented. Then also P^{∗} = Hom_{R}(P, R), P ⊗_{R} P, Λ^{i}P etc. are almost finitely generated projective. Furthermore there is an almost isomorphism P ⊗_{R} P^{∗} ≈ End_{R}(P), and thus a trace-map tr : 𝔪⊗_{R} End_{R}(P) → R, or equivalently End_{R}(P) → Hom(𝔪, R). Thus for f ∈ End_{R}(P) we can define a power-series

det(1 + T f) = ∑_{i=0}^{∞} tr(Λ^{i}f) · T^{i} ∈ Hom(𝔪, R)[[T]].

One checks that Hom(𝔪, R) is a ring

f ∘ g(π^{α+β}) = f(π^{α}) · g(π^{β}))

and that

det(1 + T f) det(1 + T g) = det(1 + T (f + g + T fg))(= ∑_{i=0}^{∞} tr(Λ^{i}(f + g + T fg)) · T^{i}

We say that P has rank ≤ r if Λ^{r+1}P ≈ 0. Then letting e ∈ Hom(𝔪, R) denote the coefficient of T^{r} in det(1+T ·id_{p}), one checks that e = e^{2} is an idempotent, and thus R ≈ R_{1} ×R_{2} factors (up to almost isomorphism). Also P ≈ (P ⊗_{R} R_{1})×(P ⊗_{R} R_{2}).

Over one of the factors (say R_{1})e = 1. One then checks that after base-change to R_{1} L = Λ^{r}P satisfies L ⊗_{R} L^{∗} ≈ R, i.e. L behaves like a line-bundle: There are the maps tr : End_{R}(L) ≈ L ⊗_{R} L^{∗} → R and the identity: R → End_{R}(L). We have tr(id_{p}) = e = 1. In fact for two f, g ∈ End_{R}(P) we have tr(Λ^{r}f) · tr(Λ^{r}g) = tr(Λ^{r}(fg)) = coefficient of T^{2r} in det((1 + T f)(1 + T g)). Apply this to f and g of the form

f(x) = ∑_{i=1}^{r} λ_{i}(x)μ_{i}, λ_{i} ∈ P^{∗}, μ_{i} ∈ P

g(x) = ∑_{i=1}^{r} ν_{i}(x)ϕ_{i}

to get tr(Λ^{r}f) · det(ν_{i}(ϕ_{j})) = det(ν_{i}(f(ϕ_{j}))), i.e. tr(λ^{r}f)(ν_{1} ∧ …, ∧ν_{r} | ϕ_{1} ∧ … ϕ_{r}) = (ν_{1} ∧ … ∧ ν_{r} | Λ^{r}f(ϕ_{1} ∧ … ∧ ϕ_{r})). Varying ν’s and g’s gives Λ^{r}f = tr(Λ^{r}f) is a scalar. As the Λ^{r}f (given by x ↦ ⟨λ_{1} ∧ … ∧ λ_{r} | x⟩μ_{1} ∧ … ∧ μ_{r}) generate End_{R}(L), everything follows. The same argument gives: If e = 0, then tr(Λ^{r}f) = 0 and Λ^{r}f = 0, and then Λ^{r}P = 0. Thus by induction we get an almost isomorphism

R ≈ R_{0} × … × R_{r},

such that over R_{i} Λ^{i}P is invertible, λ^{i+1}P = 0. Also over R_{i} we have a determinantfunction det(f) = tr(Λ^{i}f) : End_{R}(P) → Hom(𝔪, R_{i}) homogeneous of degree i with det(fg) = det(f) det(g), det(1) = 1.

3. Remark. I do not know whether for arbitrary finitely generated projective P ’s we have a decomposition R ≈ ∏_{i=0}^{∞} R_{i} with P ⊗_{R} R_{i} of rank i.

2. Almost étale coverings

1. Definition. A ring homomorphism A → B is called an almost étale covering if i) B is almost finitely generated projective as an A-module, of finite rank ii) B is almost finitely generated as a B ⊗_{A} B-module

2. Remarks.

a) There exists an idempotent e_{B/A} ∈ Hom(𝔪, B ⊗_{A} B) measuring where B has rank 1 as B ⊗_{A} B -module. e_{B/A} is annihilated by (b ⊗_{A} 1 − 1 ⊗_{A} b) and maps to 1 ∈ Hom(𝔪, B) under multiplication.

b) Replacing A by Hom(𝔪, A) we may reduce (in many cases) to the case where rank_{A}B = r is constant. Then there exists an almost faithfully flat base-change A → A′ (A′ is almost flat, and M ⊗_{A} A′ ≈ 0 implies M ≈ 0) with B′ = B ⊗_{A} A′ ≈ A^{′r}

Proof. If r ≥ 1, A → B is almost faithfully flat: If 𝔞 ⊆ A is an ideal with 𝔞B ≈ B, then 𝔪A ⊆ Norm_{B/A}(𝔞B) ⊆ 𝔞 Hom(𝔪, A). Furthermore after base-change to B we may split of one factor B from B ⊗_{A} B, thus reducing r to r − 1.

c) A → B is called a G-covering if a finite group G acts on B such that after almost faithfully flat base-change A → A′ we have B′ = B ⊗_{A} A′ ≈ A^{′G}. If B has constant rank r, there exists A → B → C with C/A is a Gcovering with G = γ_{r} (symm. group in r letters) C/B is an H-covering with H = γ_{r−1} ⊆ G (C ⊆ B^{⊗r} is an almost direct factor given by the idempotent ∏_{i<j} λ_{ij}(1 − e_{B/A}), λ_{ij} : B^{⊗2} → B^{⊗r} the canonical maps).

2a. Hochschild-Cohomology and Lifting

Recall that for an A-algebra B (with unit) we can form a complex of B-bimodules (B ⊗ B^{op} -modules)

C(B/A) : B ⊗_{A} B ← B ⊗_{A} B ⊗_{A} B ← B ⊗_{A} B ⊗_{A} B ⊗ −AB ←,

with differential

d_{n} : C_{n}(B/A) = B^{⊗n+2} → C_{n−1}(B/A) = B^{⊗n+1}

d_{n}(b_{0} ⊗ b_{1} ⊗ … ⊗ b_{n+1}) = ∑_{i=0}^{n}(−1)^{i}b_{0} ⊗ … ⊗ b_{i} · b_{i+1} ⊗ … ⊗ b_{n+1}

This complex has an augmentation C_{0}(B/A) → B and is (with B in degree −1) nullhomotopic as complex of left respectively right B-modules. For a B-bimodule M define

H^{∗}(B/A, M) = (Hom_{B⊗B^{op}}(C_{∗}(B/A), M))

Thus H^{0}(B/A, M) = M^{B} = {m ∈ M/bm = mb (all b ∈ B)} H^{1}(B/A, M) = { biderivations B → M}/{ inner derivations b → [b, m] = bm−mb} etc. For any B-module N the bimodule M = Hom_{A}(B, N) has

H^{i}(B/A, M) = \{ \begin{matrix}0, i > 0 \\ N, i = 0 \end{matrix}

For example if B is a projective B ⊗_{A} B -module then for any bimodule M we have that M = Hom_{B}(B, M) (using say left - B -module structures) is a direct summand in Hom_{B}(B⊗_{A}B, M) = Hom_{A}(B, M), and thus Hochschild-cohomology vanishes in positive degrees. By the same argument B almost B ⊗_{A} B -projective gives H^{i}(B/A, M) ≈ 0 for i > 0.

Now suppose that I ⊆ A is an ideal with square zero: I^{2} = 0.

If B is an A-algebra which is almost unramified (B is an almost projective B⊗_{A}B -module), C an A-algebra, f̄: B → C̄ = C/IC an A-algebra morphism (or almost morphism), then there is at most one way to lift f̄ to f : B → C (up to almost equality): For two lifts f_{1}, f_{2} the difference is an almost derivation B̄ → I · C, thus almost zero is H^{1}(B̄/Ā, I · C) ≈ 0. If B is also an almost projective A-module, then in fact an almost lifting exists: For each α > 0 π^{α}f̄ lifts to an A-linear map f_{α} : B → C. Consider g_{α}(x, y) = π^{2α}f_{α}(xy) − π^{α}f_{α}(x)f_{α}(y) ∈ I · C. Then f_{α}, g_{α} defines a cochain in Hom_{B⊗_{A}B}(C_{2}(B̄/Ā, I · C)). We claim that it is a cocycle. To check that compute the boundary

dg_{α}(x, y, z) = xg_{α}(y, z) − g_{α}(xy, z) + g_{α}(x, yz) − g_{α}(x, y)z

Using that multiplication by f_{α}(u) coincides with multiplication by π^{α}u in IC, this is:

f_{α}(x)(π^{α}f_{α}(yz) − f_{α}(y)f_{α}(z)) − π^{2α}f_{α}(xyz) + π^{α}f_{α}(xy)f_{α}(z)
+π^{2α}f_{α}(xyz) − π^{α}f_{α}(x)f_{α}(yz) − (π^{α}f_{α}(xy) − f_{α}(x)f_{α}(y))f_{α}(z) = 0.

Thus π^{α}g_{α} is a coboundary,

π^{α}g_{α}(x, y) = xh_{α}(y) − h_{α}(xy) + h_{α}(x)y,

and f̃_{α} = π^{3α}f_{α} + h_{α} lifts π^{4α} · f̄ and satisfies

π^{4α} · f̃_{α}(xy) = f̃_{α}(x)f̃_{α}(y)

Furthermore for 0 < α ≤ \frac{β}{2} < β

g_{β,α} = π^{4β}f_{β} − π^{(β−4α)}f_{α}

satisfies

g_{β,α}(xy) = f_{β}(x)f_{β}(y) − π^{8β−8α}f_{α}(x)f_{α}(y)
= f_{β}(x)(f_{β}(y) − π^{4β−4α}f_{α}(y)) + (f_{β}(x) − π^{4β−4α}f_{α}(x))π^{4β−4α}f_{α}(y)

(using the terms in brackets are in IC)

= π^{4β}x(f_{β}(g) − π^{4β−4α}f_{α}(y)) + (f_{β}(x) − π^{4β−4α}f_{α}(x))π^{4β}y
= x g_{β,α}(y) + g_{β,α}(x)y.

Thus g_{β,α} is an almost derivation, hence almost zero. Multiplying by π^{β} gives

π^{5β} · f_{β} = π^{g(β−α)}π^{5α} · f_{α}

Hence considering the π^{5α} ·f_{α} as multiplicative maps from π^{9_{α}} ·V ⊗_{V} B to C (lifting f̄), they glue to a map

f : 𝔪 ⊗_{V} B = \underrightarrow{lim}(π^{9_{α}} · V) ⊗_{V} B → C

Also f(1) ∈ Hom(𝔪, C) is an idempotent lifting 1, thus f(1) = 1.

Finally we lift almost étale coverings. First we treat the underlying projective module: Let P̄ be an almost projective Ā = A/I -module. For any α > 0 choose a free Ā -module L̄ and maps f̄_{α} : L̄_{α} → P, ḡ_{α} : P → L̄_{α} with f̄_{α} ∘ ḡ_{α} = π^{α} · id. Let ē_{α} ∈ End(L̄_{α}) denote ḡ_{α} ∘ f̄_{α}, so ē_{α}^{2} = π^{α} · ē_{α}.

Lift ē_{α} to an endomorphism f_{α} of L. Thus f_{α}^{2} − π^{α}f_{α} takes values in I · L, hence (f_{α}^{2}−π^{α}f_{α})^{2} = 0. If e_{α} = 3π^{α}f_{α}^{2}−2f_{α}^{3}, then e_{α} lifts π^{2α}e_{α} and satisfies e_{α}^{2} = π^{3α}e_{α}. Hence P_{α} = L_{α}/(π^{3α} − e_{α})L_{α} is ”projective up to π^{3α}”, i.e. π^{3α} annihilates all higher Ext_{A}^{i}(P_{α},?). Also f̄_{α} and ḡ_{α} induce maps

P̄_{α} ⟶^{f̄′_{α}} P̄ ⟶^{ḡ′_{α}} P̄_{α}

with

f̄′_{α} ∘ ḡ′_{α} = π^{α} · idP̄, π^{2α} ∘ ḡ′_{α} = π^{3α} · idP̄_{α}

For α ≤ β/2 < β,

π^{3α+4α−8β} ∘ ḡ′_{β} ∘ f̄′_{α} : P_{α} → P̄_{α} → P̄ → P̄_{β}

lifts to a map g_{α,β} : P_{α} → P_{β} with f̄′_{β} ∘ \overline{g_{α,β}} = π^{7α−7β} ∘ f̄′_{α}. Considering g_{α,β} as a map from π^{7α}V ⊗_{V} P_{α} to π^{7β}V ⊗_{V} P_{β}, we form P = \underrightarrow{lim}_{α}π^{7α}V ⊗_{V} P_{α}, for a

suitable sequence α = α_{n}, α_{n+1} ≤ \frac{α_{n}}{2} . Then the f̄_{α} define an almost isomorphism P ⊗_{A} A/I ≈ P̄. As coker ḡ_{α,β} is annihilated by π^{7α−7β+α+3β} and hence by π^{8α}, coker g_{α,β} is annihilated by π^{16α}, and π^{7α}V ⊗_{V} P_{α} → P is an isomorphism up to π^{16α}. Thus P is almost projective and lifts P̄.

Remark. If P̄ has rank ≤ r, so has P.

Now assume B̄/Ā is an almost étale covering, that B̄ is almost projective over Ā as well as over B̄ ⊗Ā B̄. We claim that there is an almost étale covering B/A with B ⊗_{A} A/I ≈ B̄: We give a sketch, as the method follows the previous but the constants become more complicated. First lift B̄ to an almost projective A-module B. Then for all α > 0 the multiplication

m̄ : B̄ ⊗_{A} B̄ → B̄

lifts after multiplication by π^{α} to

m_{α}^{0} : B ⊗_{A} B → B.

Then π^{cα}(m_{α}^{0}(m_{0}^{α}(a, b), c) = m_{α}^{0}(a, m_{0}^{α}(b, c))) (c = suitable constnat) defines a class in H^{3}(B̄/Ā, I · B) ≈ 0. After modifying m_{α} we can make it associative. Then the commutator with a fixed element defines a derivation, thus π^{cα} · m_{α} (c suitable) is commutative, and similarly 1 ∈ B̄ lifts to an idempotent which is a unit. For α ≤ \frac{β}{2} < β a suitable multiple π^{cα} · idB̄ lifts to a multiplicative map

(π^{cα}V ⊗_{V} B, m_{α}) → (π^{cβ}V ⊗_{V} B, m_{β}).

Then B = \underrightarrow{lim} π^{cα}V ⊗_{V} B with \underrightarrow{lim} m_{α} does the job.

We have shown

3. Theorem. The functor sending B to B̄ = B ⊗_{A} Ā is an equivalence of categories (almost étale coverings of A) ⟶^{∼} (almost étale coverings of Ā).

2b. The Purity Theorem

The previous theory is applied to the study of universal étale coverings. This was first done by Tate: Assume V is a ρ-adic discrete calculation-ring, with residue field k = V/πV perfect of characteristic p > 0 (π a uniformiser) and fraction field K of characteristic zero. Assume furthermore that K_{∞} is a totally ramified ℤ_{p} -extension of K, V_{∞} ⊆ K_{∞} the normalisation of V. Then for any finite extension L_{∞} of K_{∞}, the normalisation W_{∞} of V_{∞} in L_{∞} is an almost étale covering of V_{∞}. This was generalised to non-perfect residue fields as follows: (see [F2], Theorem 1.2) Assume [k : k^{p}] = p^{d} < ∞. Assume given a tower K = K_{0} ⊂ K_{1} ⊂ K_{2} ⊂ … ⊂ K_{∞} = ⋃_{n≥0}K_{n}. Let V_{n} = normalisation of V in K_{n}, Ω_{V_{n}/V} the universal differential

module. It is known that Ω_{V_{n}/V} can be generated by ≤ d + 1 elements, and that its length is the different of V_{n}/V. Now assume that for any m ≥ 0 there exists an n such that Ω_{V_{n}/V} has as quotient (V_{n}/π^{m}V_{n})^{d+1}. Then again for any finite L_{∞} ⊇ K_{∞}, W_{∞} = normalisation of V_{∞} is an almost étale covering of V_{∞}.

Now suppose that over V we have an affine normal torus-embedding T ≅ 𝔾_{m}^{d+1} ⊆ T̄ ([KKMS], § 1) If L = Hom(𝔾_{m}, T) denotes the characters of T, we furthermore assume given a rational convex polyhedron σ ⊆ L_{ℝ} not containing any line, such that T̄ = Spec(V [L^{∗} ∩ σ^{∨}]). Finally we assume given an element λ ∈ L^{∗} ∩ σ^{∨} such that σ ∩ ker(λ) is a simplex, i.e. consists of the positive linear combinations of elements ρ_{1}, …, ρ_{r} ∈ L ∩ σ which are linearly independant over ℤ. Let T̄_{λ} ⊆ T̄ denote the closed subscheme defined by the vanishing of (π − λ). Assume given a noetherian ring R and a map f : Spec(R) → T̄_{λ} which is flat, has geometrically regular fibres and is formally étale in charateristic p, i.e. for a prime 𝔭 ⊆ R(p ∈ 𝔭) with image x ∈ T̄ is R_{𝔭} formally étale over OT̄_{λ,x}. This is equivalent to the fact that Frobenius induces an isomorphism (R̂_{𝔭}/pR̂_{𝔭}) ⊗ OT̄_{λ,x} ⟶^{∼} R̂_{𝔭}/pR̂_{𝔭}. (ˆ = completion in 𝔭 -adic topology). Also for each residue-field κ of Ō_{T_{λ}} the fibre R ⊗ κ has dimension ≤ log_{p}[κ : κ^{p}] ≤ d. Denote by T̄_{n} → T̄ the covering induced by multiplication by n!, Spec(R_{n}) → Spec(R) its pullback via Spec(R) → T̄_{λ} ⊆ T̄ (then R_{n} is a module over V_{n} = V [π^{1/n!}] because of the relation λ = π). If R_{∞} = ⋃_{n>0}R_{n}, and if R_{∞} → S_{∞} is the normalisation of R_{∞} in a finite étale

covering of the preimage of T ⊆ T̄ in R_{∞}[\frac{1}{π}], we claim that S_{∞} is an almost étale covering of R_{∞}.

4. Theorem. S_{∞}/R_{∞} is almost étale.

Reduction-steps.

i) Instead of the T̄_{n} we may choose Spec V [L_{n}^{∗} ∩ σ^{∨}], where the L_{n} ⊆ L form a decreasing sequence of sublattices such that for any integer m, L_{n} ⊆ m · L for n >> 0.

ii) We may start with a fixed L_{n} (instead of L) right away. Especially we may assume that S_{∞} is defined by an R-algebra S such that S[\frac{1}{π}] is finite étale over R[\frac{1}{π}] over the preimage of T ⊆ T̄.

iii) For a suitable choice of L_{n} we can achieve that

T̄_{λ,n} = Spec(V [L_{n}^{∗} ∩ σ^{∨}]/(λ − π)) ⊇ T_{λ,n} = Spec(V [L_{n}^{∗}]/(λ − π))

is over Spec(K) an embedding of the complement of a simple normal crossings divisor (use the properties of λ), and especially T̄_{λ,n} ⊗_{V} K is regular.

By Abhyankhar’s lemma we then may assume that S[\frac{1}{π}] is finite étale over R[\frac{1}{π}].

iv) From now on it will be sufficient to work with a sequence of lattices L_{n} cofinal with p^{n}L. The more general result then follows by base-extension (once we have that S_{∞}[\frac{1}{π}] extends to an almost étale covering, this covering must be the normalisation of R_{∞}, and this remains true after base-change).

v) Next we consider what happens in the generic points in characteristic p. Suppose Z is an irreducible component T̄_{λ} ⊗_{V} k = Spec(k[L ∩ σ^{∨}]/(λ)). Then σ corresponds to an extremal ray (halfline) in σ. If it is generated by an indivisible element ρ ∈ L, and if λ(ρ) = e, then in a neighbourhood of the generic point z of Z the torus-embedding T̄ is isomorphic to T′ × 𝔸^{1}, T′ ≅ 𝔾_{m}^{d}, and T̄_{λ} has as local ring there

T̄_{λ,z} ≅ V′[t]/(t^{e} − uπ) ≅ V′[^{e}\sqrt{uπ}]

where V′ is the lcoal ring corresponding to T′, u ∈ V′ is a unit. As uπ is a uniformiser of V′ it follows that T̄_{λ,z} is a discrete valuation-ring. It follows that T̄_{λ} satisfies R_{1}. As it is Cohen-Maccaulay (T̄ is as all torusembeddings are [KKMS], § 3), T̄_{λ} is even normal. Also R and R_{n} are normal and Cohen-Macaulay.

After that interlude let us pass to the sublattices L_{n}, and choose a projective system of points z_{n} ∈ T̄_{λ,n} lifting z. We claim that the local rings OT̄_{λ,n,zn} satisfy the ramification-conditions of [Fa2], Th. 1.2: The integer d there coincides with our d. If we compute relative logarithmic differentials we obtain L_{n}^{∗}/L_{n}^{∗} ⊗ OT̄_{λ,n,zn}. The usual differentials map into these, with cokernel annihilated by ∏_{i=0}^{d+1} μ_{i}, if the μ_{i} ∈ σ^{∨} ∩ L_{n}^{∗} form a basis of L_{n}^{∗}. Now for L_{n} = p^{n}L the valuation of this product decreases like c · p^{−n}, and the claim follows for those L_{n} and thus also for any sequence.

By formal étaleness the same applies to the sequence (R_{n}, 𝔭_{n}), where 𝔭_{n} ⊆ R_{n} is a projective system of primes above a fixed 𝔭 ⊆ R which is a minimal prime divisor of pR or πR. It follows that S_{∞,p} is almost étale over R_{∞,𝔭}.

vi) It now suffices to prove that S_{∞} is almost flat over R_{∞}: There exists an idempotent e_{S_{∞}/R_{∞}} ∈ S_{∞} ⊗_{R_{∞}} S_{∞}[\frac{1}{π}] defining the diagonal (the direct summand S_{∞}[\frac{1}{π}]). For each α > 0 its product with π^{α} then lies in

⋂_{𝔭}(S_{∞} ⊗_{R_{∞}} S_{∞})_{𝔭} ∩ S_{∞} ⊗_{R_{∞}} S_{∞}[\frac{1}{π}] ≈ S_{∞} ⊗_{R_{∞}} S_{∞}

(𝔭 runs over all minimal prime-divisors of πR). Thus S_{∞} is an almost direct summand in S_{∞} ⊗_{R_{∞}} S_{∞}. Furthermore if

π^{α}e_{S_{∞}/R_{∞}} = ∑_{i=1}^{r} x_{i} ⊗ y_{i} (x_{i}, y_{i} ∈ S_{∞}),

then π^{α}x = ∑_{i=1}^{r} x_{i} · T r_{S_{∞}/R_{∞}}(y_{i}x) for all x ∈ S_{∞}, which exhibits S_{∞} as an up to π^{α} direct summand in R_{∞}^{r}. Then S_{∞} is almost projective over R_{∞}.

By localization we may thus assume that R is local with maximal ideal 𝔪, and that S_{∞} is almost flat over R_{∞} on the punctured spectrum Spec(R) − {𝔪}. It also follows that S_{∞} ⊗_{R} R̂(R̂ = 𝔪) -adic completion) is almost equal to the normalisation of R_{∞} ⊗_{R} R̂, using that R̂ is normal (f is formally smooth) and that the two can only differ almost by 𝔪 -torsion.

vii) Assume now that dim(R) = 2: Let z ∈ T̄_{λ} denote the image of {𝔪}, z_{n} ∈ T̄_{λ,n} a projective system of liftings. We claim that for a suitable choice of the lattices L_{n} the local rings of T̄_{λ,n} in z_{n} are regular. The same is then true for R_{n}. As S_{n} is normal of dimension two it is Cohen-Macaulay, and thus flat over R_{n} (depth + projective dimension = 2). Thus S_{∞} is flat over R_{∞}.

To show the claim we may assume that π ∈ 𝔪. Thus z ∈ T̄_{λ}⊗_{N} k ⊆ T̄⊗_{V} k corresponds two a codimension-two-point in T̄ ⊗_{V} k. If it is not the generic point of a codimension-two stratum each T̄_{λ,n,z_{n}} is already regular by the previous. If it is such a generic point then near z T̄_{λ,n} looks like the product of a 𝔾_{m}^{d−1} with a torus-embedding of rank 2. This torus-embedding is given by a 2-dimensional convex polyhedral cone which must be a simplex. For suitable choice of the L_{n} this simplex is spanned by two basis-vectors of L_{n}, and T̄_{n} is smooth over V in z_{n}. It then follows easily that T̄_{λ,n} is still regular in z_{n}. (The singularity looks like V_{n}[T_{1}, T_{2}]/(T_{1}^{e}T_{2}^{f} − π_{n})).

So finally now dim(R) ≥ 3. The Frobenius-mapping is surjective on R_{∞}/pR_{∞} (or R_{∞}/πR_{∞}):

This holds for OT̄_{∞}, and S_{∞} is generated by OT̄_{∞} and R^{p}.

Following Fontaine we form the projective limit R = \underleftarrow{lim}(R_{∞}/pR_{∞}) (transitionmaps are Frobenius) and A_{inf}(R) = W (R). Then A_{inf}(R)/p · A_{inf}(R) ⟶^{∼} R, A_{inf}(R) admits a Frobenius-automorphism ϕ, and there is a surjection A_{inf}(R) ↠ R̂_{∞} (ρ-adic completion) with kernel generated by a single element ξ = p − [p]. Here [p] = [p, 0, …, 0] with p = \underleftarrow{lim}(p^{1/p^{n}}) ∈ R a compatible system of p-power roots of p. Note that ϕ(ξ) = p − [\underline{p}]^{p} = p − (p − ξ)^{p} = ±ξ^{p} + p · (unit). The elements ρ and ξ form a regular sequence in A_{inf}(R), and A_{inf}(R)/ξ is complete in the ξ -adic topology. However this topology is not ϕ- invariant. We thus consider the finer topology where a fundamental system of neighbourhoods of zero is given by the ideals generated by finite products (m_{n} ≥ 0, almost all zero) ∏_{n∈ℤ}(ϕ^{n}(ξ))^{m_{n}}. Note that any product ∏_{n>0} ϕ^{n}(ξ)^{m_{n}} with positive ϕ -powers forms together with ξ a regular sequence in A_{inf}(R), and that the ideal generated by those two elements defines the same topology as (p, ξ). Thus A_{inf}(R) with the topology defined by ∏_{n=0}^{N} ρ^{n}(ξ) is the fibred product of (A, ξ -adic topology) with (A, ∏_{n=1}^{N} ϕ^{n}(ξ) -adic topology), over (A, (p, ξ) -adic topology).

Now S_{∞} defines an almost étale covering V_{∞} of U_{∞} = Spec(R_{∞})−Spec(R_{∞}/𝔪R_{∞}). It obviously has finite rank [L_{∞} : K_{∞}]. Furthermore if we reduce modulo p it is almost invariant under Frobenius, i.e. Frobenius on O_{V_{∞}}/pO_{V_{∞}} induces an almost isomorphism

O_{V_{∞}}/pO_{V_{∞}} ⊗_{O_{U∞} /pO_{U∞}} O_{U_{∞}}/pO_{U_{∞}} ≈ O_{V_{∞}}/pO_{V_{∞}} :

Reduce to the split case.

Let 𝔪_{0} ⊆ A_{inf}(R) denote the ideal generated by p and suitable lifts of generators of 𝔪. Then we can lift V_{∞} first to an almost étale covering of Spec(A_{inf}(R)/ξ^{m}) − {𝔪_{0}}, all m, which by transport of structure induces almost étale coverings on Spec(A_{inf}(R)/ϕ^{n}(ξ)^{m}) − {𝔪_{0}}(n ∈ ℤ). These are all isomorphic modulo p because of Frobenius-invariance, and thus by induction (and gluing, that is taking fibred products) we obtain almost étale coverings of

Spec(A_{inf}(R)/(∏_{n=0}^{N} ϕ^{n}(ξ))^{m}) − {𝔪_{0}}

and by applying ϕ of

Spec(A_{inf}(R)/ ∏_{n∈ℤ} ϕ^{n}(ξ)^{m_{n}}) − {𝔪_{0}}.

That is if we consider the ind-scheme defined by quotients A_{inf}(R)/ ∏_{n∈ℤ} ϕ^{n}(ξ)^{m_{n}}, then we have defined a ϕ -invariant almost étale covering of its punctured spectrum.

Notations.

X_{inf} = ” \underrightarrow{lim} ”Spec(A_{inf}(R)/ ∏_{n∈ℤ} ϕ^{n}(ξ)^{m_{n}})
U_{inf} = X_{inf} − {𝔪_{0}}
V_{inf} → U_{inf} : the almost étale covering derived from S_{∞}.

Also ”almost” should now be defined with respect to the sequence of ideals generated by \underline{p}^{1/p^{n}} = φ^{−n}(\underline{p}). Define the cohomology H^{∗}(V_{inf}, O_{V_{inf}} ) as Čech-cohomology of the affine covering defined by a system of generators of 𝔪. This is easily seen to be independant of the choice of the covering, and one has long exact cohomologysequences from short (topologically) exact sequences. I conjecture but have not checked that it coincides with the cohomology of the ind-scheme V_{inf}. It certainly admits a Frobenius-automorphism (almost) ϕ.

For i > 0 the cohomology

H^{i}(V_{inf}, O_{V_{inf}} /ξ · O_{V_{inf}} ) ≅ H_{𝔪}^{i+1}(S_{∞}),

the local cohomology (SGA2). It satisfies a certain finiteness condition: Firstly for any n the cohomology H_{𝔪}^{2}(S_{n}) is a finitely generated S_{n} -module and thus of finite length, as for any prime 𝔭 ≠ 𝔪 depth(S_{n,𝔭}) + dim(S_{n/𝔭}) ≥ 3. Furthermore there are maps

H_{𝔪}^{2}(S_{n}) ⊗_{R_{n}} R_{∞} → H_{𝔪}^{2}(S_{n} ⊗_{R_{n}} R_{∞}) → H_{𝔪}^{2}(S_{∞}).

For any positive α > 0 the kernels and cokernels of both maps are annihilated by p^{α}, for n big enough:

If R_{∞} where flat over R_{n} the first map would be an isomorphism. This is not true in general, but in the appendix 2 it is shown that for a sequence α_{n} with α_{n} → 0 p^{α_{n}} annihilates all T or_{i}^{R_{n}}(R_{m}, M), (i > 0) (m ≥ n and M any R_{n} -module). This implies the assertion for the first map.

For the second one studies S_{n}⊗_{R_{n}} R_{∞} → S_{∞}. Its kernel consists of p-torsion and is annihilated by the p^{α_{n}} from above. For the cokernel use that for each height-one prime 𝔭 the trace-form on S_{n,𝔭} has discriminant p^{β_{n}} with β_{n} → 0. Thus

p^{β_{n}} · S_{∞,𝔭} ⊆ ⋂(S_{n} ⊗_{R_{n}} R_{∞})_{𝔭}

and (using annihilation of T or′_{1}s)

p^{α_{n}+β_{n}} · S_{∞} ⊆ (⋂_{𝔭} S_{n,𝔭}) ⊗_{R_{n}} R_{∞} = S_{n} ⊗_{R_{n}} R_{∞}.

