Integral Motives and Special Values of Zeta Functions

James S. Milne and Niranjan Ramachandran

Abstract. For each field k, we define a category of rationally decomposed mixed motives with ℤ-coefficients. When k is finite, we show that the category is Tannakian, and we prove formulas relating the behaviour of zeta functions near integers to certain Ext groups.

Contents

1. The Realization Categories 3 2. Adèlic Cohomology 7 3. An Adèlic Tate Conjecture 9 4. The Adèlic Fibre Functor 12 5. The Category of Motives 14 6. Tannakian Properties 22 7. Some Spectral Sequences 27 8. Homological Algebra in the Category of Motives 32 9. Extensions and Zeta Functions: Local Case 37 10. Extensions and Zeta Functions: Global Case 45 References 51

Introduction. For each field k, there is conjecturally a category of mixed isomotives whose full subcategory of semisimple objects is the category M(k; ℚ) of pure isomotives. In this paper, we define a ℤ-category M(k; ℤ) of motives whose quotient by its subcategory of torsion objects is M(k; ℚ). Thus, M(k; ℤ) is the abelian category of mixed motives over k whose weight filtrations split modulo torsion. As Grothendieck observed, when k is finite, the weight filtrations always split modulo torsion, and so, in this case, M(k; ℤ) is the full category of mixed motives over k.

The category M(k; ℤ) depends on our choice of M(k; ℚ). For definiteness, take M(k; ℚ) to be the category of isomotives based on the smooth projective varieties over k whose Künneth projectors are algebraic with the correspondences taken to be the numerical equivalence classes of algebraic classes. This category is abelian and semisimple (Jannsen 1992), hence Tannakian (Deligne 1990), and when we assume^{1} numerical equivalence coincides with homological equivalence, it admits canonical l-adic fibre functors for each l. The category M(k; ℤ) we construct is noetherian, abelian, and, at least when k is finite, Tannakian.

May 22, 2002; submitted version.

The first author received support from the National Science Foundation and the second author from MPIM (Bonn) and a GRB Summer Grant (UMD).

The Ext groups in the full category M(k; ℤ) seem to be pathological, but those in the subcategory M^{+}(k; ℤ) of effective motives are of interest. When k is finite, we show that the Tate conjecture implies^{2} relations between these Ext groups and the behaviour of zeta functions near integers (Theorems 10.1, 10.5). Moreover, to a smooth projective variety V satisfying a certain degeneration condition on its crystalline cohomology, we are able to attach “Weil” motivic cohomology groups H_{mot}^{i}(V, ℤ(r)) satisfying the spectral sequence

H^{i}(Γ_{0}, H_{mot}^{j}(V̄ , ℤ(r)) ⟹ H_{mot}^{i+j}(V, ℤ(r))  (1)

posited by Lichtenbaum (2002), and deduce from Theorem 10.5 that

ζ(V, s) ∼ ±χ^{×}(V, ℤ(r)) · q^{χ(V,O_{V} ,r)} · (1 − q^{r−s})^{ρ} as s → r  (2)

(Theorem 10.7). Here Γ_{0} is the subgroup of Gal(𝔽/𝔽_{q}) generated by the Frobenius element (so Γ_{0} ≅ ℤ), χ^{×}(V, ℤ(r)) is the alternating product of the orders of the cohomology groups of a complex

⋯ → H_{mot}^{i}(V, ℤ(r)) → H_{mot}^{i+1}(V, ℤ(r)) → ⋯

arising from (1), χ(V, O_{V} , r) is as in Milne 1986a, and ρ = ∑(−1)^{i}i·rank H_{mot}^{i}(V, ℤ(r)).

The category M(k; ℤ) is defined in §5 as a full subcategory of the fibre product of the ℚ-category M(k; ℚ) with a certain ℤ̂-category of realizations. The construction requires that we show that the various l-adic fibre functors on M(k; ℚ) define an adèlic fibre functor — the proof of this depends crucially on the theorem of Gabber (1983). In §4, we show that the usual Tate conjecture implies an adèlic version. For example, for a variety over a finite field, if the l-adic Tate conjecture holds for a single l, then it holds l-integrally for almost all l.

The category M(𝔽_{q}; ℤ) is shown to be Tannakian in §6, and the main results on Exts and motivic cohomology groups are obtained in §10 after preliminaries in §§7–9.

This article is part of a project whose ultimate goal is to define a triangulated category of motives with t-structure whose heart is the category M(k; ℤ) defined above, and to show that the Tate conjecture for smooth projective varieties implies the formula (2) for arbitrary varieties over 𝔽_{q}.

We note that our approach to motives is opposite to that of other authors (Hanamura 1995, 1999; Levine 1998; Voevodsky 2000) who define a triangulated category of motives with the goal of putting a t-structure on it whose heart will be the abelian category of motives. While our approach can be expected to give the full category of mixed motives only when the ground field is 𝔽_{q} or 𝔽, we do expect to obtain the correct p-structure, as illustrated by (2). Most of the difficulties in the present work involve the p-part in characteristic p.

1See (5.26) for an alternative choice of M(k; ℚ) that does not require any unproven conjectures.

2When M(k; ℚ) is as in (5.26), these relations are hold unconditionally.