It follows that for each α > 0 p^{α} · H_{𝔪}^{2}(S_{∞}) is contained in a finitely generated 𝔪 -torsion S_{∞} -module. In the appendix 2 it is also shown how to define an invariant (the normalized length)

λ(p^{α} · H_{𝔪}^{2}(S_{∞})) ∈ ℝ.

λ is ≥ 0, is additive in short exact sequences, and for an 𝔪 -torsion module which is a submodule of a finitely presented 𝔪 -torsion S_{∞} -module, λ vanishes only if the module is zero. Also if M is annihilated by p^{\frac{1}{p}}, then

λ(M ⊗_{R_{φ}} R/pR) = p^{d+1}λ(M).

Now consider

M = H^{1}(V_{inf}, O_{inf}).

Then M/ξM injects into H^{1}(V_{∞}, O_{V_{∞}}) = H_{𝔪}^{2}(S_{∞}). Thus for any α > 0 p^{α}(M/ξM) as well as p^{α}(M/(ξ, p)M) have finite λ -invariant. Furthermore Frobenius induces an almost isomorphism

M/(ϕ^{−1}(ξ), p)M ⊗_{R_{φ}} R/pR ≈ M/(ξ, p)M

Thus for any β > 0 (p^{β} takes care of the ”almost”)

p^{d+1}λ(p^{α}M/(ϕ^{−1}(ξ), p)M) ≤ λ(p^{pα−β}M/(ξ, p)M)

Choose 0 < α < \frac{1}{p^{2}} , β ≤ \frac{α}{p−1}. Then p^{pα−β}M/(ξ, p)M has a filtration by submodules p^{α−β} + i(\frac{1}{p} − α)M/(ξ, p)M, 0 ≤ i ≤ p. Furthermore each subquotient is a quotient

of p^{α}M/(ρ^{−1}(ξ), p)M, via multiplication by p^{(p−1)α−β+i( \frac{1}{p} −α)}. Thus the right hand side is ≤ (p + 1)λ(p^{α}M)(ρ^{−1}(ξ), p)M). As p^{d+1} ≥ p^{3} > p + 1, we see that both sides vanishes, so λ(p^{α} · M/(ξ, p)M) = 0 (all α > 0) and λ(p^{α} · M/ξM) = 0 (p is nilpotent in M/ξM). Now for n >> 0 there is a map

M/ξM → H_{𝔪}^{2}(S_{n}) ⊗_{R_{n}} R_{∞}

with kernel annihilated by p^{α}. As λ(p^{α} · image) = 0 it follows that p^{α} · iamge = 0, thus p^{2α} annihilates M/ξM. As α was arbitrary, M/ξM ≈ 0, and by induction also any quotient M/ ∏_{n∈ℤ} ϕ^{n}(ξ)^{m_{n}} · M ≈ 0 vanishes almost.

It follows that the map Γ(V_{inf}, O_{V_{inf}} ) → S_{∞} = Γ(V_{∞}, O_{V_{∞}}) is almost surjective: Order the finite products ∏_{n∈ℤ} ϕ^{n}(ξ)^{m_{n}} in a countable sequence g_{1} = ξ, g_{2}, g_{3}, …, with g_{n+1}/g_{n} = a ϕ -power of ξ. Then H^{1}(V_{inf}, g_{n} · O_{V_{inf}} /g_{n+1} · O_{V_{inf}} ) ≈ 0, as this φ -power induces an almost isomorphism with M. For α > 0 choose a strictly increasing sequence α_{1} = 0 < α_{n} < α_{n+1} … < α. For f_{1} ∈ S_{∞} = Γ(V_{inf}, O_{V_{inf}} /g_{1} · O_{V_{inf}} ) construct inductively f_{n} ∈ Γ(V_{inf}, O_{V_{inf}} /g_{n}·O_{V_{inf}} ) with f_{n+1} lifting \underline{p}^{α_{n+1}−α_{n}}· f_{n}. This is possible because \underline{p}^{α_{n+1}−α_{n}} annihilates the obstruction to lift f_{n}. Then the p^{α−α_{n}} · f_{n} form a compatible system of lifts of \underline{p}^{α} · f_{1} = p^{α} · f_{1}. Thus finally S_{inf} = Γ(V_{inf}, O_{V_{inf}} ) is an A_{inf}(R) -algebra with S_{inf}/ξ · S_{inf} = S_{∞}. Furthermore Frobenius induces an almost automorphism of S_{inf}.

Next consider the cokernel M of the map

S_{inf} ⊗_{A_{inf} (R)} S_{inf} → Γ(V_{inf}, O_{V_{inf}} ⊗_{A_{inf} (R)} O_{V_{inf}} ) :

On it Frobenius is an almost isomorphism. On the other hand M/ξ · M (almost) injects into the cokernel of S_{∞} ⊗_{R_{∞}} S_{∞} → Γ(V_{∞}, O_{V_{∞}} ⊗_{R_{∞}} O_{V_{∞}}), that is H_{𝔪}^{1}(S_{∞} ⊗_{R_{∞}} S_{∞}).

By the same reasoning as before H_{𝔪}^{1}(S_{∞} ⊗_{R_{∞}} S_{∞}) is for each α > 0 p^{α} - isomorphic to a finitely presented torsion-module, thus λ(p^{α} · M/ξM) < ∞ (all α > 0) and M/ξM ≈ 0. Finally the canonical idempotent defining the diagonal defines an almost section of O_{V_{inf}} ⊗_{A_{inf(R)}} O_{V_{inf}} lifting the corresponding idempotent e_{S_{∞}/R_{∞}}. It is thus at least modulo ξ in the image of S_{inf} ⊗_{A_{inf(R)}} S_{inf}, and e_{S_{∞}/R_{∞}} lies almost in the image of S_{∞} ⊗_{R_{∞}} S_{∞}.

Appendix 1. Some toroidal geometry. Assume Spec(R) → T̄_{λ} is formally étale in characteristic p. We assume that R is local with maximal ideal containing π. We want to study how much of this parametrisation is canonical.

Obviously we can replace σ by its face parametrising the stratum of T̄ which contains the image of the closed point of Spec(R). Let L′ ⊆ L denote the lattice spanned by σ. Then we have a formally smooth map

Spec(R) → T̄′_{λ} ⊆ T̄′,

into the torus-embedding defined by (L′, σ). Thus change notation and assume L = L′, but the map only formally smooth (i.e. flat with geometrically regular fibres).

4. Proposition. R is normal. The complement of the open stratum in Spec(R) is the union of irreducible divisors E. They correspond one to one to the extremal rays of σ. If ρ ∈ L^{∨}∩σ^{∨} is the indivisible generator of such a ray, then for any μ ∈ L the function m ∈ R has valuation μ(ρ) in the generic point of the correspodning divisor.

Furthermore L^{∨} ∩ σ^{∨} = invertible ideals I ⊂ R with support in the complement of the open stratum, via μ ↦ R · μ.

Proof. Choose a ρ ∈ L which lies in the interior of σ. Then define a decreasing filtration on V [L^{∨} ∩ σ^{∨}] by F^{b}(V [L^{∨} ∩ σ^{∨}]) = ⊕_{μ(φ)≥n}V · μ. It induces a quotient filtration on V [L^{∨} ∩ σ^{∨}]/(π − λ) with gr_{F_{ρ}}^{∘}(V [L^{∨} ∩ σ^{∨}]/(π − λ)) ≅ k[L^{∨} ∩ σ^{∨}], and by pushout a filtration F_{ρ}^{∘} on R with gr_{F_{ρ}}(R) ≅ κ[L^{∨} ∩ σ^{∨}], κ = residue-field of R. Furthermore this filtration defines the 𝔪 -adic topology on R. As the associated graded is normal, so is R.

For each extremal ρ we obtain a divisor in T̄ isomorphic to the torus-embedding defined by (L/⟨ρ⟩, projection of σ), and the valuation of μ ∈ L^{∨} along this divisor is μ(ρ). The intersection with T̄_{λ} is as follows: If λ(ρ) = 0, it is T̄′_{λ} with T′ corresponding to L/⟨ρ⟩. Its preimage in Spec(R) is normal and thus irreducible. If λ(ρ) = n > 0, then the intersection does not meet the generic fibre of T̄_{λ}. In the special fibre T̄_{λ} ⊗_{V} k = Spec(k[L^{∨} ∩ σ^{∨}](λ)) the intersection is the boundary component of Spec(k[L^{∨} ∩σ^{∨}]) defined by ρ. Its preimage in Spec(R) is irreducible for the same reasons as before, and also the valuation of μ is as required (the intersection with T̄_{λ} is transverse). As the complement of the open stratum in Spec(R) is the union of the divisors, we have shown the second assertion.

Finally assume I = R · f ⊆ R is an invertible ideal. For any ρ ∈ L^{∨} ∩ σ^{∨} we can as before define filtrations F_{ρ} on V [L^{∨} ∩ σ^{∨}], V [L^{∨} ∩ σ^{∨}]/(π − λ) and R. Now gr_{F_{ρ}}^{0}(R) is formally smooth over gr_{F_{ρ}}^{0}(V [L^{∨} ∩ σ^{∨}]/(π − λ)) = k[L^{∨} ∩ σ^{∨} ∩ ρ^{⊥}], and gr_{F_{ρ}}(R) is an integral domain (as this holds for T̄_{λ}).

Now the conditions on I mean that f divides some μ_{0} ∈ L^{∨}∩σ_{j}^{∨}, μ_{0} = f ·g. First choose ρ ∈ σ^{∘}. Then if a and b are maximal with f ∈ F_{ρ}^{a}(R), g ∈ F_{ρ}^{b}(R), it follows by compution in gr_{F_{ρ}}(R) = κ[L^{∨} ∩ σ^{∨}] that a + b = μ_{0}(ρ), and after multiplication by units that there is a decomposition μ_{0} = μ − ν with f − μ ∈ F_{ρ}^{a+1}(R), g − ν ∈ F_{ρ}^{b+1}(R). μ and ν do not change if we vary slightly the ray ℝ_{+} · ρ. As σ^{0} is convex they are thus independant of ρ.

Next choose ρ generating an extremal ray of σ. It corresponds to an irreducible divisor E. We show that for the corresponding valuation v_{E}(f) ≤ v_{E}(μ) = μ(ρ). If this holds for all E then f divides μ (by normality) and g divides ν (by symmetry). But then I = R · μ. Next assume that v_{E}(f) > μ(ρ) = n. Thus f ∈ F_{ρ}^{n+1}(R), which is generated by finitely many χ ∈ L^{∨} ∩ σ^{∨} with χ(ρ) ≥ n + 1. Choose a ray in the interior σ^{∘} sufficiently close to ρ. It is generated by a ρ̃ such that \frac{n·ρ̃}{μ(ρ̃)} is very close to ρ. Especially we can achieve that each of the κ’s has values > n on it. Thus if ñ = μ(ρ̃), then f ∈ F ñ+1 ; ρ̃, which cannot happen. Hence we have shown surjectivity of the map.

Injectivity follows because we already know how to recover μ from I_{μ}. Conversely once we have identified σ^{∨} with the semigroup of effective Cartier-divisors supported in the boundary, we obtain L^{∨} as the group of all such divisors, and then also L and σ. Also λ corresponds to π. Now extend λ to a basis of L^{∨}, and choose equations for the remaining basis-elements. This defines a map Spec(R) → R̄_{λ} which differs from the given one by the action of T via a point z of T (R). Passing to the 𝔪 -adic completion R̂ the two maps differ by an automorphism of R̂, which is the identity on κ = R/𝔪 and multiplies μ ∈ L^{∨} ∩ σ^{∨} by μ(z).

All in all (L, σ) is essentially unique up to automorphism once given Spec(R) and its open stratum.

5. Remark. One might ask whether the fibre R⊗_{V} k is reduced. This happens if and only if this holds for T̄_{λ}, i.e. if k[L^{∨}∩σ^{∨}]/(λ) is reduced, that is if λ(ρ) ∈ {0, 1} for all extremal rays ρ of σ. We can always achieve this by replacing L by L̃ = λ^{−1}(eℤ) with e sufficiently divisible, and λ by λ/e. Choose e such that λ(ρ) divides e if λ(ρ) ≠ 0. Then ρ changes to ρ̃ = \frac{e}{λ(p)} · ρ, and λ/e(ρ̃) = 1. This operation amounts to adjoining ^{e}\sqrt{π} to V. We shall usually assume that this has been done. Then R ⊗_{V} A is normal for any flat normal V -algebra A. Especially R ⊗_{V} V̄ is normal.

There is also an intimate relation to log -structures as defined by Fontaine-Illusie-Kato: Namely on X = Spec(R)^{et} consider the sheaf of monoids P = j_{∗}(O_{X^{0}}^{∗}) ∩ O_{X} (j : X^{0} ↪ X inclusion). Its fibre at x ∈ τ− stratum is an extension 0 → O_{x,x}^{∗} → P_{x} → τ^{∨} → 0, with O_{x,x}^{∗} acting freely on P_{x}. Thus P defines a fine log -structure, log-smooth over Spec(V) with the log-structure defined by V − {0}. Conversely if X^{0} ⊆ X is such that P = j_{∗}(O_{x^{0}}^{∗}) ∩ O_{x} defines a fine log-structure log-smooth Spec(V) with V − {0}, then for each x ∈ X P_{x}((O_{x,x}^{sh})^{∗} injects into the group of Weil-divisors supported on the complement of the open stratum in Spec(O_{x,x}^{sh}). If L_{x}^{∨} = ⟨P_{x}/(O_{x,x}^{sh})^{∗}⟩^{group}, L = (L^{∨})^{∨}, then the valuations at irreducible components of the boundary induce maps L^{∨} → ℤ and thus elements ρ ∈ L, which span a polyhedral cone σ ⊆ L_{ℝ} with P_{x}/(O_{x,x}^{sh})^{∗} = L^{∨} ∩ σ^{∨}. Furthermore elements of P_{x} lifting a basis of L contained in σ^{∨} define a formally smooth map Spec(O_{x,x}^{sh}) → T̄_{λ}, where λ ∈ L^{∨} corresponds to π. λ ∈ L^{∨} may not be indivisible (it is only indivisible by p), but assume that it is. Hence our toroidal description can be translated into log -structures. However as all the important tools we need from the theory have been formulated and proved in toroidal geometry long before the advent of logstructures, we use the toroidal language.

Appendix 2: Length computations. Let us start with the torus-embedding

T = 𝔾_{m}^{d+1} ⊆ X = Spec(V [L^{∨} ∩ σ^{∨}]) = Spec(R).

For a sublattice L′ ⊆ L let X′ = Spec(R′) = Spec(V [L^{′^{∨}} ∩ σ^{∨}]). As an R-module

R′ = ⨁_{μ∈L^{′^{∨}} /L^{∨}} V [(μ + L^{∨}) ∩ σ^{∨}] = ⨁_{μ} R′_{μ},

where the isomorphism-class of R′_{μ} as R-module depends only on the integral parts [μ(l)], l ∈ L the generator of an extremal ray of σ. Namely for κ ∈ L^{v} (μ + κ) lies in σ^{∨} if any only if κ(l) ≥ −[μ(l)] for all such l. As μ can be chosen in a compact set (representatives for L_{ℝ}^{∨}/L^{∨}) the numbers [μ(l)] are bounded. One derives that there are onyl finitely many isomorphism classes of R′_{μ}’s, independant of the choice of L′. Furthermore if for some integer N L′ ⊆ N · L, then the multiplicity of a given isomorphism-class of R_{μ}’s is [L : L′] · (constant + O(\frac{1}{N})). Also the constant is strictly positive for the multiplicity of the trivial R-module R_{μ} ≅ R. Now firstly if λ ∈ L^{∨} is such that σ ∩ ker(λ) is a simplex spanned by a partial basis of L, then R_{λ} is regular and R′_{λ} is flat over R_{λ} (being finite and Cohen-Macaulay). Thus if P_{·} → P′ is a resolution of R by finitely generated projective R-modules there exists on N such that λ^{N} is nullhomotopic on P_{1}, i.e. λ^{N} · id = d ∘ s + s ∘ d for suitable s : P_{1} → P_{2}, P_{0} → P_{1}.

Thus λ^{N} annihilates T or_{1}^{R}(R′, M) for all R-modules M, and also all higher T or’s. It now follows that N can be chosen independantly from L′. The same holds after flat pushout. This already shows one of our claims: If we let L′ = L_{n} (with L_{n} as before) and if σ ∩ ker(λ) is a simplex, then for a suitable choice we obtain first on the level of torus-embeddings and then for R ⊆ R_{n} ⊆ R_{∞}, that p^{α} · T or_{1}^{R_{n}}(R_{∞}, any R_{n} -module) = 0 for n >> 0.

Now for the definition of λ(): We assume that for a cofinal sequence of L_{n}’s L_{n} = N · L with N → ∞. Let d_{n} = [L : L_{n}] denote its index. Then after pushout we obtain that for all n (in our cofinal sequence)

R_{n+m} = ⨁_{μ∈L_{n+m}/L_{n}^{∨}} R_{n+m,μ}

where only finitely many isomorphism types of direct summands occur, and the relative frequency of each summand converges (for m → ∞) to a constant, which is non-zero for the type R_{n+m,μ} ≅ R. Moreover this summand occurs with multiplicity ≥ c · (d_{n+m}/d_{m}), c > 0 a constant independant of n and m.

Now assume M_{∞} is a finitely presented R_{∞} -module which is 𝔪 -torsion. (𝔪 = maximal ideal of R). Then M_{∞} = M_{n} ⊗_{R_{n}} R_{∞} for a finitely presented R_{n} -module M_{n} which is also 𝔪 -torsion and thus of finite length. Define

λ(M_{∞}) = lim_{m→∞} \frac{d_{n}}{d_{n+m}} length_{R}(M_{n} ⊗_{R_{n}} R_{n+m})

The limit exists because the relative multiplicities of the R_{n+m,μ} converge. Furthermore length_{R}(M_{n} ⊗_{R_{n}} R_{n+m}) ≥ c · d_{n+m} · λ(M_{∞}) for all m. Especially M_{∞} vanishes if λ(M_{∞}) = 0. In more generality we have: Let M_{n} ↠ M̃_{n} denote a surjection, inducing M_{∞} ↠ M̃_{∞}. Then for all m

length(kernel(M_{n} ⊗_{R_{n}} R_{n+m} → M̃_{n} ⊗_{R_{n}} R_{n+m}) ≥ c · d_{n+m} · (λ(M_{∞}) − λ(M̃_{∞}))

Next assume M_{∞} is only finitely generated but still M torsion. Choose a finite set of generators and denote by M_{n} the R_{n} -submodule generated by them. Then M_{n+m} is a quotient of M_{n} ⊗_{R_{n}} R_{n+m}, and the sequence λ(M_{n} ⊗_{R_{n}} R_{∞}) is decreasing.

6. Claim.

lim_{n→∞} \frac{1}{d_{n}} length_{R}(M_{n}) = lim_{n→∞}λ(M_{n} ⊗_{R_{n}} R_{∞})

Define this limit as λ(M_{∞}).

Proof of claim. For ε > 0 choose n such that λ(M_{n} ⊗_{R_{n}} R_{∞}) ≤ λ(M_{∞}) + ε. Then if N_{n+m} = ker(M_{n} ⊗_{R_{n}} R_{n+m} → M_{n+m}), we have that for l ≥ 0

length_{R} ker(M_{n}⊗_{R_{n}}R_{n+m+l} → M_{n+m}⊗_{R_{n+m}}R_{n+m+l}) ≥ c·(d_{n+m+l}/d_{n+m})·length_{R}(N_{n+m})

Passing to the limit l → ∞ we obtain

\begin{aligned}length_{R}(N_{n+m}) & ≤ \frac{d_{n+m}}{c} (λ(M_{n} ⊗_{R_{n}} R_{∞}) − λ(M_{n+m} ⊗_{R_{n+m}} R_{∞})) \\ & ≤ \frac{d_{n+m} · ε}{c} \end{aligned}

Thus 0 ≤ \frac{1}{d_{n+m}} (length_{R}(M_{n} ⊗_{R_{n}} R_{n+m}) − length_{R}(M_{n+m})) ≤ \frac{ε}{c}, which implies the claim.

Next for an arbitrary 𝔪 -torsion R_{∞} -module M define

λ(M) = sup{λ(M′) | M′ ⊆ M finitely generated submodule},

so 0 ≤ λ(M) ≤ ∞.

7. Lemma (additivity). Suppose 0 → M′ → M → M′′ → 0 is an exact sequence of 𝔪 -torsion R_{∞} -modules. Then

λ(M) = λ(M′) + λ(M′′)

Proof. We easily reduce to M and M′′ finitely generated. For a finite system of generators of M let M_{n} denote the R_{n} -module generated by them, M′′_{n} its image in M′′, and M′_{n} its intersection with M′. Then we have

lim_{n→∞} \frac{1}{d_{n}} length_{R}(M′_{n}) = λ(M) − λ(M′′).

For any finite set of elements of M′ the R_{n} -module M̃′_{n} generated by them is for big n contained in M′_{n}, and has length ≤ length_{R}(M′_{n}). It follows that

λ(M′) ≤ λ(M) − λ(M′′).

On the other hand for ε > 0 choose n such that

λ(M′′) ≥ λ(M′′_{n} ⊗_{R_{n}} R_{∞}) − ε.

Then for m ≥ 0 the kernel of M′′_{n}⊗_{R_{n}} R_{n+m} → M′′_{m} has length bounded by \frac{ε}{n} ·d_{n+m}, by the previous arguments. From the commutative diagram with exact rows

M′_{n} ⊗_{R_{n}} R_{n+m} ⟶ M_{n} ⊗_{R_{n}} R_{n+m} ⟶ M′′_{n} ⊗_{R_{n}} R_{n+m} ⟶ 0
↓  ↓  ↓
0 ⟶  M′_{n+m}  ⟶  M_{n+m}  ⟶  M′′_{n+m}  ⟶ 0

we obtain a surjection

ker(M′′_{n} ⊗_{R_{n}} R_{n+m} → M′′_{n+m}) ↠ M′_{n+m}/R_{n+m} · M′_{n}

Thus length_{R}(M′_{n+m}) ≤ length_{R}(R_{n+m} · M′_{n}) + \frac{ε}{c} · d_{n+m} and in the limit m → ∞

λ(M) − λ(M′′) ≤ λ(R_{∞} · M′_{n}) + \frac{ε}{c} ≤ λ(M′) + \frac{ε}{c} ≤ λ(M′) + \frac{ε}{c}.

As ε was arbitrary we derive the lemma. Finally if M is finitely presented and annihilated by π^{1/p}, then λ(M ⊗_{R_{∞}φ} R_{π}R_{∞}) = p^{d+1} · λ(M): This follows because first on the level of torus-embeddings pushout be Frobenius induces an isomorphism

k[L_{n}^{∨} ∩ σ^{∨}]/(λ^{1/p}) ⟶^{∼} k[pL_{n}^{∨} ∩ σ^{∨}]/(λ)

Thus for modules annihilated by π^{1/p} or λ^{1/p} pushout by Frobenius replaces L_{n} by \frac{1}{p}L_{n}, hence d_{n} by p^{−(d+1)}d_{n}, and the assertion follows. It then also holds for finitely generated M by passage to the limit from M_{n} ⊗_{R_{n}} R_{∞}.

Finally if M′ is submodule of a finitely presented M_{1} and λ(M′) = 0, then M′′ = M/M′ is also finitely presented, with λ(M) = λ(M′′). Then length_{R}(ker(M_{n} → M′′_{n})) ≤ \frac{1}{c}(λ(M) − λ(M′′)) = 0, hence M′ = (0).

Remark. There exist finitely generated M ≠ (0) with λ(M) = 0, for example the residue field of R_{∞}.

2c. Galois cohomology

Suppose G is a finite group, A a commutative ring (with unit), B an A-algebra with G-action. Assume furthermore that for b ∈ B tr(b) = ∑_{g∈G} g(b) lies in the image of A in B. If M is a B-module with semilinear G-action, we may compute the cohomology H^{∗}(G, M) using a G-acyclic resolution by such B − G -modules. If b ∈ B has image a = tr(b) ∈ A, then a annihilates all higher cohomology:

For m ∈ M we have tr(b · m) = ∑_{g∈G} g(b · m) ∈ M^{G}, and if m ∈ M^{G} then tr(b · m) = a · m. Applying this to a resolution of M the claim follows.

If we also assume that there exist elements b_{i}, c_{i} ∈ B (1 ≤ i ≤ r) such that for any b ∈ B

\begin{matrix}a = ∑_{i=1}^{r} b_{i} · c_{i} \\ 0 = ∑_{i=1}^{r} b_{i} · g(c_{i}) \end{matrix}

for g ∈ G, g ≠ 1 we can define a map

\begin{matrix}M → B ⊗_{A} M^{G} \\ m ⟼ ∑_{i=1}^{r} b_{i} ⊗_{A} tr(c_{i} · m) \end{matrix}

The composition either way of this map with the natural

B ⊗_{A} M^{G} → M

will be a. For example if A ⊆ B is an almost étale Galois-covering with group G, then we apply this to a = π^{α} (any α > 0), ∑_{i=1}^{r} b_{i} ⊗ c_{i} = π^{α} · e_{B/A}, and obtain that

\begin{matrix}H^{i}(G, M) ≈ 0 (i > 0) \\ M ≈ B ⊗_{A} M^{G} \end{matrix}

Passing to the limit this will also hold if B is an inductive limit of almost étale coverings and G the corresponding profinite group, i.e. for the extensions R_{∞} ⊆ R̄. Here R̄ denotes the normalisation of R in the maximal étale covering of the preimage of T ⊆ T̄ in Spec(R), that is the union of all finite extensions R ⊆ S with S normal and S[\frac{1}{π}]/R[\frac{1}{π}] ramified only along the divisor ”at infinity”. For simplicity we assume that R is an integral domain (otherwise we would take everywhere products of integral domains). If V̄ denotes the integral closure of V in K̄, R̄ contains as subring the normalisation of the image of R ⊗_{V} V̄ → R̄. If λ = n · λ_{0} with λ_{0} indivisible in L then firstly R contains the subring V′ = V [^{n}\sqrt{π}]. Replacing V by V′ we then may assume that λ = λ_{0}. In this case the normalisation of V in R is unramified over V. Thus if the residue field k is algebraically closed it coincides with V_{j} and R ⊗_{V} V̄ is an integral domain. In any case we denote by Δ the Galoisgroup of R̄ over the normalisation of the image of R ⊗_{V} V̄. If R ⊗_{V} V̄ is an integral domain this is Δ = π_{1}(Spec(R ⊗_{V} K̄)^{0}). The coverings defined by the R_{n} give a map

Δ → L ⊗ ℤ̂(1) = L̂(1)(profinite completion).

Let Δ_{∞} denote its image and Δ_{0} ⊆ Δ its kernel. As R is formally étale over T̄_{λ} and the R_{n} are totally ramified over R at all generic characteristic p-points if [L : L_{n}] is a p-power, there is a surjection Δ_{∞} ↠ L ⊗ ℤ_{p}(1), with kernel of order prime to p. For most arguments we can in fact replace Δ_{∞} by L ⊗ ℤ_{p}(1).

Now suppose M̄ is a p-torsion R̄ -module with a semilinear continuous Δ -action. To study the cohomology H^{∗}(Δ, M) we use the spectral sequence

E_{2}^{a,b} = H^{a}(Δ_{∞}, H^{b}(Δ_{0}, M)) ⇒ H^{a+b}(Δ, M).

The extension governed by Δ_{0} is a composition of almost étale extensions. Thus all higher Δ_{0} -cohomology almost vanishes,

M ≈ (M^{Δ_{0}}) ⊗ R̄,

and H^{∗}(Δ, M) ≈ H^{∗}(Δ_{∞}, M^{Δ_{0}}). As Δ_{∞} has cohomological dimension d this vanishes in degree > d. Also after choice of topological generators δ_{1}, …, δ_{d} of Δ_{∞} the cohomology is represented by the Koszul-complex of δ_{1} − 1, …, δ_{d} − 1 acting on M^{Δ_{0}}. This commutes with formally étale base change R → R′.

For example we can study the cohomology of quotients R̄/p^{s}·R̄. By the above we may (up to almost isomorphism) replace R by the affine ring of the torus-embedding T̄_{λ}, and have to study the cohomology of ℤ_{p}(1)^{d} acting on

V [σ^{∨} ∩ L^{∨}[\frac{1}{p}]]/(λ − π) ⊗_{V} V̄ /p^{s}V̄

This module is free over V̄ /p^{s}V̄, where a basis can be obtained as follows:

λ acts freely on the monoid σ^{∨} ∩ L^{∨}[\frac{1}{p}]. For each orbit \{μ, μ + λ, μ + 2λ … \} choose the minimal representative μ. Then these μ form a basis.

All the basis-elements μ are eigenvectors for ℤ_{p}(1)^{d}, with character e^{μ} : ℤ_{p}(1)^{d} → μ_{p^{∞}}. If this character has order p^{t} the corresponding cohomology is annihilated by ζ_{p^{t}} − 1, ζ_{p^{t}} a primitive p^{t} − th root of unity. Thus up to terms annihilated by ζ_{p} − 1 only the eigenspaces with trivial e^{μ} contribute to cohomology. e^{μ} is trivial if and only if μ | ker(λ) is in ker(λ)^{∨}. Then there exists a unique α ∈ ℤ[\frac{1}{p}], 0 ≤ α < 1, with μ − αλ ∈ L^{∨} integral. Let R = ⨁_{α,μ} V π^{α} · μ (sum over such α, μ). Then R = R_{∞} ∩ R ⊗_{V} K_{∞} is the normalisation of R ⊗_{V} V_{∞}, and πR ⊆ R ⊗_{V} V̄. Furthermore (as H^{∗}(ℤ_{p}(1)^{d}, V) = Λ^{·}(V (−1)^{d}))

H^{∗}(Δ, R̄/p^{s}R̄) ≈ R ⊗ Λ^{·}(Hom(Δ_{∞}, V̄ /p^{s}V̄)) ⊕ (terms annihilated by ζ_{p} − 1).

Especially we get compatible almost maps

tr : H^{d}(Δ, R̄/p^{s}R̄) → R ⊗_{V_{∞}} V̄ /p^{s}V̄ (−d).