Notations. Throughout, l is a prime number, p is the characteristic (or characteristic exponent) of the ground field, and ℓ is a prime number ≠ p. The l-adic valuations | |_{l} and ord_{l} are normalized so that |l|_{l} = l^{−1} and ord_{l}(l) = 1. The expression “for almost all l” means “for all but (possibly) a finite number of l”.

The profinite completion of ℤ is denoted ℤ̂ and 𝔸_{f} = ℚ ⊗ ℤ̂. Thus, ℤ̂ = ∏ ℤ_{l}, and we let ℤ^{p} = ∏_{l≠p} ℤ_{l} and 𝔸_{f}^{p} = ℚ ⊗ ℤ^{p}. When p = 1, ℤ^{p} = ℤ̂ and 𝔸_{f}^{p} = 𝔸_{f}.

We use M_{m} and M^{(m)} respectively to denote the kernel and cokernel of multiplication by m on M. For a prime l, M(l) = ⋃ M_{l^{n}} is the l-primary component of M, and M_{tors} = ⋃ M_{m} is the torsion subobject.

For a module M endowed with an action of a group Γ, M^{Γ} (resp. M_{Γ}) is the largest submodule (resp. quotient module) of a Γ-module M on which Γ acts trivially.

Lattice always means full lattice: when R is a ring and V is a free ℚ ⊗ R-module of finite rank, an R-lattice in V is a finitely generated R-submodule Λ of V such that the inclusion Λ ↪ V induces an isomorphism ℚ ⊗ Λ → V.

For a field k, k^{al} denotes an algebraic closure of k, k^{sep} a separable (algebraic) closure of k, and k^{pf} a perfect closure of k (for example, k^{pf} = k^{p^{−∞}} ⊂ k^{al}).

For a perfect field k of characteristic p, W (k) denotes the ring of Witt vectors with coefficients in k, and B(k) denotes the field of fractions of W (k). The unique lift of the automorphism x ↦ x^{p}: k → k to W (k) is denoted σ. We let A denote the skew polynomial ring W [F, σ] (polynomials in F with the relation F · a = σ(a) · F, a ∈ W).

Generally, rings are assumed to be commutative. For a ring R, Mod(R) is the category of R-modules and Modf(R) the category of finitely presented R-modules, and for a field k, Vec(k) is the category of finite dimensional k-vector spaces. We usually abbreviate M ⊗_{R} S to M_{S}. For an R-linear category (alias, R-category) M, M_{S} denotes the category with the same objects as M, but with Hom_{M_{S}}(X, Y) = Hom_{M}(X, Y)_{S}.

For an affine group scheme G over a noetherian ring R, Rep(G; R) denotes the category of linear representations of G on finitely generated R-modules.

Given functors F: A → C and G: B → C, the fibre product category A ×_{C} B has as objects the triples (A, B, γ) with A an object of A, B an object of B, and γ: F (A) → G(B) an isomorphism; the morphisms are the pairs a: A → A′ and b: B → B′ giving rise to a commutative diagram in C.

A variety V is a geometrically reduced separated scheme of finite type over a field. The group of algebraic cycles of codimension r on V is denoted Z^{r}(V), and Z_{∼}^{r}(V) is the quotient of Z^{r}(V) by some adequate relation ∼. We let H^{∗}(V, F) = ⊕_{i}H^{i}(V, F).

An equivalence class containing x is denoted [x]. We also use [S] to denote the cardinality of a finite set S. For a homomorphism f: M → N of abelian groups whose kernel and cokernel are finite, z(f) is defined to be [Ker(f)]/[Coker(f)].

Isomorphisms are denoted ≈ and canonical isomorphisms ≅.

1. The Realization Categories

The categories R away from p. Throughout this subsection, k is a field of characteristic exponent p, and R is a topological ring, for example, R = ℤ, ℤ_{ℓ}, ℚ_{ℓ}, ℤ^{p}, or 𝔸_{f}^{p} with its natural topology (discrete for ℤ).

Definition 1.1. (a) For a profinite group Γ, R(Γ; R) is the category of continuous linear representations of Γ on finitely presented R-modules.

(b) When k is finitely generated over the prime field,

R(k; R) = R(Γ; R), Γ = Gal(k^{sep}/k).

Otherwise, R(k; R) = \underrightarrow{lim} R(k′; R) where k′ runs over the subfields of k finitely generated over the prime field.

Remark 1.2. Let Γ be a profinite group. Two continuous linear representations ρ_{1} and ρ_{2} of open subgroups U_{1} and U_{2} of Γ on an R-module M are said to be related if they agree on an open subgroup of U_{1} ∩ U_{2}. This is an equivalence relation, and an equivalence class will be called a germ of a continuous linear representation of Γ. A morphism (M, [ρ_{U}]) → (M′, [ρ′_{U′}]) of germs is an R-linear map α: M → M′ that is equivariant for some open subgroup of U ∩ U′. When k is a separable closure of a field k_{0} finitely generated over the prime field, an object of R(k; R) can be identified with a germ of a continuous linear representation of Gal(k/k_{0}).

Lemma 1.3. For each object M of R(Γ; ℤ_{ℓ}), there is an open subgroup U ⊂ Γ such that M decomposes into the sum of a torsion object and a torsion-free object in R(U; ℤ_{ℓ}). Consequently, when k is separably closed, for each M in R(k; ℤ_{ℓ}), M_{tors} is a direct summand of M.

Proof. Let M = M_{tors} ⊕ M_{1} as ℤ_{ℓ}-modules, and let γ ∈ Γ act on M as the matrix ( \begin{matrix}a(γ) & b(γ) \\ 0 & c(γ) \end{matrix} ). The map γ ↦ a(γ) is the homomorphism describing the action of Γ on M_{tors}. After replacing Γ with an open subgroup U, we may suppose this action to be trivial. The map γ ↦ c(γ) defines an action of U on M_{1}, and hence an action of U on Hom(M_{1}, M_{tors}), which, again, we may suppose to be trivial. Then γ ↦ b(γ) is a homomorphism U → Hom(M_{1}, M_{tors}), and after we replace U with the kernel of this map, M = M_{tors} ⊕ M_{1} will be a decomposition of U-modules. □

For M in R(Γ; ℤ_{ℓ}), set M^{∨} = Hom_{ℤ_{ℓ}}(M, ℤ_{ℓ}) with γ ∈ Γ acting according to the rule (γf)(m) = f(γ^{−1}m). We say M is reflexive if the canonical map M → M^{∨∨} is an isomorphism.

Proposition 1.4. (a) An object of R(Γ; ℤ_{ℓ}) is reflexive if and only if it is torsionfree.

(b) Every object of R(Γ; ℤ_{ℓ}) is a quotient of a reflexive object.

Proof. (a) As M^{∨} is torsion-free, the condition is necessary, and it is obviously sufficient.

(b) If M decomposes in R(Γ; ℤ_{ℓ}) into the direct sum of a torsion module M_{t} with trivial Γ-action and a torsion-free module M_{1}, then M will be the quotient of ℤ_{ℓ}^{r} ⊕M_{1} for some r. According the lemma, there is an open subgroup U of Γ such that M has such a decomposition in R(U; ℤ_{ℓ}), and hence there will be a surjection ℤ_{ℓ}^{r} ⊕M_{1} → M in R(U; ℤ_{ℓ}) with M_{1} free. Now

Ind_{U}^{Γ}(ℤ_{ℓ}^{r} ⊕ M_{1}) ↠ Ind_{U}^{Γ}(M) \underrightarrow{φ↦∑_{s∈Γ/U} sφ(s^{−1})} M

realizes M as the quotient of a torsion-free Γ-module. Here Ind_{U}^{Γ}(N) denotes the induced module. □

Remark 1.5. (a) The category R(k; ℚ_{ℓ}) is denoted \underline{Tate}(k) in Saavedra 1972, VI A4.2, A4.3, and its objects are called Tate modules. It is a neutral Tannakian category over ℚ_{ℓ}.

(b) The functor M ↦ (M_{ℓ})_{ℓ≠p}: R(Γ; ℤ^{p}) → ∏_{ℓ≠p} R(Γ; ℤ_{l}) is exact, full, and faithful, with essential image the objects (M_{ℓ})_{ℓ≠p} such that dim_{𝔽_{ℓ}} M_{ℓ}/ℓM_{ℓ} is bounded.

The categories R at p. An F -isocrystal over a perfect field k is a finite dimensional B(k)-vector space M endowed with a σ-linear bijection F_{M}: M → M. With the obvious structures, the F -isocrystals over k form a Tannakian category Isoc(k) over ℚ_{p}, and the forgetful functor is a B(k)-valued fibre functor (Saavedra 1972, VI 3.2). The identity object is 11 = (B(k), σ), and the Tate object is 𝕋 = (B(k), p^{−1} · σ).

An F -isocrystal (M, F_{M}) is effective if F_{M} stabilizes a W (k)-lattice in M. Let Isoc^{+}(k) denote the full subcategory of Isoc(k) whose objects are the effective F - isocrystals. Every object of Isoc(k) is of the form X ⊗𝕋^{⊗m} with X effective. In other words, for each (M, F_{M}) in Isoc(k), there exists an m ∈ ℤ such that p^{m}F_{M} stabilizes a lattice (Saavedra 1972, VI 3.1.3, 3.2.1).

An F -crystal^{3} over a perfect field k is a finitely generated W (k)-module Λ endowed with a σ-linear map F_{Λ} : Λ → Λ such that the kernel of F_{Λ} is torsion. The F -crystals over k form an abelian tensor category Crys^{+}(k) over ℤ_{p}. For an F -crystal (Λ, F_{Λ}), (Λ, F_{Λ})_{ℚ} =_{df} (Λ_{ℚ}, F_{Λ} ⊗ 1_{ℚ}) is an F -isocrystal, and the functor

(Λ, F_{Λ}) ↦ (Λ, F_{Λ})_{ℚ}: Crys^{+}(k)_{ℚ} → Isoc(k)

is fully faithful with essential image Isoc^{+}(k).