(Compatibility forces them to vanish on the second direct summand). The target can be more canonically identified with Ω_{R/V}^{d,log} ⊗_{R} R ⊗_{V_{∞}} V̄ /p^{s}V̄ (−d). Here Ω_{R/V}^{log} denotes the logarithmic module of differentials, for the log -structures on T̄_{λ} and Spec(V) given by σ^{∨} ∩ L^{∨} and ℕ (μ ∈ σ^{∨} ∩ L^{∨} maps to μ, and 1 ∈ ℕ to π). By etaleness

Ω_{R/V}^{log} ⊗ V/p^{s}V ≅ ker(λ)^{∨} ⊗ R

Of course Ω_{R/V}^{d,log} is the d′th exterior power. At least modulo p^{s} it is free of rank one. Checking all the identification we get almost maps

tr : H^{d}(Δ, R̄/p^{s}R̄) → Ω_{R/V}^{d,log} ⊗_{R} R ⊗_{V_{∞}} V̄ /p^{s}V̄ (−d)

which are independant of the choice of generators δ_{1}, …, δ_{d}. In fact they only depend on R (with its log-structure) and not on the formally étale map Spec(R) → T̄_{λ}. This follows (as in [F2], Sect. I, 4) from the following facts, which can be checked by direct calculation for T̄_{λ} and then also hold for R by formal étaleness and almost étaleness:

i) The sequence

0 → Ω_{V̄ /V}^{log} ⊗V̄ R̄ → Ω_{R̄/R}^{log} → Ω_{R̄/R⊗V̄}^{log} → 0

is almost exact ii)

Ω_{V̄ /V}^{log} ≅ V̄ [\frac{1}{p}]/ρ^{−1}V̄ (1)

for an element ρ ∈ V̄ , ρ ≠ 0. The isomorphism is induced from d log : μ_{p^{∞}} → Ω_{V̄ /V}^{log} .

iii) Consider the exact sequence

Ω_{R/V}^{log} ⊗_{R} R̄ → Ω_{R̄/V̄}^{log} → Ω_{R̄/R⊗_{V} V̄}^{log} → 0

Then the kernel of the first map is almost contained in ⋂_{s≥0} p^{s}·(Ω_{R/V}^{log} ⊗_{R}R̄). Furthermore the induced map

Hom(\frac{1}{p^{s}} ℤ/ℤ, Ω_{R̄/R⊗_{V} V̄}^{log} ) → Ω_{R/V}^{log} ⊗_{R} R̄/p^{s}R̄

is an almost isomorphism. iv) Applying Hom(\frac{1}{p^{s}} ℤ/ℤ,) to the exact sequence in i) gives an almost exact

sequence

0 → ρ^{−1}(R̄/p^{s}R̄(1)) → middle term → Ω_{R/V}^{log} ⊗_{R} R̄/p^{s}R̄ → 0,

thus a map

Ω_{R/V}^{log} → H^{1}(Δ, (R̄/p^{s}R̄)(1))

By skew-commutativity of the cup-product we obtain

Ω_{R/V}^{d,log} ⊗_{V} V̄ /p^{s}V̄ (−d) → H^{d}(Δ, R̄/p^{s}R̄)

which is ρ^{d} · d! -times the map constructed before.

v) As ρ^{d} · d! -times our maps are canonical, compatibility for varying s implies the assertion.

Remark. So from now on we assume that R = R ⊗_{V} K̄, equivalent to the conecondition above ((ℚ^{+} · λ + L^{∨}) ∩ σ^{∨} = ℚ^{+} · λ + (L^{∨} ∩ σ^{∨})). Other equivalences: if μ + αλ(μ ∈ L^{∨}, α ∈ ℚ, 0 ≤ α < 1) lies in σ^{∨}, then so does μ. Also: The special fibre T̄_{λ} ⊗_{V} k is reduced.

Before we start to treat duality we have to introduce coefficients. Suppose 𝕃 is a finite ℤ_{p} -module with a continuous action of Δ. We denote by 𝕃^{t} = Hom(𝕃, ℚ_{p}/ℤ_{p}) its dual. Next recall that σ ∩ ker(λ) was supposed to be a simplex spanned by a partial basis ρ_{1}, …, ρ_{r} of L. On the generic fibre of T̄_{λ} these define normal crossings divisors D_{1}, …, D_{r}, which are cut out by a (partial) dual basis of L^{∨}. Choose a subset I of {1, …, r} and denote by D_{I} the union of the {D_{i} | i ∈ I}. Then for each n the preimage of D_{I} in T̄_{λ,n} ⊗_{V} K is a divisor. Let J_{I,n} ⊆ V [L_{n}^{∨} ∩ σ^{∨}]/(λ − π) denote the ideal of its reduced closure. That is J_{I,n} is spanned by all μ ∈ L_{n}^{∨} ∩ σ^{∨} with μ(ρ_{i}) > 0 for i ∈ I. Finally J_{I,∞} ⊆ Γ(T̄_{λ,∞}, O) denotes the union of the J_{I,n}. One checks that for any μ ∈ L^{∨} ∩ σ^{∨} with μ(ρ_{i}) > 0 for i ∈ I, μ(ρ_{i}) = 0 for i ∉ I, J_{I,∞} is almost generated by the fractional powers μ^{1/p^{n}}. (Use that any μ′ vanishing on σ ∩ ker(λ) divides a power of π). Thus J_{I,∞} is almost a union of principal ideals and almost flat.

Finally we apply pushout to R. If we assume π is contained in the Jacobsen radical of R (for example if R is π -adically complete) then R is normal, and R[\frac{1}{π}] is regular with the pullback of the D_{i} defining a normal crossings divisor. Then one checks that the ideal J_{I,∞} ⊗R̄[\frac{1}{π}] defines the reduced pullback-divisor and thus only depends on the pullback of D_{I} to Spec(R[\frac{1}{π}]), not on the particular representation Spec(R) → T̄_{λ}. Finally be almost étaleness of R̄ over Γ(T̄_{λ,∞}, O) the ideal J_{I,∞} · R̄ is almost equal to the ideal defining the closure, thus also as canonical as the divisor D_{I}. Call it J̄_{I}.

Now our aim is to study the cohomology of Δ with coefficients 𝕃 ⊗ J̄_{I}. This depends only on the image of J̄_{I} in the π -adic completion of R, so that everything is canonical even if π should not lie in the Jacobsen radical of R. Next comes finiteness. As R ⊗_{V} V̄ is the inductive limit of noetherian rings (R ⊗_{V} V′, V′ a finite extension of V) with flat transition-maps, it is coherent. That is any finitely presented R ⊗_{V} V̄ -module has a projective resolution by finitely generated free R ⊗_{V} V̄ -modules, and the finitely presented R ⊗_{V} V̄ -modules form an abelian subcategory of all R ⊗_{V} V̄ -modules.

We say that an R ⊗_{V} V̄ -module M is almost finitely presented if for each α > 0 there exists a finitely presented M_{α} and a π^{α} -isomorphism f : M_{α} → M i.e. kernel and cokernel of f are annihilated by π^{α}. The almost finitely presented R ⊗_{V} V̄ -modules also form an abelian subcategory. Now suppose 𝕃 is a finite ℤ_{p} -module with a continuous Δ -action.

8. Proposition. The cohomology-groups H^{i}(Δ, 𝕃⊗J̄_{I}) are almost finitely presented over R ⊗_{V} V̄.

Proof. By devissage we may assume that p · 𝕃 = (0). We also may replace R by its p-adic completion, so Spec(R) is formally étale over T̄_{λ}. Especially we have that R ⊗_{V} K is regular, and the complement of T ⊆ T̄ defines a normal crossing divisor. Replacing L by a sublattice L′ ⊆ L we may assume that 𝕃 is unramified over Spec(R ⊗_{V} K̄). (If Δ′ is the group corresponding to L′, use the Hochschild spectral sequence for 0 → Δ′ → Δ → Δ/Δ′ → 0). Next by almost faithfully falt descent (for R̄/R_{∞}) (𝕃 ⊗ R̄)^{Δ_{∞}} is almost finitely presented. Thus for each α > 0 there exists an integer n, a finitely presented R_{n} -module M_{α} action, and a π^{α}-isomorphism

M_{α} ⊗_{R_{n}} R_{∞} → (𝕃 ⊗ R̄)^{Gal(R̄/R_{∞}⊗_{V} K̄)}

By increasing n we make this Gal(R_{∞}/R_{n} ⊗_{V} V̄) -linear (trivial operation on M_{α}. It suffices if the images of generators of M_{α} are fixed.

Again by the Hochschild spectral sequence we may replace R by R_{n}, i.e. study H^{∗}(Δ_{∞}, M ⊗_{R} J_{∞,I}). Now for each n J_{∞,I} is the direct sum of a finitely presented R_{n} ⊗_{V} V̄ -module and modules on which a generator δ of Δ_{∞} acts like ζ_{p^{n}}, ζ_{p^{n}} a primitive p^{n} − th root of unity. (There are generated by μ′s, μ | ker(λ) not integral on p^{n−1}L.) For these the cohomology is annihilated by ζ_{p^{n}} −1, while the first direct summand has finitely presented cohomology.

Finally we come to duality. Recall ([H], Ch. V, § 2) that a dualising complex D^{∗} over a noetherian ring A is a complex with coherent cohomology which has an injective resolution in which each indecomposable injective appears precisely once. Equivalently for each prime 𝔭 ∈ Spec(A) there exists an integer d(𝔭) (related to the codimension) with Ext_{A_{𝔭}}^{i}(K(𝔭), D_{𝔭}^{∗}) = κ(𝔭) for i = d(𝔭), and 0 else. Then K^{∗} → ℝHom(K^{∗}, D^{∗}) defines a perfect duality on the derived category of finite complexes with finitely generated cohomology. If A is Cohen-Macaulay D^{∗} has only one non-trivial cohomology group, the dualising mdoule ω_{A}. For example for the torus-embedding T̄ a dualising module is given by ωT̄ = J ⊗Ω^{d+1,log}, where J is the ideal generated by all μ ∈ L^{∨} which are strictly positive on σ − {0}. This follows because on the smooth locus the dualising module coincides with the differentials, and in general it is the reflexive extension of its restriction to the smooth locus. Also for T̄_{λ} the dualising module is Ext_{OT̄}^{1}(OT̄_{λ}, ωT̄) ≅ J ⊗ OT̄_{λ}.

It is the ideal in OT̄_{λ} (or better in Ωd,log ; T̄_{λ}) generated by all μ ∈ L^{∨} which are strictly positive on σ − {0}. Finally if π is contained in the Jacobson-radical of R the map OT̄_{λ} → R is flat with Gorenstein-fibres. Thus ωT̄_{λ} ⊗T̄_{λ} R = ω_{R} is a dualising module on R. It is isomorphic to J. Similarly the dualising modules on R_{n} are isomorphic to J_{n}. By the general theory ω_{R_{n}} ≅ Hom_{R}(R_{n}, ω_{R}). This isomorphism is determined by the evaluation at 1 ∈ R_{n}, which defines a trace-map

tr : ω_{R_{n}} → ω_{R}.

It is obtained by pushout from the trace-map on OT̄ or OT̄_{λ}. The latter sends to character μ ∈ L_{n}^{∨} to itself if μ ∈ L^{∨}, and to zero else. Similarly we can define a dualising module ω_{R⊗_{V} V̄} = ω_{R} ⊗_{V} V̄.

By passing to the limit over finite extensions of V,

K^{∗} → D(K^{∗}) = ℝ Hom_{R⊗_{V} V̄} (K^{∗}, ω_{R} ⊗_{V} V̄)

still defines a perfect duality for finite complexes with finitely presented (= coherent) cohomology, or an almost perfect duality for finite complexes with almost finitely presented cohomology. Again ω_{R⊗_{V} V̄} ⊆ R ⊗_{V} V̄ is the ideal J generated by all μ ∈ L^{∨} which are strictly positive on σ − {0}. Similarly ω_{R_{n}⊗_{Vn} V̄} ≅ J_{n} ⊆ R_{n} ⊗_{V_{n}} V̄ is the ideal generated by μ ∈ L_{n}^{∨} strictly positive on σ − {0}, and we have the union J_{∞} ⊆ R_{∞} ⊗_{V_{∞}} V̄. It coincides almost with the ideal generated by μ ∈ σ^{∨} which are strictly positive on σ ∩ ker(λ)) − {0}:

For such a μ μ + α · λ is strictly positive on σ − {0}, for each α > 0. That is J_{∞} almost coincides with the ideal previously named by the same symbol. Finally the trace-map becomes more complicated:

As V_{n} = V [^{e}\sqrt{π}] = V [π_{n}] (for some e)

\frac{1}{e} · tr_{V_{n}/V} : \frac{π_{n}}{π} · V_{n} → V

defines an isomorphism

\frac{π_{n}}{π} V_{n} ≅ Hom_{V} (V_{n}, V).

Thus our old trace defines a new trace from J_{n} to J ⊗_{V} (\frac{π_{n}}{π} V_{n}) and by ⊗_{V_{n}} V̄ a trace-map

tr : J_{n} ⊗_{V_{n}} V̄ → J ⊗_{V} (\frac{π_{n}}{π} V̄) = \frac{π_{n}}{π} ⊗ (J ⊗_{V} V̄)

It sends a μ ∈ L_{n}^{∨} strictly positive on σ−{0} to itself or zero, depending on whether μ ∈ ker(λ)^{∨} + ℚ · λ or not. Obviously these combine to define

tr : R_{∞} ⊗_{V_{∞}} V̄ → \frac{1}{π} (J ⊗_{V} V̄)

As on finite levels we obtain isomorphisms

J_{n} ⊗_{V} V̄ ≅ Hom_{R⊗_{V} V̄} (R_{n} ⊗_{V} V̄ , \frac{π_{n}}{π} · (J ⊗_{V} V̄)),

we have in the limit for each μ ∈ ker(λ)^{∨} ⊗ ℚ/ℤ an almost isomorphism of μ -components

(J_{∞} ⊗_{V} V̄)_{μ} ≈ Hom_{R⊗_{V} V̄} (R_{∞} ⊗_{V} V̄ , \frac{1}{π} (J ⊗_{V} V̄))_{μ}

Of course the map

J_{∞} ⊗_{V} V̄ → Hom_{R⊗_{V} V̄} (R_{∞} ⊗_{V} V̄ , \frac{1}{π} J ⊗_{V} V̄)

cannot be an almost-isomorphism because of the difference between direct sum and product (over all μ -components).

However this difficulty disappears if we take Δ_{∞} -cohomology: If μ has denominator p^{n} the cohomology of the μ -component is annihilated by ζ_{p^{n}} − 1. It thus follows that we obtain almost-isomorphisms on cohomology. The same holds after tensoring with (𝕃 ⊗ R̄)^{Gal(R̄/R_{∞}⊗_{V} V̄)} (π^{α} -isomorphic to M_{α} ⊗_{R_{n}} R_{∞} as before). Thus (using almost étaleness).

9. Proposition. tr induces almost isomorphisms

H^{i}(Δ, 𝕃 ⊗ R̄) ≈ H^{i}(Δ, 𝕃 ⊗ Hom_{R̄⊗_{V} V̄} (R̄, \frac{1}{π} J ⊗_{V} V̄))

Finally cohomology can be (almost) computed by taking invariants under Gal(R̄/R_{∞}⊗_{V} V̄) (an almost exact operation) and then applying the Koszul-complex with (δ_{1} − 1), …, (δ_{d} − 1), δ_{i} topological generators of Δ_{∞} ≅ ℤ_{p}(1)^{d}. However the dual of the Koszul-complex is again a Koszul-complex shifted by d. We thus obtain:

10. Theorem. Assume 𝕃 is annihilated by p^{s}, and let 𝕃^{t} = Hom(𝕃, ℤ/p^{s}ℤ) denote the dual. Then

i) ℝΓ(Δ, 𝕃 ⊗ R̄) is almost isomorphic to a finite complex of R ⊗_{V} V̄ -modules concentrated in degrees [0, d] with almost finitely presented cohomology (ℝΓ(Δ, ·) denotes one of the canonical complexes computing group-cohomology)

ii) The previously defined trace-maps tr : H^{d}(Δ, R̄/p^{s}R̄) → R/p^{s}R induce maps tr : H^{d}(Δ, J̄/p^{s}J̄) → \frac{1}{π} ω_{R} ⊗ V̄ /p^{s}V̄ (R is contained in \frac{1}{π} R ⊗_{V} V̄)

iii) tr defines an almost isomorphism

ℝΓ(Δ, 𝕃 ⊗ J̄)(d)[d] ≈ ℝHom_{R⊗_{V} V̄ /p^{s}V̄} (ℝΓ(Δ, 𝕃^{∨} ⊗ R̄), \frac{1}{π} ω_{R} ⊗ V̄ /p^{s}V̄)

Proof. Only some minor details need to be checked: Firstly our trace map (from duality) J_{∞}/(δ_{i} − 1) → J ⊗_{V} \frac{1}{π} V̄ induces on cohomology the previously defined trace, (obviously direct checking), and secondly we may replace R by its π -adic completion so that the duality-theory works.

Variant. We may also use coefficients J̄_{I}; I ⊆ {1, …, r} a subset. The ideals J_{I} ⊆ R and J_{I,n} ⊆ R_{n} are Cohen-Macaulay:

Reduce to the torus-embedding T̄. If (say) δ_{1} ∈ I, let L̃ denote the torusembedding defined by L̃ = L/ℤ · δ_{1} and the image σ̃ of σ in L̃_{ℝ}. Then σ̃ has extremal rays generated by δ_{2}, …, δ_{s}. Let Ĩ = I ∩ {δ_{2}, …, δ_{r}}. Then there is an exact sequence

0 → J_{I} → J_{I−{1}} → J̃Ĩ → 0.

If the other two terms are Cohen-Macaulay (of relative dimension d + 1 respectively d), then J_{I} also is. Now duality holds between J_{I,n} and Hom_{R}(J_{I,n}, ω_{R}) = Hom_{R_{n}}(J_{I,n}, ω_{R_{n}}), and similarly between J_{I,n} ⊗_{V_{n}} V̄ and Hom_{R⊗_{V} V̄} (J_{I,n} ⊗_{V} V̄ , ω_{R} ⊗_{V} V̄). However one checks that the union

⋃_{n} Hom_{R⊗_{V} V̄} (J_{I,n} ⊗_{V} V̄ , ω_{R} ⊗_{V} V̄) ≈ J_{I^{c},∞}

is almost J_{I,∞}^{c}, I^{c} ⊆ {1, …, r} the complement of I:

The product J_{I,∞} × J_{I^{c},∞} → J_{∞} shows that the union contains J_{I^{c},∞}. On the other hand it is contained in R_{∞}, and any character μ in it must be strictly positive on δ_{i} ∈ I^{c} (choose ν ∈ L^{∨} ∩ σ^{∨} which is strictly positive on I but vanishes on δ_{i}. Then ν ∈ J_{I}, and μ + ν must be strictly positive on δ_{i} too). However these μ’s are almost in J_{I^{c},∞}. With the same proofs as before we then have

10’. Theorem.

ℝΓ(Δ, 𝕃 ⊗ J̄_{I^{c}})(d)[d] ≈ ℝHom_{R⊗_{V} V̄ /p^{s}V̄} (ℝΓ(Δ, 𝕃^{∨} ⊗ R̄), \frac{1}{π} ω_{R} ⊗ V̄ /p^{s}V̄)

Last we need a local Künneth-formula. Suppose we have given two lattices L′, L′′ with cones σ′ ⊆ L′_{ℝ}, σ′′ ⊆ L′′_{ℝ} and elements λ′ ∈ L^{′∨}, λ′′ ∈ L^{′′∨}. Then in general the fibered product T̄_{λ′} ⊗_{V} T̄_{λ′′} will not be normal. However we can achieve this by adjoining roots of π. Namely pass to V [^{e}\sqrt{π}] (replacing L′ by λ^{′−1}(eℤ), L′′ by λ^{′′−1}(eℤ)) such that R′ = R′ ⊗_{V} V̄ , R′′ = R′′ ⊗_{V} V̄. That means for any linear combination μ′ + αλ′ ∈ σ^{′∨}(μ′ ∈ L′, 0 ≤ α < 1) we have μ′ ∈ σ^{′∨}, and similarly for μ′′ + αλ′′.

11. Lemma. Under the conditions T̄_{λ′} ⊗_{V} T̄′′_{λ′′} is normal, and equal to T̄_{λ} where the torus-embedding T̄ is defined by L = ker(λ′, −λ′′) ⊆ L′ × L′′, σ = L_{ℝ} ∩ σ′ × σ′′, λ = (λ′, 0) | L = (0, λ′′) | L

Proof. There is a map

T̄_{λ} → T̄′_{λ′} ⊗_{V} T̄′′_{λ′′}

induced from the tensor-product

V [σ′ ∩ L^{′∨}] ⊗_{V} V [σ′′ ∩ L^{′′∨}] → V [σ ∩ L^{∨}],

by dividing by (π − λ′, π − λ′′). The rest follows from remark 5.

Now assume that this condition holds. Suppose R is formally étale over R′_{1}⊗_{V} R′′. Furthermore assume given representations 𝕃′ of Δ′, L′′ of Δ′′, and let 𝕃 = 𝕃′ ⊗ 𝕃′′. We assume that 𝕃′ and 𝕃′′ are free ℤ/p^{s}ℤ -modules (to avoid using derived tensorproducts). Furthermore choose finite subsets I′ ⊆ {1, …, r}, I′′ ⊆ {1, …, r′′}. As σ ∩ ker(λ) = (σ′ ∩ ker(λ′) × (σ′′ ∩ ker(λ′′)), I = I′ × I′′ identifies with a subset of the vertices of σ ∩ ker(λ). Then pullback and cup-product define a map

ℝΓ(Δ′, J̄_{I′} ⊗ 𝕃′) ⊗_{V̄ /p^{s}V̄}^{𝕃} ℝΓ(Δ′′, J̄_{I′′} ⊗ 𝕃′′) ⊗_{R_{1}⊗_{V} R_{2}}^{𝕃} R → ℝΓ(Δ, J̄_{I} ⊗ 𝕃)

12. Theorem. This induces almost isomorphisms on cohomology.

Proof. Reduce to Δ′_{∞}, Δ′′_{∞}, Δ_{∞} = Δ′_{∞} × Δ′′_{∞} acting on R′_{∞}, R′′_{∞}, R_{∞}, and apply the obvious isomorphism on the level of Koszul-complexes.

3. Global Cohomology

Assume X → Spec(V) is proper and flat. Then there exists a dualising complex D_{X} on X, such that D_{X}(K) = ℝHom_{O_{X}} (M, D_{X}) defines a perfect duality on bounded complexes with coherent cohomology. Furthermore (if X has say pure relative dimension d) ℝΓ(X, D_{X}(M)) ≅ ℝHom_{V} (ℝΓ(M, R), V)[d], induced from a trace-map

tr : ℝΓ(X, D_{X}) → V [−d]

(see [H], Ch. VII, Th. 3.3). By base-extension ⊗_{V} V̄ this extends to X⊗_{V} V̄ , D_{X} ⊗_{V} V̄, and bounded complexes with coherent cohomology. (O_{X⊗_{V} V̄} is still coherent). Only slightly less trivial is the extension to complexes M^{·} such that for any α > 0 there exists a bounded complex M_{α}^{·} with coherent cohomology, and maps M_{α}^{·} ⟶^{β_{α}} M^{·} ⟶^{λ_{α}} M_{α}^{·} with β_{α}∘λ_{α} = π^{α}·id, λ_{α}∘β_{α} = π^{α}·id. There duality holds up to almost quasi-isomorphisms (the induced maps on cohomology are almost isomorphic).

1. Lemma. A complex M^{·} with bounded cohomology satisfies these conditions (i.e. \underline{M}^{·} is π^{α} -isomorphic to a coherent M_{α}^{·}, for each α > 0) if and only if the direct sum ⊕\underline{H}^{i}(M^{·}) is almost quasicoherent and locally (over each affine Spec(R)) almost finitely presented.

Proof. The non-trivial direction is local ⇒ global. Assume first that M^{·} is given by an almost quasicoherent sheaf M. Replacing M by 𝔪M/𝔪 -torsion we can assume that M is an honest quasicoherent sheaf. Then M is the filtering inductive limit of coherent sheaves M_{i}, M = \underrightarrow{lim} M_{i}. For given α > 0 and i big enough the cokernel of M_{i} → M is annihilated by π^{α}. We thus may replace M by the image of M_{i}, as this also satisfies the local condition (the coherent R ⊗_{V} V̄ -modules form an abelian category, and so do the almost finitely presented modules). Also the kernel N of M_{i} → M satisfies the local condition. Applying the same argument we may assume that it is also annihilated by π^{α}. Then M_{i} → M is an π^{2α} -isomorphism.

For general M choose i minimal with H^{i}(M) ≠ (0). Then M is an extension (i.e. one has an exact triangle)

0 → H^{i}(M)[−i] → M → M′ → 0,

which up to isomorphism is determined by a class c ∈ H^{1}(ℝHom(M′, H^{i}(M))[−i]). By pullback with β′_{α} we get a class c_{α} ∈ H^{1}(ℝHom(M′_{α}, N))(N = H^{i}(M)[−i]), defining 0 → N → M_{α} → M′_{α} → 0 and a map M_{α} → M. Further pullback (via λ′_{α}) to H^{1}(ℝHom(M′, N)) gives π^{α} · c, which also can be obtained from c via pushforward on N by π^{α} · id. We thus obtain a map M → M_{α} inducing λ′_{α} on M′ and π^{α} · id on N. The two compositions of M → M_{α} → M differ from π^{α} · id by maps M′ → N or M′_{α} → N (respectively), which are nullhomotopic for degreereasons. Thus M′_{α} approximates M up to π^{α}. With the same type of argument we can replace N by an approximation N_{α}.

We want to apply this theory to complexes made up from ℝΓ(Δ, 𝕃 ⊗ R̄)′s. For this we first have to define an appropriate topos.

For a scheme X we denote by X^{Zar} resp. X^{et} the Zariski respectively étale topos. Furthermore X^{et,lc} denotes the topos defined by finite étale covers of X. If X is connected and x a geometric point, X^{et,lc} is equivalent to the topos of sets with continuous action of π_{1}(X, x) (SGAI). In any case for noetherian X X^{et,lc} is isomorphic to the topos of étale sheaves which are inductive limits of locally constant sheaves. Also there is a continuous ρ_{X} : X^{et} → X^{et,lc} with ρ_{X,∗}(F) = restriction of F to finite étale covers, ρ_{X}^{∗}(𝕃) = 𝕃, for 𝕃 locally constant on X.

We say X is a K(π, 1) if for each 𝕃 H^{∗}(X^{et,lc}, 𝕃) ≅ H^{∗}(X^{et}, ρ_{X}^{∗}𝕃).

Next assume given X → Spec(V), and an open subset X^{0} ⊆ X ⊗_{V} K. Define a situs whose objects consists of pairs (U, V), where U → X is étale and V → (U ∩ X^{0}) ⊗_{K} K̄ is a finite étale covering. As maps we use compatible pairs of maps, and coverings are just pairs of surjective maps. Let X 0 ; K̄ denote the corresponding topos. Its sheaves consist of ind-locally constant sheaves 𝕃_{U} on (U ∩X^{0})⊗_{K} K̄, for each U as above, together with transition maps (for U′ → U) satisfying the usual cocycle condition. Furthermore for surjective f : U′ ↠ U 𝕃_{U} has to be the universial ind-locally constant sheaf mapping to the equaliser of f_{∗}𝕃_{U} ⇉ f_{∗}(𝕃U′ ×_{U} U′)). Equivalently sections 𝕃_{U}(V) can be glued for étale coverings of V induced from U. There exist a continuous

u_{X} : (X^{0} ⊗_{V} K̄)^{et} → X 0 ; K̄,

with u_{X,∗}(F)_{U} = ρ_{U,∗}(F | (U ∩ X^{0}) ⊗_{V} K̄).

Finally assume that X ⊗_{V} K = ∏ X_{α} is stratified, locally like by a ”simple” normal crossings-divisor. Especially the closure X_{α} of each stratum is smooth over K. Define an order on indices by

X̄_{α} − X_{α} = ∏_{β>α} X_{β}.

If j_{α} : X_{α} → X denotes the inclusion, it is known that for 𝕃_{α} locally constant on X_{α} and β > α, the restrictions j_{β}^{∗}ℝ^{ν}j_{α,∗}𝕃_{α} are locally constant on X_{β}. This follows from Abhyankhar’s lemma. Now define a topos XK̄ as sheaves on the following site:

Objects are triples (U, α, V), where U → X is étale V → U_{α} ⊗_{K} K̄ is finite étale (U_{α} = (U ⊗_{V} K) ×_{X} X_{α}). Maps (U′, β, V′) → (U, α, V) exist only of β ≥ α. Then they consist of pairs of compatible maps U′ → U, V′ → normalisation of V in U′ ×_{U} Ū_{α}. (It also follows from Abhyankhar’s lemma that the restriction of this normalisation to U′_{β}, with its reduced structure, defines a finite étale cover). Coverings are pairs of surjective maps with the same α. Thus sheaves in XK̄ associate to each U locally constant sheaves 𝕃_{U,α} on U_{α} ⊗_{K} K̄, with pullback-maps (for U′ → U) as before and maps 𝕃_{U,β} → j_{β}^{∗}j_{α,∗}𝕃_{U,α} for β > α satisfying transitivity for γ > β > α (j_{λ}^{∗} ∘j_{α,∗} = j_{λ}^{∗} ∘j_{β,∗} ∘j_{β}^{∗} ∘j_{α,∗}, by local calculation) and compatibility with pullbacks. Finally for fixed α the 𝕃_{U,α} have to satisfy the previous gluingcondition for coverings U′ ↠ U. There is a continuous u_{X} : (X ⊗_{V} K̄)^{et} → XK̄, with direct image u_{X,∗}(F) defined as follows: For minimal α, u_{X,∗}(F)_{U,α} is the restriction of F to finite étale covers of U_{α} ⊗_{K} K̄. If u_{X,∗}(F)_{U,α} is already defined for all α < β, then u_{X,∗}(F)_{U,β} is the universal (inductive limit of) locally constant sheaves fitting into a commutative diagram

u_{X,∗}(F)_{U,β}  ⟶  F | U_{β} ⊗_{V} K̄
↓  ↓
∏′_{α<β} j_{β}^{∗}j_{α,∗}(u_{X,∗}(F)_{U,α}) ⟶ ∏_{α<β} j_{β}^{∗}j_{α,∗}(F | U_{α} ⊗_{V} K̄)

Here in the lower-keft corner ∏′_{α<β} denotes the subset of the product consisting of compatible families (for α̃ < α < β the α -components maps to the α̃ -component via

j_{β}^{∗}j_{α,∗}(u_{X,∗}(F)_{U,α}) → j_{β}^{∗}j_{α,∗}j_{α}^{∗}j_{α̃,∗}(u_{X,∗}(F)_{U,α̃}) = j_{β}^{∗}j_{α̃,∗}(u_{X,∗}(F)_{U,α̃})

One checks without much difficulty that this defines indeed a map of topoi. Also if F is such that all restrictions F_{α} = F | X_{α} ⊗_{K} K̄ are locally constant, then u_{X,∗}(F)_{U,α} = F | U_{α} ⊗_{K} K̄. Finally there is an obvious commutative diagram of maps

(X ⊗_{V} K̄)^{et} ⟶ XK̄
↘  ↓^{v_{X}}
X^{et}

The direct image for the vertical map sends a collection (𝕃_{U,α}) to the sheaf which associates to an étale U → X compatible families of sections in ∏_{α} Γ(U_{α}, 𝕃_{U,α}). We shall need another model for extensions by zero. For α < β define a functor Ψ_{α,β} on sheaves on X 0 ; K̄ by

Ψ_{α,β}(𝕃)_{U}(V) = Γ(V̄_{β}, j_{β}^{∗}j_{α,∗}𝕃_{U} | V)

Here V̄ denotes the normalisation of U ⊗_{V} K̄ in V → U_{α} ⊗_{V} K̄, V̄_{β} its β -stratum (the preimage of U_{β}) and 𝕃_{U} | V the pullback of 𝕃_{U} to V). One checks without difficulty that this satisfies the sheaf property with respect to étale coverings of U. Also for α > β > γ there are transition maps

Ψ_{α,β}(𝕃) → Ψ_{α,γ}(𝕃)

satisfying the obvious commutativity for α < β < γ_{1}, γ_{2} < δ. To proceed further we need a number of local assumptions.