Let 𝕃 = (W (k), p · σ) be the Lefschetz F -crystal. The functor

− ⊗ 𝕃: Crys^{+}(k) → Crys^{+}(k)

is faithful (full and faithful on torsion-free objects). We define Crys (k) to be the tensor category obtained from Crys^{+}(k) by inverting 𝕃. Thus, the objects of Crys (k) are pairs (Λ, m) with Λ an F -crystal and m ∈ ℤ, and

Hom((Λ, m), (Λ′, m′)) = \underrightarrow{lim}_{N≥m,m′} Hom(Λ ⊗ 𝕃^{⊗N−m}, Λ′ ⊗ 𝕃^{⊗N−m′}).

The tensor product is defined by

(Λ, m) ⊗ (Λ′, m′) = (Λ ⊗_{W (k)} Λ′, m + m′).

The Tate object 𝕋 of Crys(k) is (11, 1).

The functor

Λ ↦ (Λ, 0): Crys^{+}(k) → Crys(k)

is faithful, and it is full on torsion-free objects. The forgetful functor defines an equivalence of the full subcategory of Crys(k) of torsion objects with the category of W (k)-modules of finite length.

Definition 1.6. Let k be a field of characteristic p ≠ 0. If k is finitely generated over 𝔽_{p},

R(k; ℚ_{p}) = Isoc(k^{pf}), R(k; ℤ_{p}) = Crys(k^{pf} ).

3Some authors use “crystal” to mean free crystal, i.e., an M that is torsion-free (as a W (k)- module).

Otherwise,

R(k; ℤ_{p}) = \underrightarrow{lim}_{k′⊂k} R(k′; ℤ_{p}), R(k; ℚ_{p}) = \underrightarrow{lim}_{k′⊂k} R(k′; ℚ_{p}),

where the limits are over the subfields of k finitely generated over 𝔽_{p}. We define R^{+}(k; ℚ_{p}) and R^{+}(k; ℤ_{p}) = \underrightarrow{lim}_{k′⊂k} Crys^{+}(k^{′pf}) similarly.

Lemma 1.7. For each object M of R^{+}(k; ℤ_{p}), there is an n such that M ⊗𝕃^{⊗n} decomposes into a direct sum of a torsion object and a torsion-free object. Consequently, for each M in R(k; ℤ_{p}), M_{tors} is a direct summand of M.

Proof. Let M = M_{tors} ⊕ M_{1} as W (k^{′pf})-modules, and let F_{M} act as ( \begin{matrix}a & b \\ 0 & c \end{matrix}). Then F_{M⊗𝕃^{⊗n}} acts as ( \begin{matrix}p^{n}a & p^{n}b \\ 0 & p^{n}c \end{matrix} ). Choose n so that p^{n}M_{tors} = 0, and then p^{n}b = 0. This proves the first statement, and the second follows because − ⊗ 𝕃 is an equivalence of categories on R(k; ℤ_{p}). □

Let (Λ, F) be an F -crystal. For some N there exists a σ^{−1}-linear map V : Λ → Λ such that F V = p^{N} = V F. Let Λ′ = Hom_{W (k)}(Λ, W (k)) and define F′: Λ′ → Λ′ by (F′f)(λ) = f(V λ). For an object (Λ, F, m) of R(k; ℤ_{p}), define (Λ, F, m)^{∨} = (Λ′, F′, N − m). We say that (Λ, F, m) is reflexive if the canonical map (Λ, F, m) → (Λ, F, m)^{∨∨} is an isomorphism.

Proposition 1.8. (a) An object of R(k; ℤ_{p}) is reflexive if and only if it is torsionfree.

(b) Every object of R(k; ℤ_{p}) is a quotient of a reflexive object.

Proof. (a) is a obvious. After (1.7) it suffices to prove (b) for a torsion F -crystal N. Such an N is a quotient M → N of a free finitely generated W -module M. Endow M with the F -crystal structure such that F_{M} = σ. After M → N has been tensored with a high power of 𝕃, it will be a morphism of F -crystals. □

Remark 1.9. (a) There is a canonical functor \underrightarrow{lim} Isoc(𝔽_{q}) → Isoc(𝔽), which is faithful and essentially surjective (Demazure 1972, p. 85) but not full (W (𝔽) ≠ \underrightarrow{lim} W (𝔽_{q})).

(b) Ekedahl 1986, p36, defines a virtual F -crystal to be a triple (M, F, Λ) with (M, F) an isocrystal and Λ a W -submodule of M such that B(k) · Λ = M; a virtual F -crystal is of finite type if Λ is finitely generated. The functor ((Λ, F_{Λ}), m) ↦ (Λ_{ℚ}, F_{Λ}⊗p^{−m}, Λ) defines an equivalence of the full subcategory of Crys(k) of torsionfree objects with the category of virtual F -crystals of finite type.

The categories R in general.

Definition 1.10. For a field k of characteristic p ≠ 0,

R(k; ℤ̂) = R(k; ℤ^{p}) × R(k; ℤ_{p}), R(k; 𝔸_{f}) = R(k; 𝔸_{f}^{p}) × R(k; ℚ_{p})
R^{+}(k; ℤ̂) = R(k; ℤ^{p}) × R^{+}(k; ℤ_{p}), R^{+}(k; 𝔸_{f}) = R(k; 𝔸_{f}^{p}) × R^{+}(k; ℚ_{p})

For a field of characteristic zero,

R(k; ℤ̂) = R(k; ℤ^{p}), R(k; 𝔸_{f}) = R(k; 𝔸_{f}^{p}).