Local Conditions LC: Let R = O_{X,x}^{sh} denote the strict henselisation of X in a point x. We assume:

a) The closures of the strata on Spec(R ⊗_{V} K̄) are normal subvarieties of Spec(R ⊗_{V} V̄), defining a stratification (again indexed by α, β, …)

b) For each α, the inclusion of the α -stratum admits a retraction

r_{α} : Spec(R ⊗_{V} V̄) → \overline{Spec(R ⊗_{V} V̄)_{α}},

sending any β -stratum with β < α into the (open) α -stratum

c) There exists l_{1}, …, l_{r} ∈ R such that the stratification on Spec(R ⊗_{V} K̄) is defined by the ideals generated by subsets of {l_{1}, …, l_{r}}

2. Lemma. Suppose LC hold. Then i) The Ψ_{α,β} are exact ii) If 𝕃 is an injective sheaf on X_{α,K̄}^{0}, then all Ψ_{α,β}(𝕃) are acyclic for the direct image v_{X} (to X^{et}). Also all j_{β}^{∗}j_{α,∗}(𝕃) are acyclic for X_{β}^{0} ⊗_{V} K̄ → X, and the direct images R^{ν}j_{α,∗}𝕃_{α}(j_{α} : X_{α}^{0} ⊗_{V} K̄ → X ⊗_{V} K̄) are defined by the functors j_{β}^{∗}R^{ν}j_{α,∗}.

Proof. The assertions are local in X^{et}. By an inductive limit argument we may replace X by Spec(R), R = O_{X,x}^{sh} as before. Then R ⊗_{V} V̄ is a finite product of strictly henselian local rings. We thus may assume that it is local (eventually replace K by a finite extension). Then by b) Spec(R ⊗_{V} V̄) is stratified with all closed strata normal and thus irreducible. Let Δ_{α} denote the fundamental group of the α -stratum. For each β > α we obtain subgroups I_{β} ⊆ D_{β} ⊆ Δ_{α}, well-determined up to conjugation, namely the inertia and decomposition-group of the β -stratum. If this is defined by (say) l_{1} = … = l_{t} = 0, then I_{β} ≅ ℤ̂(1)^{t}, via the coverings obtained by adjoining roots of l_{i}. Furthermore there is a canonical map Δ_{β} → D_{β}/I_{β} (Δ_{β} - fundamental group of β -stratum) which is an isomorphism by b). Now any locally constant sheaf 𝕃 on the generic stratum corresponds to a continuous representation of Δ. Then one checks that Ψ_{0,β}(𝕃) is defined by Ind_{D_{β}}^{Δ}(𝕃) = continuous D_{β} - linear maps Δ → 𝕃. This is an exact functor and implies the first assertion (over proper subsets U ⫋ Spec(R) one has to localize further).

For ii) we need to compute direct images for v_{X_{α}} : X_{α} ⊗_{V} K̄ → X^{et}. Again the fibre in x ∈ X is the cohomology of Spec(R ⊗_{V} V̄), that is the cohomology of the topos of locally constant sheaves on the α -stratum. (This topos is a retract of the full topos and has the same cohomology). This coincides with the groupcohomology H^{∗}(Δ, …). However (using inclusions Δ_{α} ⊇ D_{α,β} ⊇ I_{α,β} ≅ ℤ̂(1)^{t}) H^{∗}(Δ_{α}, Ind_{D_{α,β}}^{Δ_{α}}L) = H^{∗}(D_{α,β}, 𝕃) vanishes in positive degrees for injective L′s. Similar for the cohomology of j_{β}^{∗}j_{α,∗}𝕃, which corresponds to L^{I_{β}}.

Now we come to one of the main results. For a sheaf 𝕃 on X 0 ; K̄ define a complex Ψ(𝕃)

Ψ(𝕃) : 0 → 𝕃 → ⨁_{codim(X_{β})=1} Ψ_{0,β}(𝕃) → ⨁_{codim(X_{β})=2} Ψ_{0,β}(𝕃) → …

the maps being sums of restrictions with ± signs. On the other hand we have a sheaf j_{0,!}𝕃 on XK̄, which is 𝕃 on X 0 ; K̄ and trivial on all other strata.

3. Proposition. ℝv_{X,∗}(Ψ(𝕃)) ≅ ℝv_{X,∗}(j_{0,!}𝕃) canonically in the category of étale sheaves on X.

Proof. Choose an injective resolution 𝕃 → I^{·} in X 0 ; K̄. Then Ψ(I^{·}) is v_{X,∗} -acyclic, thus ℝv_{X,∗}(Ψ(𝕃)) is represented by v_{X,∗}(Ψ(I^{·})). This associated to U → X étale the complex of global sections Γ((U ∩ X_{0}) ⊗_{V} K̄, Ψ(I^{·})), i.e. the double complex made up from

⨁_{codim(β)=a} Γ((U ∩ X_{0}) ⊗_{V} K̄, Ψ_{0,β}(I^{b})) = ⨁_{codim(β)=a} Γ((U ∩ X_{β}) ⊗_{V} K̄, j_{β}^{∗}j_{0,∗}(I^{b}))

This is the direct image of the total complex associated to a double complex on X ⊗_{V} K̄ with components

⨁_{codim(β)=a} j_{β,∗}j_{β}^{∗}j_{0,∗}(I^{b}),

which is thus acyclic for v_{X,∗}. On the α -stratum its components are

⨁_{\begin{matrix}β≤α \\ codim(β)=a \end{matrix}} j_{α}^{∗}j_{β,∗}j_{β}^{∗}j_{0,x}(I^{b}) = ⨁_{\begin{matrix}β≤α \\ codim(β)=a \end{matrix}} j_{α}^{∗}j_{0,∗}(I^{b})

By one of the standard calculations in homological algebra it then follows that for codim(α) > 0 this forms an acyclic complex. As one the 0-stratum our complex gives 𝕃 it represents j_{0,!}(l) and its direct image v_{X,∗}(j_{0,!}(𝕃)). As the resolution I^{·} is unique up to homotopy, this construction is also canonical (in the derived category) and functorial.

We leave it to the reader to formulate a variant where one extends 𝕃 by zero along some of the codimension-one strata and by ℝ_{j_{0},∗}(𝕃) along the others. Of course one then uses only Ψ′_{0,β}s with the β -stratum an intersection of the codimension-one strata of the first type.

Finally we apply our theory to the previous case of schemes formally étale over the torus-embedding T̄_{λ} ⊆ T̄. We have to check the local conditions (LC): By assumption σ ∩ ker(λ) is spanned by a partial basis {ρ_{1}, …, ρ_{t}} of the lattice L. The strata are indexed by subsets I ⊆ {1, …, t}, and the ideal of the I-stratum is spanned by those μ ∈ L^{∨} ∩ σ^{∨} which are strictly positive on some ρ_{i}, i ∈ I. If L_{I} denotes the quotient-lattice L_{I} = l/⟨ρ_{i} | i ∈ I⟩ and σ_{I} ⊆ L_{I,ℝ} the image of σ, then σ_{I} defines a torus-embedding T_{I} ⊆ T̄_{I}. Furthermore there is a projection T̄ → T̄_{I} which has partial inverse given by an isomorphism T̄_{I} ⟶^{∼} I-stratum ⊆ T̄. The latter map is given on affine rings by sending μ ∈ L^{∨} ∩ σ^{∨} to either μ ∈ L_{I}^{∨} ∩ σ^{∨} or to zero (if μ ∉ L_{I}^{∨}, that is μ(ρ_{i}) > 0 for some i ∈ I). This picture persists after base change to T̄_{λ} (dividing rings by the ideal (π − λ)), and also to the strictly henselian local ring R. Furthermore by ajoining a suitable root π^{1/e} we can achieve that all strata in X ⊗_{V} V̄ are normal. Thus we have conditions a) and b).

Next we define a sheaf Ō on X^{0} ⊗_{V} K̄ as follows: To U and V → UK̄ finite étale associate the global sections of the normalisation O_{V}^{norm} of O_{U} in V. We also want to define a sheaf A_{inf}(Ō_{V}), but for this we need that the Frobenius is surjective on Ō/p · Ō. In fact we have a slightly stronger statement.

4. Lemma. Suppose U = Spec(R) → T̄_{λ} is étale over T̄_{λ}. Then Frobenius is surjective on R̄/p · R̄.

Proof. Frobenius is surjective on the affine ring of the curve T̄_{λ,∞}, and thus also on R_{∞}/p · R_{∞} by étaleness. Then for any almost étale cover S of R_{∞} it is almost surjective on S/pS (by Frobenius-invariance of almost étale coverings) and thus also almost surjective on R̄/p · R̄. Thus if x ∈ R̄ there is a y with p^{\frac{1}{2}} · x ≡ y^{p}mod p. Then y is divisible by p^{\frac{1}{2p}} and x is a p − th power mod p^{1/2}. Repeating we see that it is a p − th power mod p.

A little bit more involved is

5. Lemma. Suppose a ∈ V̄ divides p^{α} for α < 1. Then the map X ↦ X^{p} − aX is surjective on Ō/p · Ō.

Proof. It suffices to check local rings. So let R = O_{X,x}^{sh} ⊆ R̄. Obviously we may assume that p lies in the maximal ideal of R and thus also in the Jacobson-radical of R̄. For y ∈ R̄ consider S = R̄[X]/(X^{p} − aX − y). It suffices to show that S[\frac{1}{p}] is étale over R̄[\frac{1}{p}], or that X^{p} − aX − y and pX^{p−1} − a have no common zero in any extension-field L of R̄[\frac{1}{p}]. But if not, then X =^{p−1} \sqrt{a/p}, y =^{p−1} \sqrt{a/p}(\frac{a}{p} − a). Thus

\begin{matrix}y^{p−1}(\frac{p}{a})^{p} = (1 − pa)^{p} \\ 0 = (1 − pa)^{p} − y^{p−1}(\frac{p}{a})^{p} \end{matrix}

However the right hand side lies in R̄ and is ≡ 1 mod p^{α} (some α > 0), thus is a unit.

We also need to consider the kernel of the endomorphism X^{p} − aX of R̄/pR̄. Assume now that a = b^{p−1} with b^{p} dividing p^{α}, α < 1. Then X^{p} − b^{p−1}X ∈ pR̄ if and only if

(\frac{X}{b})^{p} − (\frac{X}{b}) ∈ \frac{p}{b^{p}} · R̄

This happens if and only if locally in the Zariski - or étale topology

X ∈ ℤ_{p} · b + b · \frac{p}{b^{p}} · r̄

Thus Ker(X^{p} − b^{p−1}X) = b · 𝔽_{p} · b ⊕ (\frac{p}{b^{p−1}} )R̄/pR̄ (as sheaves).

We also need cohomology with values in the projective limit R(Ō) = \underleftarrow{lim} Ō/pŌ (transition maps are Frobenius). This is a ”topological” sheaf. However we do not need continuous cohomology. Projective system F = {F_{0} ← F_{1} ← F_{2} ← … } indexed by integers form a new topos. The functor of global sections maps this to a projective system of abelian groups, whose projective limit is H^{0}(F). Deriving this functor gives higher cohomology. The derived functor of (projective systems F_{n}) → (projective systems of abelian groups) can be computed termwise. Finally the derived functors of \underleftarrow{lim}_{ℕ} are wellknown: There is only a non-zero \underleftarrow{lim}_{ℕ}^{(1)}, which

vanishes if the projective system satisfies a Mittag-Leffler-condition. By Leray we have short exact sequences

0 → \underleftarrow{lim}^{(1)} H^{i−1}(F_{n}) → H^{i}(F) → \underleftarrow{lim} H^{i}(F_{n}) → 0.

Thus we have defined the cohomology of R(Ō) = \underleftarrow{lim} Ō/p · Ō as well as direct images R^{a}v_{X,∗}(R(Ō)) (which are projective systems on X^{et}). Also choose any a ∈ R(V̄), a ≠ 0. Then a can be represented by a system of elements a_{n} ∈ ˆ V̄ , a_{n} = a_{n+1}^{p}. Also a = b^{p−1} for some b. Then for n big enough the a′_{n}s and b′_{n}s satisfy the condition in Lemma. Thus the sequence

0 → 𝔽_{p} ⊕ (\frac{p}{b_{n}^{p−1}} Ō)/pŌ → Ō/pŌ ⟶^{Φ−a} Ō/p − Ō → 0

(Φ = Frobenius) is exact.

Passing to the limit we obtain a sequence

0 → 𝔽_{p} → R(Ō) ⟶^{Φ−a} R(Ō) → 0

whose cohomology has trivial transition-maps for n >> 0, and thus will disappear if we take cohomology. In the same vein we can pass to A_{inf}(Ō) = W (R(Ō)), which a priori is a double projective limit but can be written as a single projective limit.

However we really only need the A_{inf}(Ō)/p^{s}A_{inf}(Ō) = W_{s}(R(Ō)). They fit into a sequence

0 → ℤ/p^{s}ℤ → W_{s}(R(Ō)) ⟶^{Φ−[a]} W_{s}(R(Ō)) → 0

with again essentially trivial cohomology. Finally we tensor with u_{X,∗}(𝕃), 𝕃 a locally constant ℤ/p^{s}ℤ -sheaf on (X^{0} ⊗_{V} K̄)^{et}, to obtain

0 → u_{X,∗}(𝕃) → u_{X,∗}(𝕃) ⊗ A_{inf}(Ō) ⟶^{Φ−[a]} u_{X,∗}(𝕃) ⊗ A_{inf}(Ō) → 0.

As a consequence we obtain exact sequences

0 → Coker(Φ − [a] | H^{i−1}(X^{0} ⊗_{V} K̄, u_{X}^{∗}(𝕃) ⊗ A_{inf}(Ō)) → H^{i}(X^{0} ⊗_{V} K̄, 𝕃)

→ ker(Φ − [a] | H^{i}(X^{0} ⊗_{V} K̄, u_{X,∗}(𝕃) ⊗ A_{inf}(Ō))) → 0

We also need a variant for cohomology with compact support. For each stratum α we have sheaves Ō_{α} on X_{α}^{0} ⊗_{V} K̄ as before. For α < β we define a sheaf ψ̃_{α,β}(Ō_{α}) on X_{α}^{0} ⊗_{V} K̄ as follows:

Suppose we have U → X étale and V → U_{α}⊗_{V} K̄ finite étale, with normalisation V^{norm} → Ū_{α} ⊗_{V} K̄. It is stratified with reduced strata obtained by taking closures of generic fibres. We define ψ̃_{α,β}(Ō_{α}) as the normalisation of O_{U} in V_{β}^{norm}. If U = Spec(R) is affine, S = normalisation of R in V, then there are finitely many primes 𝔭_{ν} ⊆ S corresponding to the irreducible components of V_{β}^{norm}. ψ̃_{α,β}(Ō_{α}) is then the direct sum of the normalisations of (S/𝔭_{ν}). (S/𝔭_{ν}[\frac{1}{π}] is already normal). Also for β < γ there are morphisms

ψ̃_{α,β}(Ō_{α}) → ψ̃_{α,γ}(Ō_{α})

satisfying some transitivity for α < β < γ_{1}, γ_{2} < δ.

6. Lemma.

i) Frobenius Φ is surjective on ψ̃_{α,β}(Ō_{α})/p · ψ̃_{α,β}(Ō_{α})

ii) For b ∈ V̄ with b^{p} dividing p^{α}, α < 1, Φ − b is surjective on ψ̃_{α,β}(Ō_{α})/p · ψ̃_{α,β}(Ō_{α}).

Proof. It suffices to check these over Spec(R), R = O_{X,x}^{sh}. We may assume α = 0. Then ψ̃_{α,β}(R̄) the subset of the product ∏(R̄/𝔭_{ν})^{norm} where Δ = π_{1}(Spec(R ⊗_{V} K̄)) acts continuously. The product is over all minimal primes of the β -stratum. Equivalently it is the induced module (via D_{β} ⊆ Δ) from one (R̄/𝔭_{ν})^{norm}, which by LC coincides with the version of R̄ for the β -stratum. There assertions i) and ii) are already known. In fact for i) we can prove surjectivity of Frobenius on global section over sufficiently small affines, reducing to R_{∞} and checking there directly. We now have the obvious complexes

ψ̃(Ō_{α}) : Ō_{α} → ⊕_{codimβ=1}ψ̃_{α,β}(Ō_{α}) → ⊕_{codim(β)=2}ψ̃_{α,β} (Ō_{α}) → …

R(ψ̃(Ō_{α})) and A_{inf}(ψ̄(Ō_{α})). For any 𝕃 locally constant 𝕃 on (X^{0} ⊗_{V} K̄)^{et}, annihilated by p^{s}, we obtain a complex of projective systems

0 → ψ(𝕃) → u_{X,∗}(𝕃) ⊗ A_{inf}(ψ̃(Ō)) ⟶^{Φ−[a]} u_{X,∗}(𝕃) ⊗ A_{inf}(ψ̃(Ō))) → 0

with irrelevant cohomology, and thus short exact sequences

0 → Coker(Φ − [a] | H^{i−1}(X^{0} ⊗_{V} K̄, u_{X,∗}(𝕃) ⊗ A_{inf}(ψ̃(Ō))) →

→ H_{!}^{i}(X^{0} ⊗_{V} K̄, 𝕃) → ker(Φ − [a] | H^{i}(X^{0} ⊗_{V} K̄, u_{X,∗}(𝕃) ⊗ A_{inf}(ψ̃(Ō))) → 0

There are variants with partial compact support at infinity, with obvious formulations and proofs.

Finally we get back to ”almost mathematics”. There is also a subsheaf J̄ ⊆ Ō, J̄ = kernel(Ō → ⨁_{codimβ=1} ψ̃_{0,β} (Ō)).

Lemma.

J̄ ≈ ψ̃(Ō)

is an almost isomorphism.

Proof. Again we may work with R = O_{X,x}^{sh}, Spec(R) → T̄_{λ}. The assertion holds for J_{∞} ⊆ R_{∞}: Here each stratum is irreducible, with ideal generated by the μ ∈ ⋃_{n} L_{n}^{∨} with μ(ρ_{i}) > 0 for some i ∈ I = index-set of the stratum. By direct calculation we have an isomorphism. After that R̄ ⊇ R_{∞} is the limit of almost étale coverings. Assume we have such a covering S ⊇ R_{∞} with group G. Then if 𝔮_{α} ⊆ S is the prime ideal giving an irreducible component of the α-stratum, D_{α} ⊆ G its decomposition-group, and 𝔭_{α} = 𝔮_{α} ∩ R_{∞} ⊆ R_{∞}, we have a map R_{∞}/𝔭_{α} ⊗_{R_{∞}} S → Ind_{D}^{G}((S/𝔮_{α})^{norm}), which is an isomorphism after inverting π. As the left side is almost étale over R_{∞}/𝔭_{α} and thus has non-degenerate trace-form, the map must be an almost isomorphism. Passing to the limit over all S we obtain that ψ̃(R̄) is almost isomorphic to J̄_{∞} ⊗_{R_{∞}} R̄ ≈ J̄.

If we define R(J̄) = \underleftarrow{lim} J̄/pJ̄ = ker(R(Ō) → R(Ō/J̄)) and similarly A_{inf}(J̄) = W (R(J̄)), we get almost isomorphisms

H^{∗}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ A_{inf}(J̄)) ≈ H^{∗}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ A_{inf}(ψ̃(Ō))).

Next consider the direct images R^{ν}q_{X,∗}(J̄/p^{s}J̄) which are sheaves on X^{et}. By Th. 2.10.i) there are almost zero unless 0 ≤ ν ≤ d, are almost quasi-isomorphic to bounded complexes of quasicoherent sheaves, and there is a canonical almost map

R^{d}q_{X,∗}(J̄/p^{s}J̄) → ω_{X} ⊗ V̄ /p^{s}V̄ ,

inducing almost quasi-isomorphisms

ℝq_{X,∗}(𝕃 ⊗ J̄) ≈ ℝHom_{O_{X}⊗V̄ /p^{s}V̄} (ℝq_{X,∗}(𝕃^{∨} ⊗ Ō), ω_{X} ⊗ V̄ /p^{s}V̄)[−d]

for 𝕃 locally constant annihilated by p^{s}. Furthermore there is a canonical trace-map

H^{d}(X^{et}, ω_{X}) = H^{d}(X, ω_{X}) → V

inducing global duality for coherent sheaves on X. As the direct iamges ℝq_{X,∗}(𝕃 ⊗ J̄) etc. are almost coherent these combine to produce almost quasi-isomorphisms

ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ J̄) ≈ ℝHom_{V̄ /p^{s}V̄} (ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ Ō), V̄ /p^{s}V̄)[−2d]

Remark. Previously we always worked with \frac{1}{π} ω_{X} instead of ω_{X}. This is because in the torus-embedding we identified the dualising complex for the quotient V [L^{∨} ∩ σ^{∨}]/(λ − π) as the tensor-product of that for V [L^{∨} ∩ σ^{∨}] with the quotient, instead of the Ext^{1}. Then we used the trace-map for V [L^{∨} ∩ σ^{∨}]. This differs from the trace-map used in relative duality by a factor π. That this is so can be seen easily for the affine torus-embedding V [λ], and follows in the general case by functoriality V [λ] ⊆ V [L^{∨} ∩ σ^{∨}]. It is also related to the fact that we have tacitly identified Λ^{d+1}L^{∨} and Λ^{d} ker(λ)^{∨}.

The structure of H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ A_{inf}(Ō)) is surprisingly simple. Assume for the moment that 𝕃 is annihilated by p. Then this cohomology is a module over R(V̄).

8. Theorem. For each i H^{i}(X^{0}⊗_{V} , K̄, 𝕃 ⊗ R(J̄_{I})) is almost isomorphic to a finitely generated free R(V̄)- module, namely to H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) ⊗ R(V̄) (supportcondition given by I).

Proof. Let M = H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})). Then for ξ = \underline{p} ∈ R(V̄), M/ξM ⊆ H^{i}(X^{0}⊗_{V} K̄, 𝕃⊗J̄_{I}), which is almost finitely presented over R(V̄)/ξ·R(V̄) = V̄ /pV̄. Also Frobenius Φ induces an isomorphism on M. Now for torsion V̄ -modules N we can define a normalised length λ(N) similarly as before: Write V = ∪V_{α} with V_{α} finite over V. Then for N = V̄ ⊗_{V_{α}} N_{α}

λ(N) = \frac{1}{[V_{α} : V]}length_{V} (N_{α}),

which defines λ(N) for N finitely presented. For N only finitely generated, N = V̄ · N_{α}, we have

λ(N) = lim_{β→∞} \frac{1}{[V_{β} : V]}length_{V} (V_{β} · N_{α}),

and for arbitrary N

λ(N) = sup{λ(N′))N′ ⊆ N finitely generated }.

Also λ(V̄ /𝔞) = inf{v(x) | x ∈ 𝔞}, where v is the valuation normalised such that v(π) = 1. Finally for a submodule N of a finitely presented V̄ -module, λ(N) = 0 implies N = (0).

By additivity we extend λ to ξ -torsion R(V̄) -modules. Pushout by Frobenius multiplies λ by p. Next for each α > 0

ξ^{α} · H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ J̄_{I})

has finite λ -invariant, and by devissage this holds for ξ^{α} · H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{n}), all n. Applying Frobenius-pushout to this module multiplies λ by p, and produces

ξ^{pα}H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{np}).

On the other hand filter 𝕃 ⊗ R(J̄_{I})/ξ^{np} by submodules ξ^{in}(𝕃 ⊗ R(J̄_{I}))/ξ^{np}, apply the exact cohomology-sequences and use that for an exact sequence

N_{1} → N_{2} → N_{3}

we have

λ(ξ^{α+β}N_{2}) ≤ λ(ξ^{α}N_{1}) + λ(ξ^{β}N_{3}),

with equality for all α, β > 0 only possible if

λ(ξ^{α} · ker(N_{1} → N_{2})) = λ(ξ^{β} · Coker(N_{2} → N_{3})) = 0, all α, β > 0.

It then follows that for m ≥ 0 the exact sequences

H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{m}) → H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{m+1})
→ H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{m})

satisfy this condition. If (say) N denotes the kernel of the first map, for each α, β > 0 multiplication by ξ^{β} on ξ^{α}N factors as ξ^{α}N ↠ N_{1} → ξ^{α}N with N_{1} submodule of a finitely presented V̄ /pV̄ -module. As λ(N_{1}) ≤ λ(ξ^{α} · N) = 0, we have N_{1} = (0), and N ≈ 0. Quite similarly (or from the long exact sequence in cohomology) the cokernel is almost zero. Especially the projective system of H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})/ξ^{m}) has almost surjective transition-maps, and lim^{(1)} of it is almost zero. Also if M^{i} = H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})), then H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗

R(J̄_{I})/ξ^{m}) ≈ M^{i}/ξ^{m} · M^{i}, and M = \underleftarrow{lim}_{m}M/ξ^{m}M: If z ∈ H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗

R(J̄_{I})/ξ^{m}), we show that for each α > 0 ξ^{α} · z lifts to M. For this choose a strictly decreasing sequence {α_{n}, n ≥ m}, 0 < α_{n+1} < α_{n} < α = α_{m}. Then choose z_{n} ∈ H^{i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ R(J̄_{I})(ξ^{n}) with z_{m} = z, z_{n+1} lifting ξ^{α_{n}−α_{n+1}} · z_{n}. Then ξ^{α} · z is lifted by \underleftarrow{lim}(ξ^{α_{n}} · z_{n}).

Also it follows that multiplication by ξ defines an almost isomorphism ξ^{n}M^{i}/ξ^{n+1}M^{i} ≈ ξ^{n+1}M^{i}/ξ^{n+2}M^{i}, so ξ is almost injective on M^{i}. Next if M^{i}/ξM^{i} is not almost zero, choose an element m ∈ M^{i}. m is not almost zero in M^{i}/ξM^{i}. Multiplication by m defines a map

ξ · R(V̄) ≅ R(V̄) → M^{i},

injective as m is not ξ^{l} -torsion for any l. Consider α, 0 ≤ α ≤ 1, such that this map extends to ξ^{α} · R(V̄) (this is not possible for α < 0, as m ∉ ξ · M^{i}). Such an extension is almost unique. Thus if \underline{α} is the infimum of these α’s, the map extends uniquely to the union

𝔞(\underline{α}) = ⋃_{α<\underline{α}} ξ^{α} · R(V̄).

35

Hence we get an injection 𝔞(\underline{α}) ⊆ M^{i}, with quotient M̃^{i} = M^{i}/𝔞(\underline{α}) again almost without ξ -torsion. If M̃^{i} is not almost zero we can repeat the construction to obtain an increasing sequences of submodules of M^{i}, with subquotients 𝔞(\underline{α}_{ν}), and almost ξ -torsion-free quotients. Especially

λ(ξ · M/ξ^{2} · M) ≥ ∑_{ν} λ(ξ · 𝔞(\underline{α}_{ν})ξ^{2} · 𝔞(\underline{α}_{ν}))

= number of \underline{α}_{ν}’s, so the process must stop. Hence we have found a finite filtration

(0) = M_{0} ⊂ M_{1}^{⊂} … ⊂ M_{r} = M = M^{i},

with M_{n}/M_{n−1} ≈ 𝔞(\underline{α}_{n}).

It then follows easily that for each α > 0 there exists a free submodule R(V̄)^{r} ⊆ M with quotient annihilated by ξ^{α}. Especially M is almost projective of finite rank r over R(V̄). Next define an R(V̄) -algebra

A = S[M]/(Φ(m) − m^{p})

(symmetric algebra divided by the relations displayed). Obviously A[ξ^{−1}] is finite flat over R(V̄)[ξ^{−1}] of rank p^{r}, generated by monomials with exponents < p in a basis of M[ξ^{−1}] = R(V̄)[ξ^{−1}]^{r}. Also as the free submodules N ≅ R(V̄)^{r} ⊆ M form a filtering inductive system with union almost M, we obtain (taking such monomials in a basis of N) a filtering inductive system of free submodules of A (of rank p^{r}) with union almost A. Thus A is almost projective of finite rank p^{r} over R(V̄), and almost finitely generated. Also assume given such a free submodule with basis {m_{1}, …, m_{r}}, quotient annihilated by ξ^{α}, and let ξ^{α} · Φ(m_{i}) = ∑_{j} a_{ij} · m_{j} with a_{ij} ∈ R(V̄). Then the matrix (a_{ij}) has an inverse up to ξ^{β} for any β > (p + 1)α, as ξ^{β}·m_{i} ∈ ξ^{α}·Φ(ξ^{α}·M). Also all monomials with exponents < p in the ξ^{α/p−1}·m_{i} = T_{i} form an subalgebra A_{α} isomorphic to

R(V̄)[T_{1}, …, T_{r}]/(T_{i}^{p} − ∑_{j} a_{ij} · T_{j}).

¿From ramification-theory it follows that the trace-form identifies the dual of A_{α} with det(a_{ij})^{−1} · A_{α}. Especially A[1/ξ] = A_{α}[1/ξ] is étale over R(V̄)[1/ξ], and the corresponding canonical idempotent e ∈ A_{α} ⊗_{R(V̄)} A_{α}[1/ξ] has denominator det(a_{ij}) which divides ξ^{rβ} for any β > (r + 1)α. Letting α → 0 we see that the canonical idempotent in A ⊗_{R(V̄)} A[1/ξ] is almost integral, thus A is almost étale over R(V̄).

Hence we can almost lift A to an almost étale cover of A_{inf}(V̄) and its quotient ˆV̄. However ˆ V̄ [\frac{1}{p}] is algebraically closed, so this cover is almost trivial. It follows that A ≈ R(V̄)^{p^{r}}. We thus obtain p^{r} different (almost) homomorphisms of A into R(V̄), corresponding to (almost) maps M → R(V̄) which are Frobenius linear. These maps almost span the (module) dual of A and thus also of M (check with free submodules). Now the 𝔽_{p} -vectorspace Hom_{Φ,R(V̄)}^{almost}(M, R(V̄)) has p^{r} elements and thus dimension r. A basis of it defines an almost map M → R(V̄)^{r} inducing an almost surjection on duals, and thus an almost isomorphism. Also the canonical 𝔽_{p}-structure on R(V̄)^{r} can be recovered as the space of Φ -invariants.

Next replace M by H^{i}(X^{0} ⊗_{V} K̄, 𝕃⊗ψ̃(R(Ō))) = M̃^{i}, which is almost the same. Then we have the exact sequences (b ∈ R(V̄) as before)

0 → Coker(Φ − b^{p−1} | M̃^{i−1}) → H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) → Ker(Φ − b^{p−1} | M̃_{i}) → 0.

But M̃^{i} ≈ R(V̄)^{r_{i}} (Φ -linear). Thus Φ − b^{p−1} is almost surjective on M̃^{i−1}. Now the above sequence comes (with tensor-products etc.) from the first line in

0 ⟶ 𝔽_{p} ⟶^{b·} R(Ō) ⟶^{Φ−b^{p−1}} R(Ō) ⟶ 0
∥  ↓_{b′}  ↓_{b^{′p}}
0 ⟶ 𝔽_{p} ⟶^{b·b′} R(Ō) ⟶^{Φ−(bb′)^{p−1}} R(Ō) ⟶ 0

Extend this as shown with b′ ∈ R(V̄), b′ ≠ 0, unit, to get

0 ⟶ coker(Φ − b^{p−1})/M̃^{i−1}) ⟶ H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) ⟶ ker(Φ − b^{p−1}/M̃^{i}) ⟶ 0
↓_{b^{′p}·}  ∥  ↓_{b′}
0 ⟶ Coker(Φ − (bb′)^{p−1}/M̃^{i−1}) ⟶ H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) ⟶ ker(ϕ − (bb′)^{p−1}/M̃^{i}) ⟶ 0

The first vertical arrow is zero, thus H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) injects into M̃_{i} for all b (we may vary b, b′). Also multiplication by b respectively b′ applied to M̃ ≈ R(V̄)^{r_{i}} induces maps 𝔽_{p}^{r_{i}} = Φ -invariants in R(V̄)^{r_{i}}

→ ker(Φ − b^{p−1}/M̃^{i})
→ ker(Φ − (bb′)^{p−1}/R(V̄)^{r_{i}}) = b · 𝔽_{p}^{r_{i}},

and one derives that indeed

M̃^{i} ≈ H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) ⊗ R(V̄).

Corollary 1. For any 𝕃 (only annihilated by p^{s})

ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ A_{inf}(J̄_{I})) ≈ ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, 𝕃) ⊗ A_{inf}(V̄)
ℝΓ(X^{0} ⊗ K̄, 𝕃 ⊗ J̄_{I}) ≈ ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, L) ⊗ V̄

Proof. Devissage in 𝕃

Corollary 2. ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, 𝕃) satisfies Künneth.

Proof. Descente from ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ J̄_{I}) (for V̄ ⊗ (a ℤ/p^{s}ℤ -module), ”almost zero” implies zero).

Also the map u_{X} : (X^{0} ⊗_{V} K̄)^{et} → X^{0} ⊗_{V} K̄ and its variants induce a map

ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, 𝕃) → ℝΓ_{(!)}((X^{0} ⊗_{V} K̄)^{et}, 𝕃).

Especially the trace-form on H_{!}^{2d}((X^{0} ⊗_{V} K̄)^{et}, ℤ/p^{s}ℤ(d)) (if X has pure relative dimension d) induces a trace-form tr^{et} on H^{2d}(X^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ(d)). We shall show later that it coincides with the restriction of the trace-form on H^{2d}(X^{0} ⊗_{V} K̄, J̄/p^{s}J̄(d)). Hence it induces by descente a perfect duality.

Corollary 3. The étale trace induces a perfect duality

ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, 𝕃) ⊗^{𝕃} ℝΓ_{(!^{c})}(X^{0} ⊗_{V} K̄, 𝕃^{∨})(d)[2d] → ℤ/p^{s}ℤ.

4. Chern-classes and Gysin-maps

Suppose L is a line-bundle on X. We define its first Chern-Class c_{1}(L) ∈ H^{2}(X^{0}⊗_{V} K̄, ℤ/p^{s}ℤ(1)) as follows: L is given by a class in H^{1}(X, O_{X}^{∗}) = H^{1}(X^{et}, O_{X}^{∗}). Pullback via v_{X}^{∗}O_{X}^{∗} → Ō^{∗} gives a class in H^{1}(X ⊗_{V} K̄, Ō^{∗}), and via the Kummersequence 0 → μ_{p^{s}} → Ō^{∗} → Ō^{∗} → 0 we obtain a class in H^{2}(X ⊗_{V} K̄, μ_{p^{s}}). It is clear that pullback to H^{2}(X ⊗_{V} K̄)^{et}, μ_{p^{s}}) maps it to the étale first Chern-class of L | X ⊗_{V} K̄.

Next we consider a closed immersion i : Y → X such that locally in the étale topology X ≅ X_{1} × 𝔸^{1}, Y ≅ X_{1} × {0}, with X_{1} having toroidal singularities, stratifications etc. as before. We intend to give a geometric description of the Gysinmap i_{∗}. We define a new stratification on X by adding Y as well as its intersections with X_{α}’s to the strata, and call the new topos X̃. Thus X̃^{0} = X^{0} − Y^{0}. Next there are maps of topoi

j : (X \ Y)^{0} ⊗_{V} K̄ → X^{0} ⊗_{V} K̄
w : X̃ ⊗_{V} K̄ → X ⊗_{V} K̄
i : Y ⊗_{V} K̄ → X̃ ⊗_{V} K̄

Now a sheaf F on (X ⊗_{V} K̄)^{et} which is locally constant on all strata induces 𝕃 = u_{X,∗}(F) on X ⊗_{V} K̄ and 𝕃̃ = w^{∗}(𝕃) on X̃ ⊗_{V} K̄. Furthermore on (X ⊗_{V} K̄)^{et} there is an exact sequence (or better an exact triangle)

0 → F → ℝj_{∗}j^{∗}F → i_{∗}(F/Y)(−1)[−1] → 0,

whose ℝu_{X,∗} receives a map from

0 → 𝕃̃ → ℝj_{∗}j^{∗}𝕃̃ → i_{∗}(𝕃̃/Y)(−1)[−1] → 0.

It is wellknown that the Gysin-homomorphism in étale topology is defined by the connecting map

ℝΓ((Y^{0} ⊗_{V} K̄)^{et}, F(1))[−2] → ℝΓ((X^{0} ⊗_{V} K̄)^{et}, F)

from the first triangle, and similar for variants with (partial) compact support.

Sketch. Taking cup-product with classes in ℝΓ_{!}((X^{0} ⊗_{V} K̄)^{et}, F^{∨}) we reduce to constant coefficients and cohomology in highest degrees. Then the assertion follows from the explciit description of the cohomology-class of a point in Y^{0}.

Hence the second triangle defines the Gysin-homomorphism for i : Y ⊗_{V} K̄ → X̃ ⊗_{V} K̄.

Finally we come to coefficients 𝕃 ⊗ Ō.

1. Proposition. There is an almost exact sequence (better triangle)

0 → 𝕃 ⊗ Ō_{X} → ℝj_{∗}(𝕃 ⊗ ŌX̃) → i_{∗}(𝕃 ⊗ Ō_{Y})(−1)[−1] → 0

The maps can be described as follows:

i) The first is induced from

Ō_{X}/p^{s} · Ō_{X} → j_{∗}ŌX̃/p^{s}ŌX̃.

We claim that this is an almost isomorphism. Applying this to all strata and tensoring with 𝕃 then gives an almost isomorphism of H^{0}’s.

ii) Let ι denote the stratum Y in X̃. Then there is a map ℝj_{∗}(𝕃 ⊗ ŌX̃) → ℝ^{1}j_{∗}(𝕃 ⊗ ψ̃_{0ι}(ŌX̃)) Furthermore we claim that

j_{∗}ψ̃_{0,ι}(ŌX̃) = i_{∗}(Ō_{Y})

and that the cup-product

ℝ^{1}j_{∗}ℤ/p^{s}ℤ × j_{∗}(ψ̃_{0,ι}(ŌX̃) → R^{1}j^{∗}(Ō_{Y} /p^{s}Ō_{Y})

is an almost isomorphism. Again this persists after tensoring with 𝕃 iii) The higher direct images ℝ^{ν}j_{∗}ŌX̃/p^{s}OX̃ ≈ 0 for ν ≥ 2, and also after

tensoring with 𝕃.

The assertions in i), ii), iii) imply almost exactness of the sequences as well as that the corresponding sequence with constant coefficients maps to it. If we show

iv) The connecting homomorphism in our exact sequence computes i_{∗},

we also get the required compatability for i_{∗}.

Now to the required assertions. For i), ii), iii) we have already reduced to constant coefficients. That j_{∗}(ψ̃_{0,ι}(ŌX̃)) = i_{∗}(OȲ) follows from the definitions:

To evaluate the left hand side on a finite étale covering V → U^{0} ⊗_{V} K̄ (U ⊆ X étale) we first restrict to the preimage of X \ Y, then extend by normalisation to recover V, next restrict to the preimage of Y ⊗_{V} K̄, and then form global sections of the normalisation. But this gives the right hand side.

For the remaining assertions it suffices to check them on stalks Spec(R), R = O_{X,x}^{sh}. Let T ∈ R define the stratum Y. Extending V we may assume that R ⊗_{V} V̄

is local and normal. Consider the normalisations R̄ and ¯ R̃ of R in the maximal étale cover of Spec(R)^{0} ⊗_{V} K̄ respectively of the complement of Y in it. By the theory of almost étale extension R̄ and ¯ R̃ are limits of almost étale covers of R_{∞} respectively R̃_{∞}, which in turn are induced by toroidal coverings. As locally in the étale topology X ≅ 𝔸^{1} × X_{1} we may use for R̃_{∞} the toroidal covering induced from toroidal coverings of X_{1} and from adjoining roots of T, and for R_{∞} we use the toroidal covering induced from X_{1} as well as from p-power roots of a suitable unit, say 1 + T. Especially ¯ R̃ is almost étale over the subsextension obtained by adjoining to R̄ all roots of T (and normalising the result). Now it Δ and Δ̃ denote the relevant fundamental groups, and Δ_{∼} ⊆ Δ̃ the kernel of the surjection to Δ, we

have to compute the cohomology H^{∗}(Δ_{∼}, ¯ R̃/p^{s} ¯R̃): We need that

i) H^{0}(Δ_{∼}, ¯ R̃/p^{s} ¯R̃) ≈ R̄/p^{s}R̄

ii) H^{1}(Δ_{∼}, ¯ R̃/p^{s} ¯R̃) ≈ (R̄/T R̄)^{norm} ⊗ ℤ/p^{s}ℤ(−1)

iii) H^{ν}(Δ_{∼}, ¯ R̃/p^{s} ¯R̃) ≈ (0) for ν ≥ 2.

By the theory of almost étale extensions we may first replace R̄ by R_{∞} and ¯ R̃ by the subextension corresponding to the quotient Δ_{∼} → ℤ̂(1), i.e. where we adjoin roots of T. Furthermore as groups of order prime to p have trivial cohomology, we reduce to the quotient Δ_{∼} → ℤ_{p}(1). Let S_{∞} ⊆ ¯ R̃ denote the corresponding normalisation,

S_{∞}/ ¯J̃_{∞} ≈ (R_{∞}/T · R_{∞})^{norm} ≅ Hom_{cont}(ℤ_{p}(1), V̄)

The assertions we have to show are i) H^{0}(ℤ̂(1), S_{∞}/p^{s} · S_{∞}) ≈ R_{∞}/p^{s} · R_{∞} ii) H^{1}(ℤ̂(1), S_{∞}/p^{s} · S_{∞}) ≈ (R_{∞}/T R_{∞})^{norm} ⊗ ℤ/p^{s}ℤ(−1) iii) H^{ν}(ℤ̂(1), S_{∞}/p^{s} · S_{∞}) = 0 for ν > 1.

Here iii) is clear because if cohomological dimension one. Also we may replace ℤ̂(1) by the quotient ℤ_{p}(1), as the kernel ∏_{l≠p} ℤ_{l}(1) has trivial cohomology. This amounts to changing S_{∞} and R̃_{∞} by adjoining only p-power roots of T. We do this, but retain the notation for simplicity. Now write S_{∞} = ⋃_{n} S_{n}, with

S_{n} = R_{∞}[T^{1/p^{n}}]^{norm}

Note that S_{n}[\frac{1}{T}] is almost étale over R_{∞}[\frac{1}{T}], by the purity theorem. Also R_{∞} is the inductive limit of regular rings R_{m} = V_{m}[(1 + T)^{p^{−n}}]^{sh}, and thus S_{n} = ⋃_{m} S_{n,m} with S_{n,m} = R_{m}[T^{p^{−n}}]^{norm} a free R_{n} -module of rank q = p^{n}. The group Δ_{n} = ℤ/p^{n}ℤ(1) operates on S_{n,m}. We have H^{0}(Δ_{n}, S_{n,m}) = R_{m}. For the higher cohomology consider the infinite complex

K_{n,m}^{·} :→ S_{n,m} ⟶^{(σ−1)} S_{n,m} ⟶^{tr} S_{n,m} ⟶^{(σ−1)} S_{n,m} ⟶^{tr} S_{n,m}

where tr = ∑_{i=0}^{q−1} σ^{i}.

Its cohomology H^{i}(K_{n,m}) = H̃^{i}(Δ_{n}, S_{n,m}) is called the Tate-cohomology (i ∈ ℤ). It coincides for i > 0 with H^{i}(Δ_{n}, S_{n,m}), while H̃^{0}(Δ_{n}, S_{n,m}) = R_{m}/tr(S_{n,m}). Obviously H̃^{i} is periodic with period two. If S_{n,m}^{∨} = Hom_{R_{m}}(S_{n,m}, R_{m}), we can similarly compute H̃^{i}(Δ_{n}, S_{n,m}^{∨}) from the version of K_{n,m} where S_{n,m} is replaced by duals. However this is isomorphic to (K_{n,m}^{·})^{∨}[1]. As R_{m} has finite cohomological dimension two we thus obtain a spectral sequence

E_{2}^{a,b} = Ext_{R_{m}}^{a}(H̃^{−b}(Δ_{n}, S_{n,m}^{∨}), R_{m}) ⇒ H̃^{a+b+1}(Δ_{n}, S_{n,m})

with E_{2}^{a,b} = 0 unless 0 ≤ a ≤ 2.

Now we pass to the limit m → ∞. The trace-form defines embeddings S_{n,m} ↪ S_{n,m}^{∨}. If we localise at height one primes 𝔭 of R_{m} these are determined by the different of the extension. It is a unit except if 𝔭 divides p or T. However at the prime-divisors of P the p-valuation of the different converges to zero because S_{n}[\frac{1}{T}] is almost étale over R_{∞}[1/T]. For 𝔭 dividing T the different is generated by T^{\frac{q−1}{q}}. It thus follows that

T^{\frac{q−1}{q}} · S_{n,m} ⊂ S_{n,m}^{∨},

with the quotient annihilated by p^{α_{m}}, lim_{m→∞}α_{m} = 0. It also follows that p^{α_{m}} annihilates S_{n}/S_{n,m} ⊗_{R_{m}} R_{∞}. Hence we obtain:

a) S_{n} is almost projective finitely generated /R_{∞}

b) The trace map factors

tr : J_{n} = T^{\frac{1}{q}} · S_{n} → T · R_{∞},

and \frac{1}{T} · tr induces an almost isomorphism

J_{n} ≈ Hom_{R_{∞}}(S_{n}, R_{∞})

Especially tr(J_{n}) ≈ T · R_{∞}.

c) The H̃^{i}(Δ_{n}, S_{n}) are almost finitely presented. There is a spectral sequence (up to almost zero modules)

E_{2}^{a,b} = Ext_{R_{∞}}^{a}(H̃^{−b}(Δ_{n}, J_{n}), R_{∞}) ⇒ H̃^{a+b+1}(Δ_{n}, S_{n})

with E_{2}^{a.b} ≈ 0 unless 0 ≤ a ≤ 2.

((c) follows because S_{n,m} ⊗_{R_{m}} R_{∞} → S_{n} is a p^{α_{n}} -isomorphismus)

Now H̃^{∗}(Δ_{n}, J_{n}) is annihilated by p^{n} as well as by tr(S_{n}) which almost contains T. As (p^{n}, T) form a regular sequence in R_{∞}, Ext_{R_{∞}}^{a}(M, R_{∞}) = (0) for a ≤ 1 if M is annihilated by p^{n} and T. Thus the spectral sequence almost degenerates to give (using period two)

H̃^{a+1}(Δ_{n}, S_{n}) ≈ Ext_{R_{∞}}^{2}(H̃^{a}(Δ_{n}, J_{n}), R_{∞})

However H̃^{0}(Δ_{n}, J_{n}) = (J_{n} ∩ R_{∞})/tr(J_{n}) ≈ T · R_{∞}/T · R_{∞} ≈ 0, and thus H̃^{even}(Δ_{n}, J_{n}) ≈ H̃^{odd}(Δ_{n}, S_{n}) ≈ (0). Passing to the limit n → ∞ implies

H^{i}(Δ_{∞}, J_{∞}) = (0), i > 0 even
H^{i}(Δ_{∞}, S_{∞}) = (0), i > 0 odd
H^{0}(Δ_{∞}, J_{∞}) ≈ T · R_{∞}
H^{0}(Δ_{∞}, S_{∞}) ≈ R_{∞}

Also all cohomology vanishes in degree > 1 for p-torsion modules).

Finally we compute

H^{∗}(Δ_{∞}, S_{∞}/J_{∞}) ≈ H^{∗}(ℤ_{p}(1), (R_{∞}/T R_{∞})^{norm}).

Here the action of ℤ_{p}(1) on (R_{∞}/T R_{∞})^{norm} is trivial. Thus the module coincides with H^{0}. Also H^{1} are the continuous homomorphisms ℤ_{p}(1) → (R_{∞}/T · R_{∞})^{norm}. As the target has no torsion these vanish. Finally H^{2} can be computed by tensoring with 0 → ℤ_{p} → ℚ_{p} → ℚ_{p}/ℤ_{p} → 0. The result:

H^{0}(Δ_{∞}, (R_{∞}/T · R_{∞})^{norm}) = (R_{∞}/T · R_{∞})^{norm}
H^{1}(Δ_{∞}, (R_{∞}/T · R_{∞})^{norm}) = (0)
H^{2}(Δ_{∞}, (R_{∞}/T · R_{∞})^{norm}) = (R_{∞}/T · R_{∞})^{norm} ⊗ ℚ_{p}/ℤ_{p}(−1)

Next using 0 → J_{∞} → S_{∞} → S_{∞}/J_{∞} → 0 we compute

H^{0}(Δ_{∞}, S_{∞}) = R_{∞}
H^{1}(Δ_{∞}, S_{∞}) ≈ (0)
H^{2}(Δ_{∞}, S_{∞}) ≈ (R_{∞}/T · R_{∞})^{norm} ⊗ ℚ_{p}/ℤ_{p}(−1)

Finally 0 → p^{s} · S_{∞} → S_{∞} → S_{∞}/p^{s} · S_{∞} → 0 gives

\begin{matrix}H^{0}(Δ_{∞}, S_{∞}/p^{s} · S_{∞}) ≈ R_{∞}/p^{s} · R_{∞} \\ H^{1}(Δ_{∞}, S_{∞}/p^{s} · S_{∞}) ≈ (R_{∞}/T · R_{∞})^{norm} ⊗ ℤ/p^{s}ℤ(−1) \end{matrix}

Thus we have shown our claim (the maps are easily seen to be a described there). Apply ℝq_{X,∗} to the almost exact sequence

0 → Ō_{X}/p^{s} · Ō_{X} → ℝj_{∗}ŌX̃/p^{s} · ŌX̃ → i_{∗}(Ō_{Y} /p^{s} · Ō_{Y})[−1](−1) → 0

to obtain in highest degree

ℝ^{d}q_{X,∗}(Ō_{X}/p^{s}Ō_{X}) → ℝ^{d}q_{X̃,∗}(ŌX̃/p^{s}ŌX̃) → ℝ^{d−1}q_{Y,∗}(Ō_{Y} /p^{s} · Ō_{Y})(−1) → 0

on X^{et}. These map to the exact sequence

0 → ω_{X}(−d) → ω_{X}(Y)(−d) = ωX̃(−d) → i_{∗}ω_{Y} (−d) → 0

with the resulting two squares commutative:

This is an easy local calculation. The first square commutes on X \ Y which is dense. For the second choose a local equation T for Y and extend to local parameter X_{1} = T, X_{2}, …, X_{d}. Then we obtain a ℤ_{p}(1)^{d} = Δ -extension of OX̃ by adjoining p-power roots of the X_{i}. Then the trace-map sends the generator of H^{d}(Δ, ℤ_{p}(d)) ≅ ℤ_{p} to d log X_{1} ∧ … ∧ d log X_{d}. When mapping to the ”Y -terms” the first goes to the canonical generator of H^{d−1}(Δ_{Y} , ℤ_{p}(d − 1)) = ℤ_{p}, where Δ_{Y} ≅ ℤ_{p}(1)^{d−1} corresponds to adjoining p-power roots of X_{2}, …, X_{d}. In turn this goes to d log X_{2} ∧ … ∧ d log X_{d}, which is the residue of d log X_{1} ∧ … ∧ d log X_{d}. However the direct image in coherent cohomology is defined by the exact sequence of ω’s.

By taking cup-products with classes of complementary degree it follows that the Gysin-map

i_{∗} : ℝΓ_{(!)}(Y, 𝕃 ⊗ Ō_{Y})(−1)[−2] → ℝΓ_{!}(X, 𝕃 ⊗ Ō_{X})

is the connecting map from the exact sequence (triangle)

0 → 𝕃 ⊗ Ō_{X} → ℝj_{∗}(𝕃 ⊗ ŌX̃) → i_{∗}(𝕃 ⊗ Ō_{Y})(−1)[−1] → 0

Also by direct computation at stalks there is an exact triangle

0 → 𝕃 → ℝj_{∗}(𝕃) → i_{∗}(𝕃)(−1)[−1] → 0

mapping to the previous one (onto the almost invariants of Frobenius). It then induces an ”adjoint”

ℝΓ_{(!)}(Y, 𝕃)(−1)[−2] → ℝΓ_{(!)}(X, 𝕃)

for the trace-maps induced from cohomology with values in Ō. If we know that these trace-maps take values in ℤ/p^{s}ℤ we could use them to define a Poincaréduality and i_{∗} would be the adjoint to i^{∗}. In any case u_{X}^{∗} maps our exact sequence to its pendant in (X ⊗_{V} K̄)^{et}, which defines the direct image in étale cohomology. Thus one checks that our triangle defines an adjoint i_{∗} for the inner product defined by tr^{étale}.

The upshot of this discussion:

There exists an adjoint i_{∗} for the tr^{étale} -theory inducing the adjoint in Ō - cohomology.

Next let us compute in some examples that the comparison-maps H^{i}(X ⊗_{V} K̄, ℤ/p^{s}ℤ) → H^{i}((X ⊗_{V} K̄)^{et}, ℤ/p^{s}ℤ) tend to be isomorphisms:

Example 1. Let X = ℙ^{n}, X^{0} = 𝔾_{m}^{d} = ℙ^{d} \ ∪_{i=0}^{d}H_{i}, the standard-hyperplanes {T_{i} = 0}. Then X can be covered by standard affines 𝔸^{n} with fundamental group π_{1}(𝔾_{m}^{d} ⊗_{V} K̄) = ℤ̂ corresponding to n-power coverings ℙ^{d} → ℙ^{d}. Hence

H^{∗}(X^{0} ⊗_{V} K̄, Ō/p^{s}Ō = ℝΓ(ℤ̂(+1)^{d}, ℝΓ(ℙ^{d}, O/p^{s}O) ⊗_{V} V̄)

= (exterior algebra in ℤ_{p}(−1)^{d}) ⊗ V̄ /p^{s}V̄.

Taking Frobenius-invariants

H^{∗}(X^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ) = exterior algebra in ℤ/p^{s}ℤ(−1)^{d}
= H^{∗}((X^{0} ⊗_{V} K̄)^{et}, ℤ/p^{s}ℤ)

By using the fundamental triangle for the embeddings H_{i} ↪ ℙ^{d} ↩ ℙ^{d} − H_{i} one obtains by induction that

H^{∗}(X ⊗_{V} K̄, ℤ/p^{s}ℤ) = H^{∗}((X ⊗_{V} K̄)^{et}, ℤ/p^{s}ℤ) = ⨁_{i=0}^{d} ℤ/p^{s}ℤ · ξ^{i}, ξ = c_{1}(O(1)).

Example 1*. Let E denote a vectorbundle on X of rank r + 1, Y = ℙ_{X}(E) ⟶^{pr} X. Then H^{∗}(Y^{0} ⊗_{V} K̄, pr^{∗}(𝕃) ⊗ Ō_{Y}) = ⨁_{i=0}^{r} H^{∗−2i}(X^{0} ⊗_{V} K̄, 𝕃 ⊗ Ō_{X}) · ξ^{i} with ξ = c_{1}(O(1)), pr^{∗}(E) → O(1) the universal quotient. Also H^{∗}(Y^{0} ⊗_{V} K̄, pr^{∗}𝕃) = ⨁_{i=0}^{r} H^{∗−2i}(X^{0} ⊗_{V} K̄, 𝕃) ∘ ξ^{i}.

Proof. Both assertions are equivalent. We prove the corresponding equality for the derived image ℝpr_{∗} ∘ ℝq_{X,∗}, that is étale locally in X^{et}. But over some Spec(O_{X,x}^{sh}) Y is a product X × ℙ^{r}, and the result follows from Künneth.

In example 1 the tr : H^{2d}(ℙ^{d}, Ō/p^{s}Ō(d)) → V̄ /p^{s}V̄ sends c_{1}(O(1))^{d} to 1: This follows by induction from (i = inclusion H_{0} ↪ ℙ^{d})

i_{∗}c_{1}(O(1))^{d−1} = c_{1}(O(1))^{d}.

The formula holds in étale cohomology, and we already know that both versions of i_{∗} are compatible.

By checking fibrewise we derive that in example 1* pr_{∗}(ξ^{r}) = 1. Also one defines the Chern-classes c_{i}(E) ∈ H^{2i}(X ⊗_{V} K̄, ℤ/p^{s}ℤ(i)) by the usual formula ∑_{i=0}^{r+1}(−1)^{i}c_{i}(E) · ξ^{r−i} = 0. They map to the corresponding Chern-classes in étale cohomology.

Example 2. Let X̃ → ℙ^{d} = X denote the blow-up in the point Y = (1 : 0 …). Then X̃ is a ℙ^{1} -bundle over ℙ^{d−1}, with exceptional divisor E ≅ ℙ^{d−1}. Using example 1^{∗} one checks that for each i the sequence 0 → H_{(!)}^{i}(X^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ) → H_{(!)}^{i}(X̃^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ) ⊕ H_{()}^{i}(Y^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ)

→ H_{(!)}^{i}(E^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ) → 0

is exact.

Example 2*. Let Y ⊆ X denote a closed subvariety such that locally in the étale topology (X, Y) ≅ X_{1} × (𝔸^{t}, 0), with X_{1} toroidal etc. Let X̃ denote the blow-up of X in Y, E ⊆ X̃ the exceptional divisor, E ≅ ℙ_{Y} (J_{Y} /J_{Y}^{2} ). Then for each i the sequence

0 → H_{(!)}^{i}(X^{0}⊗_{V} K̄, 𝕃) → H_{(!)}^{i}(X̃^{0}⊗_{V} K̄, 𝕃)⊕H_{(!)}^{i}(Y^{0}⊗_{V} K̄, 𝕃) → H_{(!)}^{i}(E^{0}⊗_{V} K̄, 𝕃) → 0

is exact.

Proof. As in example 1* one checks this by a localisation in X^{et}. This would give a priori a long exact sequence in cohomology, but for example the pullback p_{X}^{∗} : H_{(!)}^{i}(X^{0} ⊗_{V} K̄, 𝕃) → H_{(!)}^{i}(X̃^{0} ⊗_{V} K̄, 𝕃) is injective, with left-inverse p_{X,∗}.

Thus consider a product X_{1} ×𝔸^{t} = X and the blow-up X̃ in X_{1} ×{0}. 𝔸^{t} as well as its blow-up in zero obtain the structure of a torus-embedding from 𝔾_{m}^{t} ⊆ 𝔸^{t}. On 𝔸^{t} the codimension one strata are the coordinate hyperplanes, while on the blow-up we have their proper transforms as well as the exceptional divisor (≅ ℙ^{t−1}).

Now the blow-up induces an almost isomorphism on the cohomology of the stratum X_{1} × 𝔾_{m}^{t}, using that we retain the same fundamental group and the theory of toroidal embeddings. Also the other strata can be treated by induction.

Remark. For constant coefficients we also could have used local Künneth. The above procedure also works if 𝕃 is not induced from the factor X_{1} in X_{1} × 𝔸^{t}.

Finally we need the self-intersection formula:

2. Lemma. For the inclusion i : Y ↪ X of a divisor i^{∗}i_{∗}(z) = −z ∪ c_{1}(J_{Y} /J_{Y}^{2} ) = z ∪ c_{1}(N_{Y}).

Proof. This amounts to computing the extension class of the pullback by i^{∗} of the exact sequence

0 → ℤ/p^{s}ℤ → ℝj_{∗}ℤ/p^{s}ℤ → i_{∗}ℤ/p^{s}ℤ(−1)[−1] → 0

Cover X by affines U_{ν} on which Y has a local equation f_{ν}. Then the p^{s} − th roots of f_{ν} form a μ_{p^{s}} -torsor on U_{ν} ∩ (X \ Y), hence give a class in Γ(U_{ν} ⊗_{V} K̄, ℝ^{1}j_{∗}μ_{p^{s}}) which maps to 1 ∈ i_{∗}(ℤ/p^{s}ℤ). Thus the restriction to U_{ν}∩Y trivializes the extension there. The trivialisation is not global. Namely one intersections U_{ν} ∩ U_{μ} we get a difference given by the μ_{p^{s}}-torsor of roots of u_{νμ} = (f_{ν}/f_{μ}) | Y. However this defines c_{1}(J_{Y} /J_{Y}^{2} ), and the negative sign arises from sign conventions.

Finally we treat immersions of higher codimensions. Let Y ⊆ X be a closed immersion such that locally in the étale topology (X, Y) ≅ X_{1} × (𝔸^{t}, 0). Let X̃ = blow-up of X in Y, E = exceptional divisor = ℙ_{Y} (J_{Y} /J_{Y}^{2} ).

E ↪^{ĩ} X̃
↓^{P_{Y}}  ↓^{P_{X}}
Y ↪^{i} X

On E we have the vectorbundle F = pr_{Y}^{∗} (J_{Y} /J_{Y}^{2} )^{∗}/O(−1). F has rank t − 1, and from the usual formulas one computes that c_{t−1}(F) = ∑_{i=0}^{t−1} c_{r−1}(pr_{Y}^{∗} (J_{Y} /J_{Y}^{2} )^{∗}) ∪ ξ^{i}. Especially p_{Y,∗}(c_{t−1}(F)) = 1 (in all cohomology-theories).

3. Proposition. p_{X}^{∗} ∘ i_{∗}(z) = ĩ_{∗}(c_{t−1}(F) ∪ p_{Y}^{∗} (z)) for z ∈ H^{∗}(Y ⊗_{V} K̄, 𝕃).

Proof. The right-hand side lies in the image of pr_{X}^{∗}: To check this it suffices to check the restriction to E. By the self-intersection formula this is c_{1}(O(−1)) ∪ c_{t−1}(F) ∪ p_{Y}^{∗} (z) = pr_{Y}^{∗} (c_{t}(J_{Y} /J_{Y}^{2} ) ∪ z) and indeed induced from Y. It now suffices to apply p_{X,∗} :

p_{X,∗}(ĩ_{∗}(c_{t−1}(F) ∪ p_{Y}^{∗} (z))
= i_{∗} ∘ p_{Y,∗}(c_{t−1}(F) ∪ p_{Y}^{∗} (z))
= i_{∗}(z).

Remark. The proof actually shows the formula in the derived category.

4. Corollary. The transformation

ℝΓ(X^{0} ⊗_{V} K̄, 𝕃) → ℝΓ((X^{0} ⊗_{V} K̄)^{et}, 𝕃)

commutes with i_{∗}.

Proof. It commutes with everything else in the formula.