Note that R(k; 𝔸_{f}) ≅ R(k; ℤ̂)_{ℚ}. For notational convenience, we sometimes use R^{+}(k; ℤ_{l}) to denote R(k; ℤ_{l}) when l ≠ p.

The motivic subcategory of R(k; ℚ_{ℓ}). Let M(k; ℚ_{ℓ}) be the strictly full subcategory of R(k; ℚ_{ℓ}) whose objects are subquotients of a Tate twist of a tensor power of an ℓ-adic étale cohomology group of a smooth projective variety over k. When k = 𝔽_{q}, M(k; ℚ_{ℓ}) consists of the semisimple representations of Γ = Gal(k^{al}/k) for which the eigenvalues of the Frobenius generator of Γ are Weil q-numbers: the Weil conjectures (proved by Grothendieck and Deligne) imply that every object of M(k; ℚ_{ℓ}) has this property, but all such representations arise already in the cohomology of abelian varieties (cf. Milne 1994, 3.7). When k is a number field, Conjecture 1 of Fontaine and Mazur (1995) suggests a description of M(k; ℚ_{ℓ}).

2. Adèlic Cohomology

Modules over 𝔸_{f}^{p} and lattices. Recall that ℤ^{p} = ∏_{ℓ≠p} ℤ_{ℓ} and 𝔸_{f}^{p} is the restricted direct product 𝔸_{f}^{p} = ∏_{ℓ≠p}(ℚ_{ℓ}, ℤ_{ℓ}).

2.1. Endow (𝔸_{f}^{p})^{m} with its topology as the restricted direct product of the spaces ℚ_{ℓ}^{m} relative to the subspaces ℤ_{ℓ}^{m}. Every 𝔸_{f}^{p}-linear bijection (𝔸_{f}^{p})^{m} → (𝔸_{f}^{p})^{m} is bicontinuous. This allows us to endow any free 𝔸_{f}^{p}-module M of finite rank with a topology by choosing an isomorphism α: M → (𝔸_{f}^{p})^{m}.

2.2. Let M be a free 𝔸_{f}^{p}-module of finite rank m, so that M_{ℓ} =_{df} M ⊗_{𝔸_{f}^{p}} ℚ_{ℓ} is a ℚ_{ℓ}-vector space of dimension m for all ℓ ≠ p. A lattice in M is a finitely generated ℤ^{p}-submodule Λ such that the inclusion Λ ↪ M induces an isomorphism Λ_{𝔸_{f}^{p}} → M; then Λ_{ℓ} =_{df} Λ⊗_{ℤ^{p}}ℤ_{ℓ} is a ℤ_{ℓ}-lattice in M_{ℓ} for all ℓ ≠ p, and Λ is the free ℤ^{p}-submodule spanned by a basis for M. Clearly, every lattice in M is compact and open, and so any two lattices Λ and Λ′ in M are commensurable, that is, both quotients Λ/Λ ∩ Λ′ and Λ′/Λ ∩ Λ′ are finite (hence, Λ_{ℓ} = Λ′_{ℓ} for almost all ℓ). Let M_{0} be a ℚ-structure on M. There is a canonical one-to-one correspondence between the lattices Λ in M and the lattices Λ_{0} in M_{0}, namely,

Λ ↦ Λ ∩ M_{0}, Λ_{0} ↦ its closure in M.

2.3. Let (M_{ℓ})_{ℓ≠p} be a family of vector spaces over the fields ℚ_{ℓ}, all of the same finite dimension. Two families (Λ_{ℓ})_{ℓ≠p} and (Λ′_{ℓ})_{ℓ≠p} of lattices in the M_{ℓ} will be said to be equivalent if Λ_{ℓ} = Λ′_{ℓ} for almost all ℓ. To give a free 𝔸_{f}^{p}-module of finite rank is the same as to give a family (M_{ℓ})_{ℓ≠p} together with an equivalence class of families (Λ_{ℓ})_{ℓ≠p}: given a family (M_{ℓ}, Λ_{ℓ})_{ℓ≠p}, let M = ∏(M_{ℓ}, Λ_{ℓ}); given M, choose an isomorphism M → (𝔸_{f}^{p})^{m}, and let Λ_{ℓ} be the inverse image of ℤ_{ℓ}^{m}. Let M and M′ be the free 𝔸_{f}^{p}-modules of finite rank defined by systems (M_{ℓ}, Λ_{ℓ})_{ℓ≠p} and (M′_{ℓ}, Λ′_{ℓ})_{ℓ≠p} respectively. A family of ℚ_{ℓ}-linear maps α_{ℓ}: M_{ℓ} → M′_{ℓ} defines an 𝔸_{f}^{p}-linear map α: M → M′ if and only if α_{ℓ}(Λ_{ℓ}) ⊂ Λ′_{ℓ} for almost all ℓ, in which case α is injective if and only if α_{ℓ} is injective for all ℓ and it is surjective if and only if α_{ℓ} is surjective for all ℓ and maps Λ_{ℓ} onto Λ′_{ℓ} for almost all ℓ.