5. Corollary. The two trace-maps on H_{!}^{2d}(X^{0} ⊗_{V} K̄, ℤ/p^{s}ℤ(d)) coincide.

Proof. We may assume that the generic fibre X ⊗_{V} K̄ is irreducible (extend K if necessary). As the special fibre X ⊗_{V} k is generically reduced, there exists (after perhaps again extending K) a section i : Y = Spec(V) ↪ X with image in the smooth locus. Then both traces are characterised by the fact that they take the value 1 on i_{∗}(1).

We intend to apply this theory to the diagonal embedding

δ : X ↪ X ×_{V} X.

As this does not satisfy the conditions on the nose, we shall modify it to

δ̃: X̃ → (X ×_{V} X)^{∼}.

This modification is done using toroidal geometry. We assume given a finite number of convex polyhedral cones {σ_{ν}}, σ_{ν} ⊆ L_{ℝ}, L = ℤ^{d+1}, and also an indivisible λ ∈ L^{∨}. We assume that each face of a σ_{ν} also appears among them, and that σ_{ν} ∩ ker(λ) is a simplex spanned by a partial basis of L.

Furthermore we assume given an étale covering of X by X(σ_{ν})’s, and étale maps X(σ_{ν}) → T̄_{λ,σ_{ν}} (the torus-embedding defined by σ_{ν}). Also for τ a face of σ, X(τ) should be equal to the preimage of the open T̄_{λ,{τ}} ⊆ T̄_{λ,{σ_{ν}}}, with the induced map.

Each X(σ) is stratified, with strata indexed by faces τ of σ. If x_{1} ∈ X(σ_{1}) and x_{2} ∈ X(σ_{2}) lie over the same point x ∈ X, and in the closed σ_{1} -stratum respectively σ_{2} -stratum, then there exists an automorphism (⟨σ_{i}⟩ ⊆ L, the sublattice spanned by σ_{i}) (⟨σ_{1}⟩, σ_{1}, λ) ≅ (⟨σ_{2}⟩, σ_{2}, λ), such that the two induced isomorphisms between O_{X,x}^{sh} = O_{X(σ_{1}),x_{1}}^{sh} = O_{X(σ_{2}),x_{2}}^{sh} and the strict henselisations of T̄_{λ,{σ_{1}}} respectively T̄_{λ,{σ_{2}}} differ by the induced automorphism and the action of an element of T.

We construct for each σ_{ν} a finite collection M(σ_{ν}) of T -invariant ideals I ⊆ V [L^{∨} ∩ σ_{ν}^{∨}], as follows: First, for any integer n, consider the set of elements μ ∈ L^{∨} ∩ σ_{ν}^{∨} such that

i) μ(ρ) ≤ n for each generator ρ of an extremal ray of σ_{ν} ii) If λ(ρ) = 0 for such a ρ, then μ(ρ) = 0 except for at most one such ρ, for

which μ(ρ) = 1. Choose n so big that the elements generate the dual of the subgroup ⟨σ_{ν}⟩ ⊆ L generated by σ_{ν}, for all ν. Then define M_{0}(σ_{ν}) as the (finite) set of principal ideals V [L^{∨} ∩ σ_{ν}^{∨}] · μ generated by such μ. If τ is a face of σ then each ideal in M_{0}(σ) generates an ideal in M_{0}(τ) in V [L^{∨} ∩ τ^{∨}] ⊇ V [L^{∨} ∩ σ^{∨}]. Also the set of ideals M_{0}(σ) is stable under isomorphisms (⟨σ_{1}⟩, σ_{1}, λ) ≅ (⟨σ_{2}⟩, σ_{2}, λ). We define M(σ) as the set of all ideals of the form

ker : V [L^{∨} ∩ σ^{∨}] → V [L^{∨} ∩ τ^{∨}]/I,

with I ∈ M_{0}(τ), τ a face of σ. Then:

i) M(σ) is stable under isomorphisms of (⟨σ⟩, σ, λ).

ii) For τ a face of σ, the map I ↦ I · V [L^{∨} ∩ τ^{∨}] induces a surjection M(σ) ↠

M(τ). iii) If σ ⊆ ker(λ), then M(σ) consists of the unit ideals and the ideals defining

the irreducible components of the boundary-divisor iv) M(σ) contains principal ideals generated by μ, for μ a system of generators

of ⟨σ⟩^{∨}.

Now define new torus-embeddings T ⊆ T^{≃}, T^{≃}_{λ} by normalised blow-up of all ideals I_{1} + I_{2}, for I_{1}, I_{2} ∈ M(σ). By ii) the commutes with open immersions defined by faces τ ⊆ σ, and by i) the induced modifications of X(σ) descend to a proper f : X̃ → X inducing X̃^{0} ⟶^{∼} X^{0}. By iii) on the generic fibre X̃ ⊗_{V} K → X ⊗_{V} K is common blow-up of all intersections of pairs of boundary-divisors. Similarly on X ×_{V} X we blow-up (and normalise) all ideals pr_{1}^{∗}(I_{1}) + pr_{1}^{∗}(I_{2}), pr_{1}^{∗}(I_{1}) + pr_{2}^{∗}(I_{2}), pr_{2}^{∗}(I_{1}) + pr_{2}^{∗}(I_{2}) with I_{1} ∈ M(σ_{1}), I_{2} ∈ M(σ_{2}) (over X(σ_{1}) × X(σ_{2})). Then the diagonal extends to Δ̃ : X̃ → X{ \atop ×_{V} X}. Note that normalised blow-up has the following toroidal description: Each I ∈ M(σ) defines a convex piecewise linear function ψ_{I} on σ by ψ_{I}(ρ) = Min{μ(ρ) | μ ∈ I ∩ σ^{∨}}. Then the normalised blow-up amounts to subdividing σ (in a minimal way) such that ψ_{I} becomes linear on each stratum.

Especially on X ×_{V} X (defined locally by L^{(2)} = ker(λ, −λ) ⊆ L × L, σ^{(2)} = σ × σ ∩ L_{ℝ}^{(2)}), the functions Min(μ(x), μ(y)) are linear on the strata of the subdivision, for μ in a system of generators of ⟨σ⟩^{∨} (condition iv)). It follows that on each open stratum meeting the diagonal we have μ(x) = μ(y), and thus this stratum is totally contained in the diagonal. Also on the diagonal we obtain the subdivision defining X̃. Thus the diagonal embedding T̄ ↪ T̄ ×_{V} T̄ extends to an open immersion (T × 1) × T^{≃} ⊆ (T̄ ×_{V} T̄)^{∼}. Hence locally in the étale cohomology (X{ \atop ×_{V} X}, X̃) ≅ X_{1} × (𝔸^{d}, 0).

Next consider support-conditions at infinity for a toroidally defined X̃ → X. If J ⊆ O_{X} partially defines the boundary, there are two canonical choices for an ideal J̃ ⊆ OX̃, J̃ ⊇ f^{∗}(J). Either J̃ defines the total preimage of V (J) (the minimal choice), or the union of boundary divisors D̃ not projecting onto irreducible components outside V (J).

6. Lemma.

i) X̃ ⊗_{V} K has normal crossings singularities

ii) For either choice of J̃ ℝf_{∗}J̃ = J.

Proof.

i) On the generic fibre we have to treat the torus-embedding defined by σ ∩ ker(λ), which up to a torus-factor is 𝔾_{m}^{d} ⊆ 𝔸^{d}. Then X̃ is the union of d! embeddings 𝔾_{m}^{d} ⊆ 𝔸^{d}. One of them maps to X via (X_{1}, X_{1} · X_{2}, …, X_{1} · . . . · X_{d}), the others by permuting variables.

ii) If ρ_{1}, …, ρ_{r} are the extremal rays for σ, those for the cones of σ̃ are sums ρ_{s} = ∑_{i∈S} ρ_{i} for non-empty subsets S ⊆ {1, …, d}. Those lie in a common simplex if and only if the S’s form a chain. Assume J is defined by {1, …, a}. Then the bigger transform J̃ corresponds to S ⊆ {1, …, a}, while the total transform to S ∩ {1, … a} ≠ ϕ.

By [KKMS], §, 3, Th. 12 for any μ ∈ L^{∨} the μ -component in H^{i}(T^{≃}; J̃) is H_{A}^{i}(σ, k) for a certain closed subset A ⊆| τ |, namely the union of all faces τ with μ ∈ Γ (τ -open, J̃). This is also equal to the relative cohomology H^{i}(σ, σ − A; k). First choose J̃ minimal. Now μ ∉ Γ(τ -open, J̃) if either not μ ∈ τ^{∨}, or μ ∈ τ^{∨} and μ(ρ_{s}) = 0

for some ρ_{s} belonging to J̃ with ρ_{s} ∈ τ, i.e. S ∩ {1, …, a} ≠ ϕ. Choose a δμ ∈ L^{∨} with δμ(ρ_{i}) < 0 for i = 1, …, a, δ_{μ}(ρ_{j}) = 0 for j = a + 1, …, r, δμ(ρ) ≥ 0 for any extremal ρ of a face τ with λ(ρ) > 0. Then for N >> 0 μ ∈ Γ(τ -open, J̃) ⇔ Nμ + δμ ∈ τ^{∨}, thus the μ -component becomes the (Nμ + δμ) component in

H^{i}(T^{≃}, O) which is known to be 0 unless i = 0 and Nμ+δμ ∈ σ^{∨}, which means μ ∈ J. For the maximal J choose δμ as before except that δμ(ρ_{j}) > −a · min{∂μ(ρ_{i}) | 1 ≤

i ≤ 0} for a+1 ≤ j ≤ r. This proves the assertion ii) for T^{≃} → T̄, and by basechange also for X̃ → X.

As an application let as before R_{∞} denote the normalisation of R ⊗_{V} V̄ in the extension defined by adjoining all p-power roots of μ ∈ L^{∨}, X̃_{∞} → X̃ the corresponding normalised pullback to f : X̃ → X = Spec(R), with Galois-group Δ_{∞} = ℤ_{p}(1)^{d}. If M_{∞} denotes an R_{∞} -module with continuous semilinear Δ_{∞} -action, and J_{∞} ⊂ R_{∞} a support-ideal extending (in two possible ways) to J̃_{∞} ⊂ OX̃_{∞}, then ℝf_{∞,∗}(M_{∞} ⊗^{𝕃} J̃_{∞}) = M_{∞} ⊗^{𝕃} J_{∞} (by basechange), and this equality persists after applying ℝΓ(Δ_{∞}, ). However by the almost-étale theory these are cohomologies of 𝕃 ⊗ Ō, if M = (R̄ ⊗ 𝕃)^{Gal(R̄/R_{∞})} for a locally constant p-torsion sheaf 𝕃 on Spec(R)^{0} ⊗_{V} K̄. That is we know that for the map q_{X} : X^{0} ⊗_{V} K̄ → X^{et} Rq_{X,∗}(𝕃 ⊗ J̄) is almost isomorphic to a complex of quasicoherent sheaves, and the above assertion means that the map

ℝq_{X,∗}(𝕃 ⊗ J̄) → ℝf_{∗} ∘ ℝq_{X̃,∗}(𝕃 ⊗ J^{≃})

is an almost isomorphism. Thus:

7. Corollary. ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ J̄) ≈ ℝΓ(X̃^{0} ⊗_{V} K̄, 𝕃 ⊗ J^{≃})

ℝΓ_{(!)}(X^{0} ⊗_{V} K̄, 𝕃) = ℝΓ_{(!)}(X̃^{0} ⊗_{V} K̄, 𝕃)

In the second equation (!) always denotes the relevant support-condition at infinity. The second assertion follows from the first by descente. Also the corresponding étale analogue holds:

8. Lemma. ℝΓ_{(!)}((X^{0} ⊗_{V} K̄)^{et}, 𝕃) = ℝΓ_{(!)}((X̃^{0} ⊗_{V} K̄)^{et}, 𝕃)

Proof. Let X − X^{0} = A ∪ B denote the decomposition of the boundary in union of irreducible components, such that we have compact support along B. Similarly X̃ − X̃^{0} = Ã ∪ B̃, with either Ã = f^{−1}(A) or B̃ = f^{−1}(B). As Poincaré-duality exchanges A and B (and Ã, B̃) we may assume that Ã = f^{−1}(A). Now with j_{A} : X − A ↪ X denoting the inclusion, we have ℝf_{∗} ∘ ℝjÃ = ℝj_{A} ∘ ℝf_{∗}. Thus it suffices if

ℝf_{∗} ∘ ℝj_{B̃,!}(𝕃) = ℝj_{B,!} ∘ ℝf_{∗}(𝕃) on X − A.

This assertion is local in X − A. We thus reduce to X^{0} = 𝔾_{m}^{d} ⊂ X = 𝔸^{d}, A = ϕ, B = X − X^{0} and thus B̃ = X̃ − X̃^{0}. As f is proper ℝf_{∗} commutes with duality which exchanges j_{B,!} and ℝj_{B,∗}, and similar for B̃. This reduces us again to A = X − X^{0}, B = ϕ, and then to A = B = ϕ.

Now the main application: The diagonal-embedding

δ : X ↪ X × X

induces

δ̃: X̃ ↪ { \atop (X × X)}.

This embedding is such that locally in the étale topology ((X{ \atop ×_{V} X}), X̃) ≅ X_{1} × (𝔸^{d}, 0). Now choose a decomposition of the boundary

X − X^{0} = A ∪ B,

and extend to X̃ − X̃^{0} = Ã ∪ B̃ with Ã = full preimage of A. Then define ℝΓ_{A,B}((X^{0} ⊗_{V} K̄)^{et}, 𝕃) as cohomology with compact support along B, and similarly for the other cohomology-theories or for X̃. On X ×_{V} X we consider the pair

(A^{(2)}, B^{(2)}) = (A ×_{V} X ∪ X ×_{V} B, B ×_{V} X ∪ X ×_{V} A),

and similarly Ã^{(2)} = full preimage of A^{(2)} in (X{ \atop ×_{V} X}), etc. Assume first that 𝕃 is annihilated by p. We then have a commutative diagram

ℝΓ((X̃^{0} ⊗_{V} K̄)^{et}, End(𝕃)) ⟶^{δ̃_{∗}} ℝΓ_{Ã^{(2)},B̃^{(2)}}(((X{ \atop ×_{V} X})^{0} ⊗_{V} K̄)^{et}, 𝕃 ⊗ 𝕃^{∨})(−d)[−2d]
↑  ↑
ℝΓ(X̃^{0} ⊗_{V} K̄, End(𝕃)) ⟶^{δ̃_{∗}} ℝΓ_{Ã^{(2)},B̃^{(2)}}((X { \atop ×_{V} X})^{0} ⊗_{V} K̄, 𝕃 ⊗ 𝕃^{∨})(−d)[−2d],

which can be identified with

ℝΓ((X^{0} ⊗_{V} K̄)^{et}, End(𝕃)) ⟶^{δ_{∗}} ℝΓ_{A^{(2)},B^{(2)}}((X ×_{V} X)^{0} ⊗_{V} K̄)^{et}, 𝕃 ⊗ 𝕃^{V})(−d)[−2d]
↑  ↑
ℝΓ(X^{0} ⊗_{V} K̄, End(𝕃)) ⟶^{δ_{∗}} ℝΓ_{A^{(2)},B^{(2)}}((X ×_{V} X_{0})^{0} ⊗_{V} K̄, 𝕃 ⊗ 𝕃^{V})(−d)[−2d]

Apply this to the canonical class 1 ∈ H^{0}(X^{0} ⊗_{V} K̄)^{et}, End(𝕃)) or its variants. Its image in

ℝΓ_{A^{(2)},B^{(2)}}(((X ×_{V} X)^{0} ⊗_{V} K̄)^{et}, 𝕃 ⊗ 𝕃^{∨})(−d)[−2d]
≅ ℝΓ_{A,B}((X ⊗_{V} K̄)^{et}, 𝕃) ⊗ ℝΓ_{B,A}((X ⊗_{V} K̄)^{et}, 𝕃^{∨})(−d)[−2d]

corresponds via Poincaré-duality to the identity, and similarly for ℝΓ(X^{0}⊗_{V} K̄, …). As the comparison-map ℝΓ_{A,B}(X^{0}⊗_{V} K̄, 𝕃) → ℝΓ_{A,B}((X^{0}⊗_{V} K̄)^{et}, 𝕃) respects the trace and cup-product it induces an isometry on cohomology, and thus an injection. That it also preserves δ_{∗}(1) means that its adjoint also preserves products, thus is also an injection. Hence the comparison-map is a quasi-isomorphism. By devissage this also holds for general p-torsion 𝕃’s.

9. Theorem. H_{A,B}^{i}(X^{0}⊗_{V} , 𝕃) = H_{A,B}^{i}((X^{0} ⊗_{V} K̄)^{et}, 𝕃).

Remark. We have given a ”global” proof. In many cases one can also proceed ”locally”: Namely if for each x ∈ X all strata in Spec(O_{X,x}^{sh} ⊗_{V} K̄) are K(π, 1)’s in the étale topology, then the stalks of ℝq_{X,∗}𝕃 coincide for q_{X} : X^{0} ⊗_{V} K̄ → X^{et} respectively (X^{0} ⊗_{V} K̄)^{et} → X^{et}.

5. Crystalline Cohomology

Let X → Spec(V) etc. be as before. Choose a uniformiser π of V. Let V_{0} = W (k) ⊆ V denote the maximal unramified subring of Witt-vectors over k. Then

V = V_{0}[π] = V_{0}[t]/(f(t)),

where f(t) = t^{e} + a_{e−1} · t^{e−1} + … + a_{0} is an Eisenstein polynomial: All a_{i} are divisible by p, and a_{0}/p is a unit in V_{0}. D_{(f(t))}(V_{0}[t]) denotes the divided powerhull of (f(t)), obtained by adjoining divided powers f(t)^{n}/n! or t^{en}/n!, and R_{V} its p-adic completion. R_{V} ⊂ K_{0}[[t]] consists of powerseries ∑_{n=0}^{∞} a_{n} · t^{n} with

[\frac{n}{e}]! · a_{n} ∈ V_{0} (all n), lim_{n→∞}([\frac{n}{e}]! · a_{n}) → 0

It has the PD-filtration F^{n}(R_{V}) = ideal generated by divided powers \frac{f^{m}}{m!} , m ≥ n. R_{V} /F^{1}(R_{V}) = V. We want to study the logarithmic crystalline topics of X/R_{V}. For lack of suitable reference we first tensor with a fixed ℤ/p^{s}ℤ. So let X_{s} = X ⊗_{V} V/p^{s}V. Then the logarithmic crystalline site ([B], [K]) (X_{s}/R_{V,s})_{cris} consists of PD-embeddings

U ↪ V

where U, V are schemes/ R_{V,s}, U → X_{s} is étale and V has a ”logarithmic structure” which should mean the following:

The covering by X(σ) → X induces an étale covering by U(σ) → U, with U(σ) → T̄_{λ,σ} ⊆ T̄. T̄ is a scheme over V_{0}[t] with t corresponding to λ. As the ideal of U in V is nilpotent U(σ) extends canonically to V (σ) → V étale, with U(σ) ↪ V (σ). Then the ”logarithmic structure” on V is an étale local extension of U(σ) → T̄ to V (σ) → T̄ (as schemes over V_{0}[t]), unique up to homomorphisms V (σ) → T_{λ} = ker(λ : T → 𝔾_{m}). That is the extensions exist on étale cover on V′(σ) ↠ V (σ) and the two pullbacks to V′′(σ) = V′(σ) ×_{V (σ)} V′(σ) differ by a map to V′′(σ) → T_{λ} satisfying the cocycle condition. It then defines a class in H^{1}(V (σ), T_{λ}) = (P ic V (σ))^{d}. Hence we only need a Zariski-covering of V (σ).

Maps f : (U_{1} ↪ V_{1}) → (U_{2} → V_{2}) are pairs of maps f_{U} : U_{1} → U_{2}, f_{V} : V_{1} → V_{2} with the usual commutation rule, and ρ : V_{1}(σ) → T_{λ} such that

V_{1}(σ) ⟶ V_{2}(σ)
↘  ↓
T̄

commutes up to the action of ρ. (Composition is obvious) Coverings in the site are induced by étale covers of U. A crystal (of quasicoherent sheaves) on (X_{s}/R_{V,s})_{crys} is a sheaf E such that for U ↪ V, E_{V} is a quasicoherent sheaf on V. Also for any map (U_{1} ↪ V_{1}) → (U_{2} ↪ V_{2}), E_{V_{1}} should be isomorphic to the pullback of E_{V_{1}}, via the pullback map. A filtered crystal is an E as above together with a decreasing sequence of subsheaves F^{n}(E) ⊆ E (indexed by n ∈ ℤ) such that on each U ↪ V, the F^{n}(E)_{V} are quasicoherent subsheaves with F^{[a]}(O_{V}) · F^{b}(E)_{V} ⊆ F^{a+b}(E)_{V} (F^{[a]}(O_{V}) : P D -filtration on the ideal of U ↪ V), and for f : (U_{1}, V_{1}) → (U_{2}, V_{2}) F^{n}(E)_{V_{1}} ⊆ E_{V_{1}} is the subsheaf generated ∑_{a+b≥n} F^{[a]}(O_{V_{1}})·f_{V}^{∗} (F^{b}(E)_{V_{2}}). Obviously any (unfiltered) E can be filtered by the F^{[n]}(J_{U/V})·E_{V}. An example if a filtered crystal is O_{X}^{crys}{a}, with underlying sheaf E_{V} = O_{V} and F^{n}(E)_{V} = F^{a+m}(O_{V}). A filtered crystal of vectorbundles (E, F^{·}) is called locally filtered free if for any U ↪ V (E_{V} , F^{·}(E_{V})) is locally isomorphic to a direct sum of O_{crys}{a}’s (no compatability with pullbacks required). Now choose an étale covering of X by X(σ)’s, with étale maps X(σ) → T̄_{λ}. These lift to étale maps of formal schemes (with adic topology defined by π−λ)

{ \atop X(σ)} → T̄^{∧} = formal completion of T̄ along T̄_{λ}. Thus we can find an étale covering

Y ↠ X and a lift to an étale scheme Ŷ over T̄^{∧}, namely for example Y = ⨿X(σ). Fix such a pair (Û, U). We need the logarithmic products Ŷ^{n+1,log}. These will be formal schemes with underlying topological space Y ×_{X}Y ×…×_{X}Y (n+1) -times), defined by étale glueing from the following: For T̄ the (n+1) -fold log -product is the formal completion of T̄×_{V_{0}} T_{λ}^{n} along (T̄×{1, …, 1}). Its projections to T̄ are given by pr_{0}(z, g_{1}, …, g_{n}) = z, pr_{i}(z, g_{1}, …, g_{n}) = g_{i}·z. Now locally in the étale topology

Y = X = T̄_{λ}, Ŷ = T̄^{∧}, and we use pullback. It follows that via any of the projections pr_{i} Ŷ^{n+1,log} is isomorphic to a product of formal schemes Ŷ^{n+1,log} ≅ Ŷ × T̂_{λ}^{n}(T̂_{λ} : formal completion along origine). This log -product corresponds to the ”exactified” product in the Fontaine-Illusie-Kato theory. The Ŷ^{n+1,log} form a simplicial formal scheme. Next if we reduce modulo some fixed p^{s} we can form the PD-hulls of the embeddings Y ×_{X} Y … ×_{X} Y ↪ Ŷ^{n+1,log} and obtain schemes (not just formal) schemes over R_{V} /p^{s}R_{V}, or (again) formal simplicial schemes over R_{V}, with the p-adic topology:

Y_{X}^{·} ↪ D(Ŷ^{·,log}).

Any crystal E mod p^{s} can be evaluated on D(Ŷ^{·;log}) ⊗_{R_{V}} R_{V} /p^{s}R_{V}, so gives E_{n} on each D(Ŷ^{·;log})_{n} ⊗_{R_{V}} R_{V} /p^{s}R_{V} with pullback-maps which are all isomorphisms. As usual this is equivalent to giving E_{0} on D(Û^{·;log})_{0} together with an isomorphism of the two pullbacks to D(Ŷ^{·;log})_{1}, satisfying a cocycle condition, or to give E_{0} with an integrable log-connection ▽ : E_{0} → E_{0} ⊗Ω_{Ŷ ,log}^{1}, where Ω_{Ŷ ,log}^{1} are the logarithmic differential forms, dual to the Lie-algebra of T_{λ}. The relation is given by Taylor’s formula. With this connection we define the de Rham complex

DR(E) = (E ⊗ Ω_{Ŷ ,log}^{·}, ▽ + d)

on D(Ŷ^{·,log})_{·}. On D(Ŷ^{·,log})_{n} its underlying sheaves are pr_{0}^{∗}(E)⊗Λ⊕_{i=0}^{n}pr_{i}^{∗}((Ω^{1}1_{Ŷ ,log}), with

(▽ + d)(pr_{0}^{∗}(e) ⊗ α) = pr_{0}^{∗}(▽e) · α + e ⊗ dα.

For any PD-thickening U ↪ V of U → X étale we find étale locally a map (U, V) ⟶^{f} (Y, Ŷ). Thus E_{V} is determined by pullback. In fact one can form the log-product (V × Ŷ_{·})^{log}, a formal simplicial scheme with same underlying topological space as the graph of f_{V}. Etale locally it is isomorphic to V × T̄_{λ}. As usual modulo p^{s} the divided power hull of this is a scheme, and we obtain a diagram

(U, V) ⟵^{pr_{1}} D((V × Ŷ^{·})^{log}) ⟶^{pr_{2}} D(Ŷ^{·})

Then it is known that

ℝpr_{1,∗}pr_{2}^{∗}(DR(E))

globalises (using stratifications) and forms an acyclic resolution of E. Thus ℝu_{X,∗}(E) on X^{et} pulls back on (U_{X}^{·})_{·} to DR(E). By cohomological descente ℝu_{X,∗}(E) is represented by ℝpr_{∗}(DR(E)_{·}), pr : U_{X} → X the projection. The same holds in the filtered context, that is for E filtered the de Rham complexes give filtered resolutions. In local coordinates D((Y × Ŷ_{0})^{log})_{n} corresponds to a free PD-algebra O_{V} {t_{ν}} in (n + 1) · d variables t_{ν} (= local coordinates near 1 in T_{λ}^{n+1}), and the de Rham complex is the tensorproduct of E_{V} and the de Rham complex in the {t_{ν}}, with its product filtration.

Basically we are interested in the cohomology of crystals of vectorbundles. However we also need boundary conditions. For this we assume (for simplicity) that the special fibre X ⊗_{V} k is reduced, or equivalently that X ⊗_{V} V̄ is normal. We can always achieve this by adjoining a root π^{1/e}. Then locally the extremal rays of σ are generated by ρ_{i} with λ(ρ_{i}) ∈ {0, 1}. Any subset I of them defines a reduced divisor with support in the boundary, and thus an ideal J_{I} ⊆ O_{X}. For I = {ρ | λ(ρ) = 1} we have J_{I} = π · O_{X}. Now restrict to subsets I with λ(ρ) = 0 for ρ ∈ I, and let I^{c} denote the complement of I in the set of all such ρ’s, which defines the ideal J ⊆ O_{X}. Then J · Ω_{X/Y}^{d,log} is the relative dualising complex, J_{I_{c}} = Hom_{O_{X}} (J_{I}, J). Globally we assume given an ideal J_{I} which locally is of the type above. For example this is always possible for O_{X} or J. Also if J_{I} is defined so is J_{I^{c}}.

The ideals J_{I} extend naturally to T̄ and thus to any object (U, V) of the log-crystalline topos, and form crystals. We are interested in the cohomologies ℝΓ(X^{crys}/R_{V} , E ⊗ J_{I}), E a crystal of vectorbundles on X^{crys}. As it stands this is defined for crystals modulo a fixed power p^{s}, and then can be represented by the hypercohomology of E ⊗ J_{I} on D(Ŷ^{·,log}). If we assume that Y is affine (which we may do) then so is each Y_{n} = Spec(S_{n}), D(Ŷ_{n}) = Spec(D(S_{n})), and the de Rham complexes (E ⊗ J_{I})(D(Ŷ_{n}) ⊗ Ω_{S_{n}}^{·,log} form (modulo p^{s}) a double complex whose associated total complex represents the crystalline cohomology. It is in fact easily seen (comparing two coverings) that this is (up to canonical isomorphism in the filtered derived category) independent of choices. Also if we have a compatible system E of crystals of vectorbundles E_{s} modulo each p^{s}, we may form the projective limit (for s → ∞) of the corresponding complexes to get the cohomology of E (in the same spirit as earlier the cohomology of A_{inf}(Ō)). As the sheaves J_{I} are modulo each p^{s} flat over R_{V} /p^{s}R_{V}, the proof in [F7] applies to show that ℝΓ(X^{crys}/R_{V} , E ⊗ J_{I}) can be represented by a finite complex of filtered free R_{V} -modules if E is locally filtered free. It is also shown there (the proof extends) that Berthelot’s construction from [B] gives a trace-form

H^{2d}(X^{crys}/R_{V} , J) → R_{V} {−d},

which induces a perfect duality (of complexes of filtered free R_{V} -modules)

ℝΓ(X^{crys}/R_{V} , E^{∨} ⊗ J_{I^{c}}) ⟶^{≈} ℝHom_{R_{V}} (ℝΓ(X^{crys}/R_{V} , E ⊗ J_{I}), R_{V} {−d}[−2d])

Also there is a Künneth-formula. In short all the usual formalism is again available.

For a closed immersion of a divisor i : Y ↪ X transversal to all strata (i.e. étale locally (X, Y) = X_{1} × (𝔸^{1}, 0)) there is an exact sequence (triangle) with X̃ corresponding to the new toroidal structure with open stratum X^{0} − Y^{0} 0 → ℝΓ(X^{crys}/R_{V} , E⊗J_{I}) → ℝΓ(X̃^{crys}/R_{V} , E⊗J_{I}) → ℝΓ(Y^{crys}/R_{V} , E⊗J_{I}){−1}[−1] → 0

(see [F3]) and the connecting map represents i_{∗}:

It suffices that the trace on H^{2d−2}(Y^{crys}/R_{V} , J) is equal to the composite of the connecting map with the trace on H^{2d}(X^{crys}/R_{V} , J). But the trace is characterised by its values on local cohomology

H_{{y}}^{2d−2}(Y^{crys}/R_{V} , J) (y ∈ Y ⊗_{V} k a closed point).

This can be computed by locally lifting to

Y = X_{1} × {0} ↪ X = X_{1} × 𝔸^{1},

with X_{1} étale over some torus-embedding T̄_{1}. The exact sequence reduces to the tensor-product of J ∘ Ω_{X_{1}}^{·,log} with

0 → Ω_{𝔸^{1}}^{·} → Ω_{𝔸^{1}}^{·}(d log 0) ⟶^{Residue} O_{{0}}{−1}[−1] → 0,

and the assertion is wellknown.

Finally for a line-bundle L on X the first Chern-class c_{1}(L) ∈ H^{2}(X^{crys}/R_{V} , F^{1}(O)) is defined via the connecting map

P ic(X) = H^{1}(X^{crys}/R_{V} , (O/F^{1}(O))^{∗}) →

H^{2}(X^{crys}/R_{V} , (1 + F^{1}(O), ·) ⟶^{log} H^{2}(X^{crys}/R_{V} , F^{1}(O)).

As usual this defines higher Chern classes, the selfintersection formula holds, and also (for immersions Y ↪ X with locally (X, Y) ≅ X_{1} × (𝔸^{t}, 0)) the formula pr_{X}^{∗} ∘ i_{∗} = ĩ_{∗}(c_{t−1}(F) ∪ pr_{Y}^{∗} ( ))

E  ⟶ X̃
↓  ↓
Y  ↪  X

So far the review.

Next relations to étale cohomology. For a ring R with Spec(R) étale over T̄_{λ}

we have defined A_{inf}(R̄), with an isomorphism A_{inf}(R̄)/ξ · A_{inf}(R̄) ⟶^{∼} R̄^{∧} (p-adic completion). We define B_{cr}^{+}(R̄) as the p-adically completed divided power hull of ξ · A_{inf}(R̄). For varying R the B_{cr}^{+}(R̄)/p^{s} · B_{cr}^{+}(R̄) form an almost sheaf of rings on X^{0} ⊗_{V} K̄. Thus we may form the cohomology ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ B_{cr}^{+}(Ō)), which however turns out to be almost equal to ℝΓ((X^{0} ⊗_{V} K̄)^{et}, 𝕃) ⊗^{𝕃} B_{cr}^{+}(V̄), as

B_{cr}^{+}(R̄)/p^{s} · B_{cr}^{+}(R̄) ≅ A_{inf}(R̄)/p^{s} · A_{inf}(R̄) ⊗_{A_{inf} (V̄)} B_{cr}^{+}(V̄).

Similarly let B_{cr}^{+}(J̄_{I}) = ker(B_{cr}^{+}(R̄) → B_{cr}^{+}(R̄/J̄_{I})) = p-adic completion of A_{inf}(J̄_{I})⊗_{A_{inf} (V̄)} B_{cr}^{+}(V̄), and then

ℝΓ_{(!)}((X^{0} ⊗_{V} K̄)^{et}, 𝕃) ⊗^{𝕃} B_{cr}^{+}(V̄) ≈ ℝΓ(X^{0} ⊗_{V} K̄, 𝕃 ⊗ B_{cr}^{+}(J̄_{I})).

Also A_{inf}(R̄) and B_{cr}^{+}(R̄) get toroidal structures by choosing for each μ in a basis of L^{∨} a compatible system of p-power roots of \underline{μ}, defining \underline{μ} = [μ] ∈ A_{inf}(R̄). Next the preimage of R in B_{cr}^{+}(R) defines a PD-thickening. Thus if E is a crystal modulo p^{s} on X^{crys}/R_{V}, we can evaluate it on this preimage to get a module with an action of Gal(R̄/R ⊗_{V} V̄). This is functorial in R, thus defines a presheaf (in fact almost a sheaf for small enough Spec(R)) on X^{0} ⊗_{V} K̄.

Now assume given a locally constant sheaf 𝕃 (annihilated by p^{s}) on (X^{0} ⊗_{V} K̄)^{et}, and a functorial map

E (preimage of R) → 𝕃 ⊗ B_{cr}^{+}(R̄),

linear over the preimage of R in B_{cr}^{+}(R̄) and commuting with the Galois-action. Then any resolution of E on the crystalline topos (like the de Rham resolution) defines by evaluation a resolution of the presheaf E on X^{0} ⊗_{V} K̄, which maps to an injective resolution of 𝕃 ⊗ B_{cr}^{+}(R̄). Hence we obtain a transformation

v_{X} : ℝΓ(X^{crys}/R_{V} , E) → ℝΓ(X^{0}⊗_{V} K̄, 𝕃⊗B_{cr}^{+}(Ō)) ≈ ℝΓ((X^{0}⊗_{V} K̄)^{et}, 𝕃)⊗B_{cr}^{+}(V̄).

If E is filtered and the basic maps to 𝕃 ⊗ B_{cr}^{+}(R̄) preserve filtrations, we obtain a transformation in the filtered derived category. An example is E = O_{X} and 𝕃 = ℤ/p^{s}ℤ. Also there is the obvious variant

v_{X} : ℝΓ(X^{crys}/R_{V} , E⊗J_{I}) → ℝΓ(X^{0}⊗_{V} K̄, 𝕃⊗B_{cr}^{+}(J̄_{I})) ≈ ℝΓ_{(!)}((X^{0})⊗_{V} K̄)^{et}, 𝕃)⊗B_{cr}^{+}(V̄).

Next we check how much the map commutes with Chern-classes and direct images. Recall that there is a canonical map

β : ℤ_{p}(1) → F^{1}(B_{crys}^{+}(V)).

If t ∈ ℤ_{p}(1) is a generator, corresponding to a compatible sequence of primitive roots ζ_{n} ∈ μ_{p^{n}}(V̄), \underline{1} = \underleftarrow{lim} ζ_{n} defines [\underline{1}] ∈ A(\underline{V}). Then β(t) = log \underline{1}.

1. Theorem.

i) For a vectorbundle E on X, c_{i}^{cr}(E) ∈ F^{i}H^{2i}(X^{crys}/R_{V} , O_{X}) maps to

v_{X}(c_{i}^{cr}(E)) = β^{⊗i} · c_{i}^{et}(E) ∈ H^{2i}((X^{0} ⊗_{V} K̄)^{et}, ℤ/p^{s}ℤ(i)) ⊗ F^{i}B_{cr}^{+}(V̄).

ii) Assume i : Y ↪ X is a closed immersion, such that locally in the étale topology (X, Y) = X_{1} × (𝔸^{t}, O). Then v_{X} ∘ i_{∗}^{cr} = β^{⊗t} · i_{∗}^{et} ∘ v_{Y}, as transformations

ℝΓ(Y^{cr}, E ⊗ J) → ℝΓ(X^{cr}, 𝕃 ⊗ B_{cr}^{+}(J̄))

Proof. For i) it suffices to treat c_{1}(L) for line-bundles L. Both Chern-classes are boundaries of the class of L in H^{1}(X, O_{X}^{∗}), with respect to either

0 → F^{1}(O_{X}^{crys}) ⟶^{exp} O_{X}^{crys,∗} → O_{X}^{∗} → 0

respectively

0 → ℤ_{p}(1) → R(Ō)^{∗} → Ō^{∧}^{∗} → 0

However evaluated on B_{cr}^{+}(Ō) the second sequence maps to the first via β respectively the map

R(Ō)^{∗} → A_{inf}(Ō)^{∗} → B_{crys}^{+}(Ō)^{∗},

hence the assertion.

Similarly by the fundamental formula i) is reduced to the case of a divisor, that is t = 1. As before we denote by X̃ X with the modified toroidal structure where Y is added to the boundary. There exists an étale covering of X by affines Spec(R) such that either Spec(R) lies over X \ Y and has an étale toric parametrisation Spec(R) → T̄_{λ}, or Spec(R) → T̄_{1,λ} ×𝔸^{1} is an étale parametrisation for the two possible toric structures (depending on whether {0} ⊆ 𝔸^{1} is added to the boundary). In the second case we also assume that R has enough units to define a good R_{∞}. For example Spec(R) could lie above T̄_{1,λ} × (𝔸^{1} − {1}). Let X′_{0} = X̃′_{0} = ⨿Spec(R) denote the corresponding affine cover of X, X̃′_{n} → X′_{n} the (n + 1) -fold logarithmic fibre-products (in the two different log -structures),

X̃^{∧}′_{n} = Spf(R̃′_{n}), X̂′_{n} = Spf(R′_{n})

affine formal lifts locally étale over T̄^{∧}^{n+1,log} (respectively (T̄_{1} × 𝔸^{1})^{∧_{n+1,log}}, and derived power hulls Spf(D(R̃′_{n})), Spf(D(R′_{n})). The crystalline cohomology ℝΓ(X^{crys}/R_{V} , E⊗ J) can be represented by the total complex associated to the de Rham complexes

(E ⊗ J)(D(R′_{n})) ⊗ Ω_{R′_{n}}^{·,log}, and similarly for X̃. However the X̃^{∧}′ are also log -smooth in the log -structure of X, i.e. without adding Y to the boundary (the logarithmic product (𝔸^{1}, {0}^{n+1,log} = 𝔸^{1} × 𝔾_{m}^{n} is smooth). Thus they may also be used to compute the crystalline cohomology of X. That is there are the subcomplexes

(E ⊗ J)(D(R̃′_{n})) ⊗ Ω_{∼}·,log ; R̃′_{n} ⊆ (E ⊗ J)(D(R̃′_{n})) ⊗R̃′_{n} Ω·,log ; R̃′_{n}

whose associated total complex represents

ℝΓ(X^{crys}/R_{V} , E ⊗ J).

The quotient represents ℝΓ(Y^{crys}/R_{V} , E ⊗J){−1}[−1], with connecting map i_{∗}^{crys}.

Also the local parametrisations give an equation f_{0} ∈ R′_{0} for the Y -stratum. By definition of the logarithmic product f_{1} = pr_{0}^{∗}(f_{0})/pr_{1}^{∗}(f_{1}) ∈ R̃_{1}^{′,∗} is a unit. d log f_{0} is a 1-cochain with residue 1 in the de Rham complex D(R̃′_{0}) ⊗ Ω·,log ; R̃′_{0}. Also the de

Rham complex for Y can be identified with the quotient of the (E ⊗ J)(D(R̃′_{n})) ⊗ Ω_{∼}·,log ; R̃′_{n} obtained by setting f_{0} = df_{0} = 0. Thus to compute i_{∗}^{crys}(z) for a class z ∈ H^{m}(Y^{crys}/R_{V} , E ⊗ J), represent z by an m-cocycle λ̄ in this quotient complex, lift to an m-cochaine λ in the complex for X, form the product (d log f_{0}, 0) ∪ λ and apply to it the differential to get a class in H^{m+2}(X^{crys}/R_{V} , E ⊗ J) representing i_{∗}^{crys}(z).

Now let us map to étale cohomology. There is a diagram of topoi

X̃ ⊗_{V} K̄ ↪^{j} X ⊗_{V} K̄ ⟵^{i} Y ⊗_{V} K̄.

We know that

j_{∗}(𝕃 ⊗ B_{cr}^{+}(J^{≃})) ≈ 𝕃 ⊗ B_{cr}^{+}(J̄))

ℝ^{1}j_{∗}(𝕃 ⊗ B_{cr}^{+}(J^{≃})) ≈ i_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄_{Y}))(−1),

and higher direct images

ℝ^{ν}j_{∗}(𝕃 ⊗ B_{cr}^{+}(J^{≃})) (ν ≥ 2) or ℝ^{ν}i_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄_{Y})) (ν ≥ 1)

are almost zero.

Next our de Rham resolutions define complexes L_{X} on X^{0}⊗_{V} K̄, LX̃ on X̃^{0}⊗_{V} K̄ and L_{Y} on Y^{0} ⊗_{V} K̄, acyclic in degrees ≠ 0, with an injection L_{X} ↪ j_{∗}LX̃. We also get a map i_{∗}L_{Y} ← (j_{∗}LX̃/L_{X})[−1]. Locally these can be described as follows: Assume that an étale Spec(R) → X lifts to Spec(R) → X′. Then L_{X} evaluated on R̄ is made up from the de Rham complexes

E ⊗ J(D(B_{cr}^{+,0}(R̄) ⊗^{log} R̃′_{n}) ⊗ Ω_{∼}·,log ; R̃_{n}

where B_{cr}^{+,0}(R̄) = preimage of R ⊆ R̄^{∧} in B_{cr}^{+}(R̄), and ”D” denotes PD-hull along

the graph of Spec(R) → X′_{n}. Similarly L̃_{X} on R^{≃} is represented by

E ⊗ J(D(B_{cr}^{+,0}(R^{≃}) ˜⊗^{log}R̃′_{n}) ⊗ Ω·,log ; R̃_{n}

with ˜⊗^{log} denoting log-product in the X̃ − log -structure. Over R̄ j_{∗}L̃_{X}^{·} is locally given by the invariants under Δ_{∼} = ker(Δ̃ → Δ).

The map j_{∗}L̃_{X} → i_{∗}L_{Y} [1] is defined by choosing an extension R^{≃} → R̄_{Y} of R →

R_{Y}, which is possible. Then the induced B_{cr}^{+}(R^{≃}) → B_{cr}^{+}(R̄_{Y}) and the residue give a map. (One can check easily that things are independent of choices, as evaluation of crystals on PD-thickenings). With this definitions obviously the diagram

0 ⟶ (DR-complex for X) ⟶ (DR-complex for X̃) ⟶^{Res} (DR-complex for Y)[−1] ⟶ 0
↓  ↓  ↓
0 ⟶ Γ(X^{0} ⊗_{V} K̄, L_{X}) ⟶ Γ(X̃^{0} ⊗_{V} K̄, LX̃) ⟶ Γ(Y^{0} ⊗_{V} K̄, L_{Y} )[−1] ⟶ 0

strictly commutes.

Next first assume 𝕃 = ℤ/p^{s}ℤ, E = O_{X}/p^{s}O_{X}, J = O_{X}. Then the canonical ele-

ment 1 ∈ H^{0}(Y^{crys}/R_{V} , O/p^{s}O) maps to an element of Γ(X̃^{0} ⊗_{V} K̄, L1 ; X̃)/Γ(X^{0} ⊗_{V}

K̄, L_{X}^{1}), and thus to a global section of

Γ(X^{0} ⊗_{V} K̄, j_{∗}L1 ; X̃/L_{X}^{1}),

and a global section of

\underline{H}^{1}(j_{∗}L1 ; X̃/L_{X}^{1}) = \underline{H}^{1}(j_{∗}L1 ; X̃).

Mapping to an injective resolution of B_{cr}^{+}(ŌX̃)/p^{s}B_{cr}^{+}(ŌX̃) we get a global section of

ℝ^{1}j_{∗}B_{cr}^{+}(ŌX̃)/p^{s} · B_{cr}^{+}(ŌX̃) ≈ i_{∗}(B_{cr}^{+}(Ō_{Y})/p^{s} · B_{cr}^{+}(Ō_{Y}))(−1).

We claim that this is given by β ∈ B_{cr}^{+}(V̄)(−1):

This is an entirely local assertion, thus can be checked over strict henselisations R = O_{X,x}^{sh}. Especially we have a lift to Spec(R) → X′. Let f_{0} ∈ R̂′_{0} denote the global equation for Y on X̂′_{0}, f_{1} = pr_{0}^{∗}f_{0}/pr_{1}^{∗}f_{0} ∈ R̂_{1}^{′∗}. Then (\frac{df_{0}}{f_{0}} , 0) defines a 1cochain in the de Rham complex for X̃, with boundary (0, \frac{df_{1}}{f_{1}} , 0) in the de Rham complex for X. It thus defines a cocycle in the quotient, with residue one. Hence we have to find its image in B_{cr}^{+}(R̄_{Y})/p^{s} · B_{cr}^{+}(R̄_{Y})[−1]. Thus consider the section (d log(1 ⊗ f_{0}), 0) of L1 ; X̃, where

1 ⊗ f_{0} ∈ D(B_{cr}^{+}(R^{≃}) ⊗ R̃^{∧}_{0})

We remark that 1 ⊗ f_{1} is a 1-unit in D(B_{cr}^{+}(R̄) ⊗ R̃^{∧}′_{1}), thus its logarithm g_{1} defines a section of L_{X}^{1}. We thus also may consider (d log(1 ⊗ f_{0}), g_{1}) which is a 1-cocycle for LX̃, and Δ̃ -invariant. We want to compute its class in cohomology, at least after mapping to j_{∗}Ĩ^{·}.

For this we not that also (1 ⊗ f_{0})/([f_{0}] ⊗ 1) = exp(g_{0}) is a 1-unit, where [f_{0}] ∈

A_{crys}(R^{≃}) is defined by an element of R(R^{≃}) made up from p-power roots of f_{0}. Moreover the differential of g_{0} is (d log(1 ⊗ f_{0}), g_{1}). However g_{0} is not Δ̃ -invariant: Namely any σ ∈ Δ̃ multiplies [f_{0}] by exp(β(σ̄)), with σ̄ ∈ ℤ_{p}(1) describing the action of σ on p-power roots of f_{0}. Thus σ(g_{0}) = g_{0} − β(σ̄) ⊗ 1, and the obstruction to make g Δ̃ -invariant is given by the 1-cocycle σ ↦ β(σ̄). Restriction to inertia (≅ ℤ_{p}(1)) at Y then implies the assertion.

For general E and J we need cup-products. The de Rham and Čech-complexes admit strictly associative differential products (in the Čech-setting, (f ∪ g)_{0...a+b} = f_{0...a} ∪ g_{a...b} for f in degree a, g in degree b). This defines such products on L_{X}(O), LX̃(O), L_{Y} (O), and makes L_{X}(E ⊗ J) etc. to differentially graded modules

over them. Furthermore the elements pr_{0}^{∗}f_{0} ∈ R̃^{∧}′_{n} generate an invertible ideal ⟨f_{0}⟩ invariant under all simplicial maps, and thus a differential ideal ⟨f_{0}, df_{0}⟩ ⊆ L_{X}(X). The quotient L_{X}(O)/⟨f_{0}, df_{0}⟩ naturally maps to i_{∗}L_{Y} (O) induced from B_{cr}^{+}(Ō_{X}) → i_{∗}B_{cr}^{+}(Ō_{Y}). Next we have defined a class ψ ∈ Γ(X^{0} ⊗_{V} K̄, j_{∗}L^{1}(OX̃]/L^{1}(O_{X})), the image of \frac{df_{0}}{f_{0}} . Under the multiplication of L^{·}(O_{X}) on j_{∗}L(OX̃)/L(O_{X}) this class is annihilated by ⟨f_{0}, df_{0}⟩. Thus cup-product defines a map

ψ∪ : L_{X}(E ⊗ J)/⟨f_{0}, df_{0}⟩ → j_{∗}LX̃(X ⊗ J)/L_{X}(E ⊗ J)[−1]

Furthermore in the derived category we had defined a canonical map

j_{∗}LX̃(E ⊗ J)/L_{X}(E ⊗ J)[−1] → ℝi_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄)),

which is compatible with the action of cup-products from L_{X}/⟨f_{0}, df_{0}⟩, acting on the right hand side via

L_{X}(E ⊗ J)/⟨f_{0}, df_{0}⟩ → i_{∗}L_{Y} (E ⊗ J) → ℝi_{∗}(𝕃 ⊗ B_{cr}^{+}(Ō_{Y}))

Remark. The maps L_{X}(E ⊗ J) → 𝕃 ⊗ B_{cr}^{+}(J̄_{X}) are a priori only defined in the derived category. However we may replace B_{cr}^{+}(Ō_{X}) by B_{cr}^{+}(Ō_{X}) ⊗_{L_{X}^{0}(O)} L_{X}(O) (which is quasi-isomorphic) and 𝕃⊗B_{cr}^{+}(J̄_{X}) by (𝕃⊗B_{cr}^{+}(J̄_{X}))⊗_{L_{X}^{0}(O)}L_{X}(O) to represent those by maps of DG-algebras respectively DG-modules. Then compatability with cup-products follows automatically, for example by computing cohomology via hypercoverings.

As ψ maps to β we see that

L_{X}(E ⊗ J)/⟨f_{0}, df_{0}⟩ ⟶^{ψ∪}  j_{∗}LX̃(E ⊗ J)/L_{X}(E ⊗ J)[+1]
↓  ↓
i_{∗}L_{Y} (E ⊗ J)  ℝj_{∗}(𝕃 ⊗ B_{cr}^{+}(JX̃))/𝕃 ⊗ B_{cr}^{+}(J̄_{X})[+1]
↓  ↓
i_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄_{Y})) ⟶^{β∘}  i_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄_{Y})(−1)

commutes up to homotopy. Now i_{∗}^{crys} is defined by the connecting map for

0 → DR_{X}(E ⊗ J) → DRX̃(E ⊗ J) → DR_{Y} (E ⊗ J)[−1] → 0.

This maps to global sections in the exact sequence

0 → L_{X}(E ⊗ J) → LX̃(E ⊗ J) → j_{∗}LX̃(E ⊗ J)/L_{X}(E ⊗ J) → 0

or also

0 → L_{X}(E ⊗ J) → L∗ ; X̃(E ⊗ J) → L_{X}(E ⊗ J)/⟨f_{0}, df_{0}⟩[−1] → 0,

the pullback via

ψ∪ : L_{X}(E ⊗ J)/⟨f_{0}, df_{0}⟩[−1] → j_{∗}LX̃(E ⊗ J)/L_{X}(E ⊗ J),

which in turn maps to

0 → 𝕃 ⊗ B_{cr}^{+}(J̄_{X}) → ℝj_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄X̃)) → i_{∗}(𝕃 ⊗ B_{cr}^{+}(J̄_{Y}))[−1](−1) → 0,

where the maps on the first two terms are the canonical comparison maps, while on the third we have the comparison-map multiplied by β. However as this last sequence defines i_{∗}^{et}, we get indeed v_{Y} ∘ i_{∗}^{crys} = β · i_{∗}^{et} ∘ v_{X}. Thus the theorem is shown.

2. Corollary. For the trace-forms on H^{2d}(X/R_{V} , J_{X}) respectively H_{!}^{2d}((X^{0} ⊗_{V} K̄)^{et}, ℤ_{p}) we have

tr^{et} ∘ v_{X} = β^{d} · tr^{crys}

Proof. Check on the class of a smooth point. Also toric blow-ups X̃ → X induce isomorphisms on crystalline cohomology. This is true on de Rham cohomology, thus modulo F^{1}(R_{V}) ⊆ R_{V}, and follows in general because F^{1}(R_{V}) consists of topologically nilpotent elements. Hence:

3. Corollary. Assume we are given functorial transformations

v_{E} : E(B_{cris}^{+,0}(R)) → 𝕃 ⊗ B_{cris}^{+}(R)
v_{E^{∨}} : E^{∨}(B_{crys}^{+,0}(R)) → 𝕃^{∨} ⊗ B_{crys}^{+}(R)
(E^{∨} = Hom(E, O_{X}/p^{s}O_{X}), 𝕃^{∨} = Hom(𝕃, ℤ/p^{s}ℤ)),

with ⟨v_{E}(e), vE^{∨}(e^{∨})⟩ = β^{⊗m} · v⟨e, e^{∨}⟩ for some fixed m.

Then

v_{X} : ℝΓ(X^{crys}/R_{V} , E ⊗ J) ⊗_{R_{V}} B_{crys}^{+}(V) → ℝΓ_{(!)}((X^{0} ⊗_{V} K̄)^{et}, 𝕃) ⊗ B_{cris}^{+}(V)

has an inverse up to β^{⊗(d+m)}, that is composition either way is multiplication by β^{⊗(d+m)}

Proof. The ”inverse” is the adjoint of the map v_{X} for E^{∨}. One has to compare the classes of the diagonals, passing to Δ̃ : X̃ → ({ \atop X × X}). (The class of the diagonal c_{δ} ∈ H_{(!)}^{2d}({ \atop X × X}, ℝ\underline{Hom}(pr_{2}^{∗}E, pr_{1}^{∗}E)) is the image of a class in

H_{(!)}^{2d}({ \atop X × X}, \underline{Hom}(pr_{2}^{∗}E, pr_{1}^{∗}E)), the direct image of the identity map on E).

Remarks.

1) If E is in fact a crystal in the crystalline topos of X relative V_{0} (that is one has integrable connections ▽ : E → E ⊗ Ω_{R′/V_{0}}^{1,log} = E ⊗ Lie(T)^{∨}), the crystalline cohomology ℝΓ(X^{crys}/R_{V} , E) inherits a logarithmic connection (as a perfect complex). Especially all cohomology-groups have such a connection. Also the Galois-group Gal(R̄/R) acts on ℝΓ(X^{crys}/R_{V} , E ⊗ J) ⊗_{R_{V}} B_{crys}^{+}, although the map R_{V} → B_{crys}^{+} is not Galois-linear. It suffices that this holds modulo F^{1}(B_{crys}^{+}). Finally if 𝕃 is defined over K, Gal(R̄/R) also operates on ℝΓ_{(!)}((X^{0} ⊗_{V} R̄)^{et}, 𝕃). And if the local comparison-maps

E(B_{crys}^{+,0}(R̄)) → 𝕃 ⊗ B_{crys}^{+}(R̄)

also respect the action of Gal(R̄/R) the comparison-map v_{X} is Gal(K̄/K) -linear.

2) Everything works in the filtered context: If E is a filtered crystal, and the comparison

E(B_{cr}^{+,0}(R)) → 𝕃 ⊗ B_{cr}^{+}(R)

preserves filtrations, that is it sends F^{i}(E(B_{cr}^{+,0}(R))) to 𝕃⊗F^{i}(B_{cr}^{+}(R)), the comparison-map

ℝΓ(X^{crys}/R_{V} , E ⊗ J) → ℝΓ_{(!)}((X^{0} ⊗_{K} V̄)^{et}, 𝕃) ⊗ B_{cr}^{+}(V)

exists in the filtered derived catgeory, and similarly for inverese ”up to β^{d+e}”.

3) If E is a Frobenius-cyrstal (in the log-sense) then E(B_{cr}^{+,0}(R)) ⊗_{B_{cr}^{+,0}(R)} B_{cr}^{+}(R) has a Frobenius-action ϕ, as B_{cr}^{+} has a canonical Frobenius-lift (which is a logarithmic: ϕ([\underline{μ}]) = [\underline{μ}]^{p}, for μ ∈ σ^{∨} ∩L^{∨}). It then makes sense to require that the comparison-map to 𝕃 ⊗ B_{cr}^{+}(R) is Frobenius-linear. In this case

v_{X} : ℝΓ(X^{crys}/R_{V} , E ⊗ J) → ℝΓ_{(!)}((X^{0} ⊗_{K} V̄)^{et}, 𝕃) ⊗ B_{cr}^{+}(V)

will also preserve Frobenius (note that F rob^{∗}(J) ⊆ J, as J is locally generated by elements μ ∈ σ^{∨} ∩ L^{∨} with F rob(μ) = μ^{p}· unit). Here the ”Frobenius” on R_{V} is Frobenius on V_{0}, and sends t to t^{p}.

4) In many cases (for example constant coefficients) one can pass to s → ∞ to obtain a ℤ_{p} -theory and then also a ℚ_{p} -theory. In more generality this is done in the context of convergent isocrystals (see [Bo], [O]):

Our base will be the ”open unit disk”: That is for each ε = \frac{1}{n} consider the algebra R_{V,ε} = V_{0}[t, t^{n}/p]^{∧}. (∧ = p -adic completion). For n >> 0 R_{V,ε} ⊆ R_{V} , R_{V,ε} has a toroidal or log -structure generated by t. An affine logarithmic enlargement (we do not need others) is given by an R_{V,ε} -algebra A (some ε > 0) with a log-structure (étale locally Spec(A) → T̄, R_{ε} -linear), and ideal I ⊂ A such that A is I-adically complete and such that some power of I is contained in pA, and an R_{ε} -linear log-map Spec(R/I) → X. The definition of maps is the obvious one. A convergent (logarithmic) isocrystal associates to any enlargement A a finitely generated projective A[\frac{1}{p}] -module E(A), with E(B) = B ⊗_{A} E(A) for maps A → B. Similarly filtered convergent isocrystals associate filtered modules, A[\frac{1}{p}] itself being filtered by powers of I.

In the usual way (compare [Fa3], IV.e.) one defines the cohomology ℝΓ(X^{crys}/R_{ε}, E⊗ J). I do not know whether this is in general representable by a perfect complex over R_{ε}[\frac{1}{p}], satisfying Poincaré-duality. However for Frobenius-crystals (Φ^{∗}E ≅ E) one can at least show that Frobenius induces a quasi-isomorphism on cohomology, that the special fibre at t = 0 is representable by a perfect complex 𝕂_{0} (i.e. has finitedimensional cohomology), and that there are Φ -linear and parallel comparsionmaps

(H^{i}(X^{crys} ⊗_{V} k/V_{0}, E ⊗ J) ⊗_{K_{0}} R_{ε}[\frac{1}{p}], d + N) → (H^{i}(X^{crys}/R_{ε}, E ⊗ J), ▽),

where N = Res(t\frac{d}{dt}) is the endomorphism defined by the logarithmic connection, satisfying Φ ∘ N = pN ∘ Φ (and thus N is nilpotent): For the finite-dimensionality one constructs integral lattices and proceeds as in [Fa7]. The comparison-map is constructed as in [Fa4]: Start with any lift of the identity mod t. Then transform it by powers of Frobenius and pass to the limit.

I expect the comparison-map to be isomorphisms. This holds if H^{i}(X^{crys}/R_{ε}, E⊗ J) is finitely generated over R_{ε}[\frac{1}{p}], and for example if E comes from a usual crystal (for example if E = constant). Then we also have Poincaré-duality (check in the fibre at t = π), and the usual construction of the diagonal-class. In any case the appendix contains more details on what can be shown and what not.

Define B_{crys,ε}^{+}(R̄) = p -adic completion of A_{inf}(R̄)[\frac{ξ^{n}}{p}] (as before, ε = \frac{1}{n}). This is an enlargement of R, thus E(B_{crys,ε}^{+}(R̄)) is defined. If we are given a smooth ℚ_{p} -adic sheaf 𝕃 on (X^{0} ⊗_{V} K̄)^{et} and functorial comparison-maps

E(B_{crys,ε}^{+}(R̄)) → 𝕃 ⊗ B_{crys,ε}^{+}(R̄)

respecting Galois-actions and compatible for varying ε’s) we obtain a comparisonmap

v_{X} : ℝΓ(X^{crys}/R_{ε}, E ⊗ J) ⊗ B_{crys,ε}^{+}(V̄) → ℝΓ_{(!)}((X^{0} ⊗_{V} K̄)^{et}; 𝕃) ⊗ B_{crys,ε}^{+}(V̄),

with the usual properties concerning duality, characteristic classes, action of Gal(K̄/K) and of Frobenius, filtrations. Also for a Frobenius-crystal we can compose with the comparison-maps to obtain

v_{X} : ℝΓ(X^{crys}⊗_{V} k/V_{0}, E ⊗J)⊗_{K_{0}} B_{crys,ε}^{+}(V̄) → ℝΓ_{(!)}((X^{0}⊗_{V} K̄)^{et}, 𝕃)⊗B_{crys,ε}^{+}(V̄)

This will respect Galois-actions if the other maps do, where however on the left hand side the Galois-action is not just via the second factor but has to be twisted, because of the connection d + N. There is a 1-cocycle Gal(K̄/K) → ℤ_{p}(1)

σ ↦ σ̄

describing the action on all p-power roots of π, or equivalently on [\underline{π}] ∈ A_{inf}(V̄). Then the Galois-action of σ on

H^{i}(X^{crys} ⊗_{V} k/V_{0}, E ⊗ J) ⊗ B_{crys,ε}^{+}(V̄)

is defined as

σ ↦ exp(β(σ̄)N) ⊗ σ

This corresponds to Fontaine’s description: If B_{st}^{+} = B_{cr}^{+}[u] with Nu = 1(u = ” log([\underline{π}]/π)”) then (E_{0} = H^{i}(X^{crys} ⊗_{V} k/V_{0}, E ⊗ J))

E_{0} ⊗ B_{cr}^{+} ≅ ker(N ⊗ 1 + 1 ⊗ N | E_{0} ⊗ B_{st})

via exp(−N ⊗ u), and as σ(u) = u + β(σ̄) this intertwines Galois-actions.