2.4. An 𝔸_{f}^{p}-linear map α: M → M′ of free 𝔸_{f}^{p}-modules of finite rank will be said to be of constant rank if the rank of α_{ℓ} =_{df} α ⊗_{𝔸_{f}^{p}} ℚ_{ℓ} is independent of ℓ. Then the kernel, cokernel, image, and co-image of α are again free 𝔸_{f}^{p}-modules of finite rank.

Adèlic cohomology (characteristic p ≠ 0). Throughout this subsection, all varieties are smooth and projective over a separably closed field k of characteristic p ≠ 0. For such a variety V, H^{r}(V, −) denotes an étale cohomology group and Λ_{ℓ}^{i}(V, r) denotes the image of H^{i}(V, ℤ_{ℓ}(r)) in H^{i}(V, ℚ_{ℓ}(r)). Thus,

Λ_{ℓ}^{i}(V, r) ≅ H^{i}(V, ℤ_{ℓ}(r))/{torsion}.

Proposition 2.5. (a) The torsion subgroup of H^{i}(V, ℤ_{ℓ}) is finite for all ℓ,

and is zero for almost all ℓ.

(b) Assume V is connected of dimension d_{V} and fix a prime ℓ ≠ p. If the cohomology groups H^{j}(V, ℤ_{ℓ}) of V are torsion-free for all j, then the pairing

H^{i}(V, ℚ_{ℓ}) × H^{2d_{V} −i}(V, ℚ_{ℓ}(d_{V})) → H^{2d_{V}} (V, ℚ_{ℓ}(d_{V})) ≅ ℚ_{ℓ}

induces a perfect^{4} pairing on the cohomology groups with coefficients in ℤ_{ℓ}.

(c) Let W be a second variety over k and fix a prime ℓ ≠ p. If the groups H^{i}(V, ℤ_{ℓ}) and H^{j}(W, ℤ_{ℓ}) are torsion-free for all i and j, then the Künneth isomorphism

H^{r}(V × W, ℚ_{ℓ}) ≅ ⨁_{i+j=r} H^{i}(V, ℚ_{ℓ}) ⊗ H^{j}(W, ℚ_{ℓ})

induces an isomorphism on the groups with coefficients in ℤ_{ℓ}.

Proof. (a) The finiteness of the torsion subgroup of H^{i}(V, ℤ_{ℓ}) is a standard result, and the rest of statement (a) is proved in Gabber 1983.

(b) The pairing on the cohomology groups with coefficients in ℤ_{ℓ} is the unique pairing for which the diagrams

H^{i}(V, ℤ_{ℓ}) × H^{2d_{V} −i}(V, ℤ_{ℓ}(d_{V})) ⟶  H^{2d_{V}} (V, ℤ_{ℓ}(d_{V})) ≅ ℤ_{ℓ}
↓  ↓  ↓
H^{i}(V, ℤ/ℓ^{m}ℤ)×H^{2d_{V} −i}(V, (ℤ/ℓ^{m}ℤ)(d_{V})) ⟶ H^{2d_{V}} (V, (ℤ/ℓ^{m}ℤ)(d_{V})) ≅ ℤ/ℓ^{m}ℤ

commute for all m. The lower pairing is perfect and the vertical maps are the cokernels of multiplication by ℓ^{m} on H^{i}(V, ℤ_{ℓ}) and H^{2d_{V} −i}(V, ℤ_{ℓ}(d_{V})) because we are assuming there is no torsion. This implies (even with m = 1) that the discriminant of the top pairing is an ℓ-adic unit, and so the top pairing is also perfect.^{5}

(c) This is a standard result. □

Definition 2.6. For V as above, H^{i}(V, 𝔸_{f}^{p}(r)) is the restricted direct product of the spaces H^{i}(V, ℚ_{ℓ}(r)) with respect to the lattices Λ^{i}(V, r). Equivalently,

H^{i}(V, 𝔸_{f}^{p}(r)) = H^{i}(V, ℤ^{p}(r)) ⊗ ℚ

where H^{i}(V, ℤ^{p}(r)) = \underleftarrow{lim}_{(n,p)=1} H^{i}(V, μ_{n}^{⊗r}).

Remark 2.7. (a) Because the Betti numbers of a smooth projective variety are independent of ℓ, H^{i}(V, 𝔸_{f}^{p}) is a free 𝔸_{f}^{p}-module of finite rank. A regular map α: V → W defines an 𝔸_{f}^{p}-linear map α^{∗}: H^{i}(W, 𝔸_{f}^{p}) → H^{i}(V, 𝔸_{f}^{p}).

(b) For a connected V, the proposition implies the following: the pairings in (b) of the proposition define a perfect pairing

4A bilinear form M ×N → R of R-modules is nondegenerate if its left and right kernels are zero, and perfect if the maps M → Hom_{R}(N, R) and N → Hom_{R}(M, R) it defines are isomorphisms.

5We thank M. Nori for this proof, which is simpler than our original.

H^{i}(V, 𝔸_{f}^{p}) × H^{2d_{V} −i}(V, 𝔸_{f}^{p}(d_{V})) → H^{2d_{V}} (V, 𝔸_{f}^{p}(d_{V})) ≅ 𝔸_{f}^{p}

of free 𝔸_{f}^{p}-modules of finite rank.

In other words, Poincaré duality holds for 𝔸_{f}^{p}-cohomology. This implies that the map α^{∗} in (a) has an 𝔸_{f}^{p}-linear adjoint

α_{∗}: H^{j}(V, 𝔸_{f}^{p}(d_{V})) → H^{j+2d_{W} −2d_{V}} (W, 𝔸_{f}^{p}(d_{W})), j = 2d_{V} − i.

(c) Part (c) of the theorem implies the following: there is a canonical isomorphism of free 𝔸_{f}^{p}-modules of finite rank

H^{r}(V × W, 𝔸_{f}^{p}) ≅ ⨁_{i+j=r} H^{i}(V, 𝔸_{f}^{p}) ⊗ H^{j}(W, 𝔸_{f}^{p}).

(d) Assume (for simplicity) that V is connected. Let f be an algebraic correspondence of degree r from V to W with rational coefficients, i.e., an element of Z^{d_{V} +r}(V × W)_{ℚ}. For all but finitely many ℓ, f will be ℓ-integral, and so its ℓ-adic cohomology classes define an element cl(f) ∈ H^{2d_{V} +2r}(V ×W, 𝔸_{f}^{p}(d_{V} +r)). Combining this remark with (b) and (c), we obtain a map (also denoted f)

x ↦ q_{W ∗}(cl(f) ∪ q_{V}^{∗} x): H^{i}(V, 𝔸_{f}^{p}(m)) → H^{i+2r}(W, 𝔸_{f}^{p}(m + r))

for all m. Here q_{W} and q_{V} are the projection morphisms.

Remark 2.8. The main theorem of de Jong 1996 shows that Gabber’s theorem (Gabber 1983) holds also for the cohomology with compact support of arbitrary varieties over separably closed fields. Thus, appropriately interpreted, so do the remaining results of this subsection.

Adèlic cohomology (characteristic zero). Let V be a variety (not necessarily smooth or projective) over an algebraically closed field k of characteristic zero. For any model V_{0}/k_{0} of V over an algebraically closed subfield k_{0} of k and embedding k_{0} ↪ ℂ,

H^{i}(V, ℤ_{l}) ≅ H^{i}(V_{0}, ℤ_{l}) ≅ H^{i}(V_{0}(ℂ), ℤ)_{ℤ_{l}}.

The group H^{i}(V_{0}(ℂ), ℤ) is finitely generated, and so H^{i}(V, ℤ_{l}) is torsion-free for almost all l. The results in the last subsection hold mutatis mutandis for the group H^{i}(V, 𝔸_{f}(r)) defined to be the restricted topological product of the H^{i}(V, ℚ_{l}) with respect to the Λ_{l}^{i}(V, r). For any choice of a pair V_{0}/k_{0}, k_{0} ↪ ℂ as above,

H^{i}(V, 𝔸_{f}) ≅ H^{i}(V_{0}, 𝔸_{f}) ≅ H^{i}(V_{0}(ℂ), ℚ)_{𝔸_{f}}.

3. An Adèlic Tate Conjecture

Throughout this section, V is a smooth projective variety of dimension d over a field k. For ℓ ≠ char(k), H^{i}(V, ℤ_{ℓ}(r)) denotes the étale cohomology group. We let k̄ = k^{sep} and V̄ = V_{/k̄}. Finally, Γ = Gal(k̄/k).

Conjectures on algebraic cycles in ℓ-adic cohomology. The group Z_{ℓ-hom}^{r}(V) is the co-image of the cycle class map Z^{r}(V) → H^{2r}(V̄ , ℤ_{ℓ}(r)), and Z_{num}^{r}(V) is the quotient of Z^{r}(V) by numerical equivalence.

Recall (Tate 1994) that, when k is finitely generated over the prime field, there are the following conjectures.

T^{r}(V/k, ℓ): The cycle map Z_{ℓ-hom}^{r}(V)_{ℚ_{ℓ}} → H^{2r}(V̄ , ℚ_{ℓ}(r))^{Γ} is surjective. E^{r}(V/k, ℓ): The quotient map Z_{ℓ-hom}^{r}(V)_{ℚ} → Z_{num}^{r}(V)_{ℚ} is an isomorphism. I^{r}(V/k, ℓ): The cycle map Z_{ℓ-hom}^{r}(V)_{ℚ_{ℓ}} → H^{2r}(V̄ , ℚ_{ℓ}(r)) is injective. S^{r}(V/k, ℓ): The map H^{2r}(V̄ , ℚ_{ℓ}(r))^{Γ} → H^{2r}(V̄ , ℚ_{ℓ}(r))_{Γ} induced by the identity

map is bijective.

An adèlic Tate conjecture.

Lemma 3.1. Assume k is finitely generated over the prime field. If T^{r}(V/k, ℓ), T^{d−r}(V/k, ℓ), and S^{r}(V/k, ℓ) hold for almost all ℓ, then the cycle class map

Z_{ℓ-hom}^{r}(V)_{ℤ_{ℓ}} → H^{2r}(V̄ , ℤ_{ℓ}(r))^{Γ}

is surjective for almost all ℓ.

Proof. According to Tate 1994, 2.9 (c)⇒(d), the conditions imply that E^{r}(V/k, ℓ) and E^{d−r}(V/k, ℓ) hold for almost all ℓ.