5^{∗}. Appendix: Cohomology of Frobenius-isocrystals

As before X is étale locally étale over some T̄_{λ}. The cohomology of convergent isocrystals can be defined via de Rham complexes on smooth local lifts, as it is done in crystalline cohomology. However one has to replace PD-hulls by open ” > ε” -neighbourhoods, as follows:

Let R_{>ε} respectively R_{≥ε} denote the rings of formal powerseries convergent in the open (respectivly closed) disk of radius p^{ε}, that is formal powerseries f = ∑_{n≥0} a_{n}t^{n} with (v_{p} = p -adic valuation).

a_{n} ∈ K_{0}, lim inf_{n→∞}(v_{p}(a_{n})n) ≥ −ε

(respectively a_{n} ∈ K_{0}, lim inf_{n→∞}(v_{p}(a_{n}) + nε) = ∞). Obviously R_{<ε} = ⋂_{δ>ε} R_{≥δ}. Also if ε = \frac{a}{b} is rational (which we assume from now on)

R_{≥ε} = K_{0} ⊗_{V_{0}} (V_{0}[t]/(t^{b} − p^{a}))^{∧}.

This is not true for R_{>ε}, but unfortunately we shall need this ring because it satisfies the Poincaré-lemma:

The de Rham complex

R_{>ε} ⟶^{d} R_{>ε} · dt

is quasi-isomorphic to K_{0}.

Next to define and compute cohomology choose ε small enough such that π has valuation > ε. Then X ⊗_{V} K is a scheme over R_{>ε} and R_{≥ε}, and in fact X itself is a scheme over R_{≥ε}^{0} = V_{0}[t]/(t^{b} −p^{a})(ε = \frac{a}{b}). If X embeds into a p-adic affine formal scheme Y over T̄ ⊗_{V_{0}} R_{≥ε}^{0}, respecting toroidal structures, let J_{X} ⊆ O_{Y} denote the ideal of X. For δ = \frac{c}{d}, 0 < δ < ε, define the closed δ -tube of X in Y as the affine Y -scheme with algebra O_{Y} [J_{X}^{d}/p^{c}]. Call it D_{X,≥δ}(Y). Define the open δ -tube as the ind-scheme

” \underrightarrow{lim}_{δ′>δ}”D_{X_{1}≥δ′}(Y) = D_{X,>δ}(Y).

As each D_{X_{1}≥δ′}(Y) is an enlargement of X/R_{≥ε} we can evaluate E on it, and form the de Rham complex E(D_{X_{1}≥δ′}(Y)) ⊗ Ω_{Y}^{·,log}. Then define ℝΓ(X/R_{≥ε}, E) as represented by the projective limit (δ′ → δ+) of the de Rham complexes. Similarly we define ℝΓ(X/R_{≥ε}, E ⊗ J), and if X cannot be embedded globally we use local embeddings any hypercoverings. One checks that the result is independant of all choices, the key being the Poincaré-lemma which allows to compare two local smooth embeddings (and there we unfortunately need open disks).

Next we first treat the case that X = Spec(R) is affine, and lifts to an étale

formal scheme over T̄^{∧}. For each ε > 0 we then obtain by pushout formal p-adic schemes Spf(S_{≥ε}) over R_{≥ε}^{0}, which are enlargements of X. Spf(S_{ε}) is formally étale over T̄ ⊗_{V_{0}[t]} R_{≥ε}^{0}. The de Rham complex computing the cohomology of X is E(S_{ε})⊗ (pullback of J ⊗ Ω·,log ; T̄_{λ}). Also S_{≥ε} has a canonical Frobenius-lift ϕ induced from T̄ (via μ ↦ μ^{p} for μ ∈ L). Thus a Frobenius Φ on cohomology, induced from Frobenius Φ^{∗} on E(S_{ε}) as well as dϕ on Ω·,log ; T̄_{λ}.

4. Lemma. Φ^{∗} induces an isomorphism on H^{∗}(X/R_{ε}, E ⊗ J).

Proof (following Mazur, Ogus). Let p^{−·} · Ω·,log ; T̄_{λ} denote the de Rham complex with Ω^{q,log} replaced by p^{−q} · Ω^{q,log}. Then ϕ^{∗} induces a quasi-isomorphism

p^{−·}Ω·,log ; T̄_{λ} ⊗ J(S_{ε}) ⊗_{ϕ} R_{ε}^{0} → Ω·,log ; T̄_{λ} ⊗ J(S_{ε}),

even modulo p^{s} for all s ≥ 0. It suffices to prove this modulo p, and for T̄ itself (étale localisation). Then the left hand side is the direct sum, over all μ ∈ L^{∨} ∩ σ^{∨} such that μ−λ ∉ σ^{∨}, and μ(ρ_{i}) > 0 for certain extremal ρ_{i} (with λ(ρ_{i}) = 0), of R_{ε}^{0}/pR_{ε}^{0}⊗ Λ^{·} ker(λ)^{∗}, with trivial differential. On the right we have the same modules but with differential equal to exterior multiplication by μ. The map between both sides is induced from μ ↦ pμ. Now on the right the μ -eigenspace has trivial cohomology unless μ ∈ p · ker(λ)^{∗}, μ = p · μ_{1}. Then μ(ρ_{i}) > 0 ⇔ μ_{1}(ρ_{i}) > 0, and μ ∈ σ^{∨} ⇔ μ_{1} ∈ σ^{∨}. Also μ − λ ∉ σ^{∨} implies μ_{1} − λ ∉ σ^{∨}. If conversely μ − λ ∈ σ^{∨}, then μ_{1} − \frac{λ}{p} = (μ_{1} − λ) + \frac{p−1}{p}λ ∈ σ^{∨}. By the condition on σ (i.e. X ⊗_{V} k is geometrically reduced) this implies μ_{1} − λ ∈ σ^{∨}. Hence the assertion.

Now choose a system of generators e_{α} for E(S_{ε}). Then

▽(e_{α}) = ∑_{β} e_{β} ⊗ w_{αβ}

with certain w_{αβ} ∈ S_{ε}[\frac{1}{p}] ⊗ Ω1,log ; T̄_{λ}. Replacing e_{α} by Φ^{N}(e_{α}) with N >> 0 we may

assume that all w_{αβ} ∈ p · S_{ε} ⊗ Ω1,log ; T̄_{λ}. Then the e_{α} span an S_{ε} -submodule E^{0}(S_{ε}), and there is a welldefined map

Φ^{∗} : E^{0}(S_{ε}) ⊗ p^{−·} · Ω·,log ; T̄_{λ} ⊗ J(S_{ε}) → E^{0}(S_{ε}) ⊗ Ω·,log ; T̄_{λ} ⊗ J(S_{ε})

which is a quasi-isomorphism modulo p and thus in general. Composing with Frobenius Φ^{∗}(E_{0}) → E_{0} and inverting p proves the lemma.

Remarks. The role of the base R_{≥ε} is rather auxiliary. One can improve the result by noting that Frobenius induces a finite flat map from R_{≥ε} to R_{≥ε/p} (which has basis {1, t, …, t^{p−1}} over ρ(R_{≥ε})). Then also we have an isomorphism

Φ^{∗} : H^{i}(X/R_{≥ε}, E ⊗ J) ⊗_{R_{≥ε}ϕ} R_{≥ε/p} ⟶^{∼} H^{i}(X/R_{≥ε/p}, E ⊗ J).

Also we might use the base R_{V}, if E is actually a crystal (in the usual crystalline topos).

5. Corollaries.

a) For any X (not necessarily affine or lifting to T̄^{∧}) Frobenius induces isomorphisms

\begin{matrix}Φ^{∗} : H^{i}(X/R_{≥ε}, E ⊗ J) ⊗_{R_{≥ε}ϕ} R_{≥ε} ⟶^{∼} H^{i}(X/R_{≥ε}, E ⊗ J) \\ H^{i}(X/R_{≥ε}, E ⊗ J) ⊗_{R_{≥ε}ϕ} R_{≥ε/p} ⟶^{∼} H^{i}(X/R_{≥ε/p}, E ⊗ J) \end{matrix}

b) Suppose E is given by an actual crystal. Then

Φ^{∗} : H^{i}(X/R_{V} , E ⊗ J) ⊗_{R_{V} ϕ}^{𝕃} R_{V} → H^{i}(X/R_{V} , E ⊗ J)

is an isogeny, that is kernel and cokernel are annihilated by some fixed power p^{N}.

These follow by cohomological descent, using an étale covering of X by good affines.

Next we show that E always satisfies the assumptions in b), that is it can be defined by a crystal E^{0}. For this choose an affine étale covering X′ = Spec(R′) → X

with X′ → T̄_{λ} étale and lift to X̂′ → T̄^{∧}. Let R̂′ define the p-adic completion of X̂′ ⊗_{V_{0}[t]} R_{V} , R̂′′ = completed PD-hull of X′ ×_{X} X′ ↪ Spf(R̂′ ˆ⊗_{R_{V}} R̂′), R̂′′′ etc. as usual. The R̂^{(m)} form a cosimplicial ring with Frobenius ϕ, and all transitionmaps flat (modulo any fixed power p^{s}, that is the sense of formal p-adic schemes). They also define affine enlargements of X. We can choose a finitely generated R̂′ -submodule E_{0}^{∗} ∈ E(R̂′) spanning E(R′), and stable under p^{N} · Φ (some fixed N) and ▽. Then pr_{0}^{∗}E_{0}^{∗} ⊆ p^{−N} ·pr_{0}^{∗}E_{0}^{∗} in E(R̂′′), after (if necessary) enlarging N. Then let

\begin{aligned}E^{0}(R̂′) & = \{e ∈ E_{0}^{∗} : pr_{1}^{∗}(e)) ∈ pr_{0}^{∗}(E_{0}^{∗})\} \\ & = \{e ∈ E(R̂′) : pr_{1}^{∗}(e) ∈ pr_{0}^{∗}(E_{0}^{∗})\} \end{aligned}

(use the section R̂′′ → R̂′). Then p^{N} · E_{0}^{∗} ⊆ E_{0}^{0}(R̂′) ⊆ E_{0}^{∗} (agian enlarging N), E_{0}(R̂′) is stable under p^{N} · Φ and ▽, and

pr_{1}^{∗}(E_{0}(R̂′)) = pr_{0}^{∗}(E_{0}(R̂′)) ⊆ E(R̂′′)

(direct check via simplicial identities, using flatness modulo p^{N} to compute in E(R̂′′′)). Thus E_{0}(R̂′) defines a crystal.

We use this as follows: For each s > 0 ℝΓ(X^{crys}/R_{V} , E^{0}/p^{s}E^{0} ⊗ J) is representable by a bounded complex over R_{V} /p^{s}R_{V}, and

p^{N} · Φ^{∗} : ℝΓ(X^{crys}/R_{V} , E^{0}/p^{s}E^{0} ⊗ J) ⊗_{ϕ}^{𝕃} R_{V} → ℝΓ(X^{crys}/R_{V} , E^{0}/p^{s}E^{0} ⊗ J)

induces on cohomology maps with kernel and cokernel annihilated by p^{2N}, N sufficiently big but independant of s.

Next we consider the special fibre, that is the pushout via R_{V} → V_{0} (sending t to 0). This computes the cohomology ℝΓ((X ×_{V} k)^{crys}/V_{0}, E ⊗ J). The restriction of E^{0} defines a crystal on the special fibre, whose evaluation on R̂′ ⊗_{V_{0}[t]} V_{0} = R̂′/tR̂′ is the tensor-product (over R_{V}) with R_{0}. If we divide by p-torsion we obtain a new crystal Ē^{0} which is flat over V_{0}. Furthermore

Ē^{0}(R̂′/tR̂′) ⊆ E(R̂′/tR̂′)

is contained in a finitely generated R̂′/tR̂′ -submodule (induced from the Ẽ^{0} used in the construction of E^{0}), thus itself finitely generated. Hence its cohomology modulo t (given by the de Rham complex on X ⊗_{V} k is finite over k. By the methods of [Fa7] it follows that ℝΓ((X ⊗_{V} k)^{crys}/V_{0}, Ē^{0} ⊗ J) is representable by a perfect complex over k. Furthermore we claim that the kernel of E^{0} ⊗_{R_{V}} V_{0} ↠ Ē^{0} as well as all higher T or_{i}^{R_{V}} (E^{0}, V_{0}) are annihilated by a fixed power p^{N}: It suffices to treat evaluations on R̂′/tR̂′. We know that E^{0}(R̂′)[\frac{1}{p}] is finitely generated projective, so there exist maps

E^{0}(R̂′)[\frac{1}{p}] ⇄_{β}^{α} (R̂′[\frac{1}{p}])^{n}

with β ∘ α = id. Also E^{0}(R̂′) is contained in a finitely generated R̂′ -submodule. This multiplying α and β by suitable p-powers we obtain

E^{0}(R̂′) ⇄_{β}^{α} (R̂′)^{n}

with β ∘ α = p^{N} · id. This p^{N} does the job.

As a corollary we see that

H^{i}(X^{crys}/R_{V} , E^{0} ⊗ J) ⊗_{R_{V}}^{𝕃} V_{0} → H^{i}((X ⊗_{V} k)^{crys}/V_{0}, Ē^{0} ⊗ J)

has kernel and cokernel annihilated by some fixed p^{N}. Let L^{∗} denote a topologically flat model of the total de Rham complex representing ℝΓ(X^{crys}/R_{V} , E^{0} ⊗J). Then L^{∗} consists of R_{V} -modules, has a Frobenius p^{N} · Φ and is p-adically complete. Furthermore if I ⊂ R_{V} denotes the kernel of R_{V} ↠ V_{0}, then L^{∗}⊗_{R_{V}} R_{V} /I represents the cohomology of the special fibre. Note that I is generated by all fractional powers t^{m}/m!. Applying Frobenius i-times to such an element gives

ϕ^{i}(t^{m}/m!) = t^{p^{i}m}/m! = (\frac{(p^{i}m)!}{m!} · t^{p^{i}m}/(p^{i}m)!

which is divisible by p^{(m+pm+...+p^{i−1}m)}. Thus ϕ^{i}(I) ⊆ p^{p^{i−1}} · I. We have seen that for each i

H^{i}(L^{∗}/I · L^{∗}) ≅ M_{0}^{i} ⊕ (p^{N} − torsion),

with M_{0}^{i} a finitely generated free V_{0}-module. Replacing Φ by p^{2N} · Φ we may assume that Φ annihilates the second summands, Φ is an isogeny on the first (i.e. an isomorphism on M_{0}^{i}[\frac{1}{p}]). We use this to lift M_{0}^{i} to a submodule of H^{i}(L^{∗}[\frac{1}{p}]), as follows: Choose a map α̃ : M_{0}^{i} → L^{i} lifting a basis of M_{0}^{i}. Then

Φ ∘ α̃ − α̃ ∘ Φ = β + d ∘ γ

with β : M_{0}^{i} → I · L^{i}

γ : M_{0}^{i} → L^{i−1}

Define α = α̃ + ∑_{m=0}^{∞} Φ^{m} ∘ β ∘ Φ^{−(m+1)}. This sum converges in Hom(M_{0}^{i}, L^{i}[\frac{1}{p}])

as Φ^{m} ∘ β is divisible by p^{p^{m−1}}, while the p-power in the denominator of Φ^{−(m+1)} grows linearly in m. Then

Φ ∘ α − α ∘ Φ = d ∘ γ,
d ∘ α = d ∘ α̃ + ∑_{m=0}^{∞} Φ^{m} ∘ d ∘ β ∘ Φ^{−(m+1)}

= d ∘ α̃ + ∑_{m=0}^{∞} Φ^{m} ∘ (d ∘ Φ ∘ α̃ − d ∘ α̃ ∘ Φ) ∘ Φ^{−(m+1)}

= d ∘ α̃ + lim_{m→∞}(Φ^{m+1} ∘ d ∘ α̃ ∘ Φ^{−(m+1)} − d ∘ α̃)

= lim_{m→∞}Φ^{m+1} ∘ d ∘ α̃ ∘ Φ^{(−m+1)} = 0,

as d ∘ α̃(M_{0}^{i}) ⊆ I · L^{i}. Hence if M_{0}^{∗} denotes the complex made up from the M_{0}^{i}, with trivial differential and with Frobenius Φ, M^{∗} = M_{0}^{∗} ⊗_{V_{0}} R_{V}, we have for N sufficiently big a map of complexes

α : M^{∗} → p^{−N} · L^{∗}

with Φ ∘ α − α ∘ Φ homotopic to zero, and the cone C(α) has the property that the cohomology of C(α)/I · C(α) is annihilated by p^{2N}. We also may extend Φ to a Frobenius on C(α). Then p^{2N} sends H^{i}(C(α)) to the image of H^{i}(I · C(α)), thus p^{2N} · Φ^{m} is divisible by p^{p^{m−1}} on H^{i}(C(α)). On the other hand we know that the cohomology of M^{∗} as well as L^{∗} some Frobenius divides some power of p. This is then also true on C(α). It follows that for some m >> 0, p^{m} divides p^{m+1} on this cohomology. Then H^{i}(C(α)) = (p^{m} -torsion) +(p-divisible submodule). Unfortunately I cannot prove in general that the second summand vanishes. What is however true in general is that the H^{i}(X ⊗_{V} k/V_{0}, E ⊗ J) still satisfy Poincaréduality. More precisely (X ⊗_{V} k/V_{0}, Ē_{0} ⊗ J_{I}) is paired with

ℝΓ(X ⊗_{V} k/V_{0}, ℝHom(Ē_{0} ⊗ J_{I}, J))[−2d]

The pairing exists because there is a map H^{2d}(X ⊗_{V} k/V_{0}, J) → V_{0}. It is nondegenerate because this is true modulo t (both objects are representable by perfect modules over V_{0}, see [Fa7]). Finally ℝHom(Ē_{0} ⊗J_{I}, J) is up to p-torsion isomorphic to Ē_{0}^{∨} ⊗ J_{I^{c}}, as Ē_{0} is up to p^{N} a direct summand in a projective module. Since there is also a Poincaré-pairing on ℝΓ(X/R_{V} , E ⊗ J) (the trace-form exists) we see that H^{i}(X ⊗_{V} k_{0}/V_{0}, E ⊗ J) ⊗_{V_{0}} R_{V} injects into H^{i}(X/R_{V} , E ⊗ J). Finally by pushout via powers of Frobenius (which map R_{V} to R_{≥ε}^{0}) we obtain the same for cohomology over R_{≥ε}, that is H^{i}(X ⊗_{V} k_{0}/V_{0}, E ⊗J)⊗_{K_{0}} R_{≥ε} ⊆ H^{i}(X/R_{≥ε}, E ⊗J). Finally mapping R_{≥ε} → K (or R_{V} → V) we know that the pushout is de Rham cohomology of E(X̂) over V. This also satisfies Poincaré-duality. By the usual calculation with characteristic of diagonals the induced maps

H^{i}(X ⊗_{V} k_{0}/V_{0}, E ⊗ J) ⊗_{K_{0}} K → H_{DR}^{i}(X, E ⊗ J)

are isomorphisms (the Hyodo-Kato isomorphisms).

Remarks. Pragmatically one could replace H^{i}(X/R_{≥ε}, E⊗J) by H^{i}(X⊗_{V} k_{0}/V_{0}, E⊗ J) ⊗_{K_{0}} R_{≥ε} and then have all the desired properties. However philosophically this looks a little bit like cheating.

6. Complements: Relative case, Fontaine-Lafaille theory

As in [Fa3] one can generalise to the case of proper maps f : X → Y with f(X^{0}) ⊆ Y^{0}. For a reasonable theory one needs some local conditions on f. We assume that étale locally f is isomorphic to a map of torus-embeddings S̄_{λ} → T̄_{λ}, where S̄_{λ} ⊆ S̄, T̄_{λ} ⊆ T̄, S̄ corresponds to a lattice L and a cone σ ⊆ L_{ℝ}, T̄ to a lattice M and a cone τ ⊆ M_{ℝ}, and the map f to a homomorphism L → M which sends σ into τ such that each face of σ maps onto a face of τ. We assume that the cokernel of L → M is torsion free. Also λ ∈ M^{∨} maps to the corresponding element λ ∈ L^{∨}. Finally σ and τ should satisfy our usual assumptions, namely that σ ∩ ker(λ) and τ ∩ ker(λ) are spanned by partial basis of L, respectively M. This implies that f_{K} : X ⊗_{V} K → Y ⊗_{V} K sends each open stratum of X smoothly onto an open stratum of Y. Also for each locally constant torsion-sheaf 𝕃 on X^{0} ⊗_{V} K̄, the direct images ℝ^{ν}f_{∗}𝕃 on Y^{0} ⊗_{V} K̄ are again locally constant. Furthermore f induces a map of topoi f : X^{0} ⊗_{V} K̄ → Y^{0} ⊗_{V} K̄:

For U → Y étale, V → U^{0} ⊗_{V} K̄ an étale covering, the pullbacks to X are in the situs defining X^{0} ⊗_{V} K̄. This defines f^{∗} on sites, etc. We intend to sketch the proof of the following

6. Theorem. Suppose 𝕃 is locally constant torsion on (X^{0} ⊗_{V} K̄)^{et}, defining such a sheaf 𝕃 on X^{0} ⊗_{V} K̄. Then the direct images ℝ^{ν}f_{∗}𝕃 under f : (X^{0} ⊗ K̄)^{et} → (Y^{0}⊗_{V} K̄)^{et} and f : X^{0}⊗_{V} K̄ → Y^{0}⊗_{V} K̄ correspond. Furthermore R^{ν}f_{∗}(𝕃⊗Ō_{X}) ≈ R^{ν}f_{∗}𝕃 ⊗ Ō_{Y}, and similarly for A_{inf}(Ō), or direct images with compact support (along boundary strata of X mapping to Y^{0}).

Proof. This is done as in the absolute case, replacing V̄ by R̄, for Spec(R) → Y étale and also Spec(R) étale over S̄_{λ}, R strictly henselian. As usual we can for many purposes replace R̄ by R_{∞}. First we have to study Poincaré-duality for the direct images ℝf_{∗}(𝕃 ⊗ Ō_{X}). There is an (almost defined) trace-map

ℝ^{2δ}f_{∗}(J̄_{X}/p^{s}J̄_{X})(δ) → J̄_{Y} /p^{s}J̄_{Y}

(δ = relative dimension) inducing an almost quasi-isomorphism

ℝf_{∗}(𝕃 ⊗ Ō_{X})[2δ](δ) ≈ ℝHomR̄(ℝf_{∗}(𝕃^{∨} ⊗ J̄_{X}), J̄_{Y} p^{s}J̄_{Y})

(and similarly for cohomology with compact support at infinity). The proofs are easy generalisations of the previous ones.

Secondly ℝf_{∗}(𝕃) is given by the ”almost Frobenius-invariants” on ℝf_{∗}(𝕃 ⊗ A_{inf}(Ō_{X})). We claim that (if R is strictly henselian)

R^{ν}f_{∗}(𝕃 ⊗ Ō_{X}) ≈ R^{ν}f_{∗}(𝕃) ⊗ R̄

(and similarly with coefficients A_{inf}(R̄)):

Reduce to 𝕃 annihilated by p, s = 1. Then choose the highest index ν for which M = R^{ν}f_{∗}(𝕃 ⊗ R(Ō_{X})) is not almost isomorphic to (almost Φ -invariants in M) ⊗ R(R̄). Then M = M/ξM is almost finitely presented over R̄/pR̄, and Frobenius is an almost isomorphism on M. Equivalently M = \underleftarrow{lim}(M, Φ), and Frobenius induces an isomorphism M/p^{1/p} · M ≈ M. Thus everything can be formulated in terms of M. Furthermore we can replace R̄ by R_{∞} and M by M_{∞} = Gal(R̄/R_{∞}) -invariants, as M ≈ M_{∞} ⊗_{R_{∞}} R̄. M_{∞} is an almost finitely presented R_{∞}/pR_{∞} -module with Frobenius Φ : M_{∞}/p^{1/p}M_{∞} ≈ M_{∞}. In the appendix it is shown that for some almost étale covering Ā of R_{∞}/pR_{∞} M_{∞} ⊗_{R_{∞}} Ā ≈ (almost Φ -invariants) ⊗A. Thus be descending induction over ν we can find such an A almost étale over R̄/pR̄ with ℝ^{ν}f_{∗}(𝕃 ⊗ Ō_{X}) ⊗R̄ Ā ≈ M_{0}^{ν} ⊗_{𝔽_{p}} Ā for certain 𝔽_{p} -

vectorspaces M_{0}^{ν}. Ā lifts to an almost étale covering Â of R̄^{∧}, the p-adic completion of R̄. Temporarily working over Â we see that the M_{0}^{ν} are the direct images of 𝕃 on the topos X^{0} ⊗_{Y} Â, the analogue of the topos X^{0} with the base V̄ replaced by Â. These map to the étale images, and these maps are isomorphisms (by duality, or basechange to discrete valuation rings). If we replace Â by a Galois-covering the

Galois-group Gal(Â/R̄^{∧}) then operates trivially on the M_{0}^{ν}, as it operates trivially on étale direct images. Hence by taking invariants

R^{ν}f_{∗}(𝕃 ⊗ Ō) ≅ M_{0}^{ν} ⊗ R̄,

and comparison is shown.

Obviously there are consequences for somparison to crystalline direct images. Finally the Fontaine-Lafaille theory from [Fa3] carries over without much change: Consider a filtered Frobenius-crystal (E, F^{i}(E)) which locally is the direct sum of shifted filtered modules O_{X}/p^{s}O_{X}{e_{i}}, with shifts 0 ≤ e_{i} ≤ p − 2. Assume that Frobenius Φ on F^{i} is divisible by p^{i}, that is there exist (assuming we evaluate on a PD-thickening with Frobenius-lift) Φ_{i} : F^{i}(E) ⊗_{ρ} O_{X} → E with Φ_{i−1} | F^{i} = p · Φ^{i}. Furthermore assume that for a local filtered basis m_{i}, of degree e_{i}, the Φ^{e_{i}}(m_{i}) also form a basis. Then 𝕃 = Hom_{F,Φ}(E ⊗ B_{cr}^{+}(R)), B_{cr}^{+}(R) ⊗ ℚ_{p}/ℤ_{p}) defines a locally constant étale sheaf on X ⊗_{V} K̄. This is shown as in [Fa3] (and uses in fact little of the structure of R̄). Finally sometimes such crystals can be constructed as crystalline direct images under f : X → Y. However for this one at least needs more stringent restrictions on f (”of relative Cartier-type” in Kato’s terminology, see also [Fa3]), and also criteria under which the derived direct image splits into the direct sum of its cohomology groups.

6^{∗}. Appendix: Some more almost mathematics

The ring R_{∞} = lim_{n→∞}R_{n} is the increasing union of noetherian rings R_{n}, such that for a sequence ε_{n} → 0 we have

p^{ε_{n}} · T or^{R_{n}}, (R_{n+m}, any R_{n} -module) = 0.

If M is any almost finitely presented R_{∞} -module, then for each α > 0, M is p^{α} -isomorphic to M_{n} ⊗_{R_{n}} R_{∞} for some finitely presented R_{n} -module. Resolve M_{n} by finitely generated free R_{n} -module, to get by base extension a complex of finitely generated free R_{∞}-modules which up to p^{α+ε_{n}} resolves M. Similarly for a map M → N of such modules. With these methods one sees easily that the almost finitely presented R_{∞} -modules form an abelian subcategory of all R_{∞} -modules, which is closed under forming T or_{i}^{R_{∞}}(M, N) or Ext_{i}^{R_{∞}}(M, N). The same holds for R_{∞}/pR_{∞} (and by the way all this remains true for R̄). Form now on assume that R is strictly henselian.

We now show that for an almost finitely presented R_{∞}/pR_{∞} -module M with a Frobenius-linear almost isomorphism Φ : M/p^{1/p}M ≈ M, for an almost faithfully flat almost étale covering A of R_{∞}/pR_{∞}, M ⊗_{R_{∞}} A ≈ M_{0} ⊗_{𝔽_{p}} A with M_{0} = Hom(M; M)^{Φ} the almost invariants under Frobenius.

We proceed by induction over dim(R) and may assume that the assertion already holds for the strict localisation in any prime 𝔭 ⊆ R, 𝔭 ≠ M. By descent it follows that the localisations M_{𝔭} are almost projective on (R_{∞}/pR_{∞})_{𝔭}. Also the assertion has already been shown for d = 0(R a discrete valuation ring), so we may assume that dim(R) ≥ 2. Replace M by Hom(M, M). Then M_{0} = M^{Φ}, frobenius is surjective on M, and M has no M-torsion. If M = \underleftarrow{lim}(M, Φ), then M has no ξ -torsion, M/ξM ≅ M, and Frobenius is bijective in the R_{∞} -module M. Next consider N = Ext_{R_{∞}}^{1}(M, R_{∞}). Then frobenius is also bijective on N, and N /ξN injects into Ext_{R_{∞}}^{1}(M, R_{∞}/pR_{∞}) = Ext_{R_{∞}}^{1}/pR_{∞}(M, R_{∞}/pR_{∞}), which is almost finitely presented and almost M -torsion. By a previously used argument (using d > 0 and λ(ξ^{α}N /ξN)) N /ξN ≈ 0, and it then follows that the maps any map M → R_{∞}/ξ^{n} · R_{∞} almost lifts to R_{∞}/ξ^{n+1} · R_{∞}, and thus in the limit to R_{∞}.

Next consider the cokernel N of

M ⊗_{R_{∞}} Hom_{R_{∞}}(M, R_{∞}) → End_{R_{∞}}(M).

Then Frobenius is almost isomorphic on N. Also N /ξ · N almost injects into the cokernel of

M ⊗_{R_{∞}} Hom_{R_{∞}}(M, R_{∞}/pR_{∞}) → End_{R_{∞}}(M).

This cokernel is almost finitely presented and almost M-torsion. By the usual argument N /ξ ·N almost vanishes, and so does N (lifting modulo ξ^{n} step by step). Hence for each α > 0, ξ^{α} · id_{N} lies in the image, thus is of the form ∑ x_{i} ⊗ y_{i} with x_{i} ∈ M, y_{i} ∈ Hom_{R_{∞}}(M, R_{∞}). Thus ξ^{d} · z = ∑ y_{i}(z) · x_{i} for each z ∈ M, and M is almost projective.

Each localisation M_{𝔭} has finite rank, for 𝔭 ≠ 𝔪. If r denotes the maximum of these ranks for 𝔭 a minimal prime divisor of pR, then M and 𝔪 have rank ≤ r. For n > r(p − 1) consider the cokernel N of the map

𝔪 ⊗ S^{n−p}(𝔫) → S^{n}(𝔫)
m ⊗ m′ → m^{p} · m!

Then N /ξN injects into the corresponding cokernel for M, which is M -torsion and almost finitely presented. Also Frobenius is an isomorphism on N, hence as usual N /ξN ≈ 0, and the map above is almost surjective (N ≈ 0). Thus

A = S^{·}(N)/⟨m^{p} − Φ(m)⟩

is almost generated by symmetric powers S^{n}(N) (n ≤ r(p − 1)), hence almost finitely presented. Also Frobenius is almost isomorphic on A, thus A itself is almost projective (same argument as for N). Finally A defines an almost étale covering of A_{inf}(R_{∞}) which is almost faithfully flat (A_{inf}(R_{∞}) is a direct summand). The universal map N → A is Φ invariant. Passing to a Galois-hull of A we obtain the desired A.

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