Note that E^{r}(V/k, ℓ) implies Z_{ℓ-hom}^{r}(V)/{torsion} ≅ Z_{num}^{r}(V), which is finitely generated (see, for example, Milne 1980, VI 11.7). Therefore, for those ℓ for which E^{r}(V/k, ℓ) and E^{d−r}(V/k, ℓ) hold,

Z_{ℓ-hom}^{r}(V)/{torsion} × Z_{ℓ-hom}^{d−r}(V)/{torsion} → ℤ

is a nondegenerate pairing of finitely generated abelian groups whose discriminant D is independent of ℓ.

Consider the diagram

Z_{ℓ-hom}^{r}(V)_{ℤ_{ℓ}} ×  Z_{ℓ-hom}^{d−r}(V)_{ℤ_{ℓ}}  ⟶^{•} ℤ_{ℓ}
↓  ↓  ∥
H^{2r}(V̄ , ℤ_{ℓ}(r))^{Γ}×H^{2d−2r}(V̄ , ℤ_{ℓ}(d − r))^{Γ} ⟶^{∪} ℤ_{ℓ}.

For those ℓ for which T^{r}(V/k, ℓ) and T^{d−r}(V/k, ℓ) hold, the vertical arrows have finite cokernel.

For those ℓ for which – the groups H^{2r}(V̄ , ℤ_{ℓ}(r)) and H^{2d−2r}(V̄ , ℤ_{ℓ}(d − r)) are torsion-free,

– E^{r}(V/k, ℓ) and E^{d−r}(V/k, ℓ) hold,

– T^{r}(V/k, ℓ) and T^{d−r}(V/k, ℓ) hold, and

– ℓ does not divide D, the vertical maps are surjective. Since these conditions hold for almost all primes ℓ, this proves the lemma. □

Theorem 3.2. Let k be as in the lemma. If T^{r}(V/k, ℓ), T^{d−r}(V/k, ℓ), and S^{r}(V/k, ℓ) hold for all ℓ ≠ p, then the cycle class map defines a bijection

Z_{num}^{r}(V)_{𝔸_{f}^{p}} → H^{2r}(V̄ , 𝔸_{f}^{p}(r))^{Γ}.

Proof. We have to show that the cycle class map Z^{r}(V) → H^{2r}(V̄ , ℤ_{ℓ}(r)) defines

(a) a bijection Z_{num}^{r}(V)_{ℚ_{ℓ}} → H^{2r}(V̄ , ℚ_{ℓ}(r))^{Γ} for all ℓ ≠ p, and
(b) a surjection Z_{num}^{r}(V)_{ℤ_{ℓ}} → Λ_{ℓ}^{2r}(V̄ , r)^{Γ} for almost all ℓ.

Applying Tate 1994, 2.9, we see that E^{r}(V/k, ℓ), I^{r}(V/k, ℓ), and T^{r}(V/k, ℓ) hold for all ℓ ≠ p. This implies (a), and (b) follows from the lemma. □

Corollary 3.3. Let V be a (smooth projective) variety over 𝔽_{q}, and let r be an integer. The following statements are equivalent:

(a) for some prime ℓ ≠ p, dim_{ℚ} Z_{num}^{r}(V) = dim_{ℚ_{ℓ}} H^{2r}(V, ℚ_{ℓ}(r))^{Γ};

(b) the order of the pole of the zeta function Z(V, t) of V at t = q^{−r} is equal to the rank of Z_{num}^{r}(V);

(c) the cycle class map defines a bijection Z_{num}^{r}(V)_{𝔸_{f}^{p}} → H^{2r}(V̄ , 𝔸_{f}^{p}(r))^{Γ}.

Proof. Apply Tate 1994, 2.9, and the theorem. □

Counterexamples to an integral Tate conjecture. Assume k is finitely generated over the prime field, and consider the cycle class map

Z^{r}(V)_{ℤ_{ℓ}} → H^{2r}(V̄ , ℤ_{ℓ}(r))^{Γ}.  (3)

Over infinite fields k there exist V ’s without a rational zero cycle of degree 1 (for example, any curve of genus zero without a rational point or any curve^{6} of genus 1 with index ≠ 1) and for such a variety (3) will fail to be surjective for r = d and suitable ℓ. The argument that Atiyah and Hirzebruch (1962) used to prove that not all torsion cohomology classes on a complex variety are algebraic (contradicting Hodge’s original conjecture) also applies over finite fields^{7} and proves that (3) is not surjective on torsion classes.

Griffiths and Harris (1985) conjecture that if V is a very general hypersurface^{8} of degree d ≥ 6 in ℙ^{4}, then d divides the degree of every curve on V. Kollár has obtained some results in the direction of this conjecture (see Ballico et al. 1992, p. 135). Thus, it is known that, if k is an uncountable algebraically closed field and ℓ > 3 is distinct from the characteristic of k, then there exist smooth hypersurfaces V ⊂ ℙ^{4} such that the cycle class map

Z^{2}(V)_{ℤ_{ℓ}} → H^{4}(V, ℤ_{ℓ}(2)) ≅ ℤ_{ℓ}

is not surjective (cf. Schoen 1998, 0.6).

We do not know of an example with k finite for which (3) is not surjective modulo torsion (however, Schoen 1998, 2.1, gives an example with V singular). For r = 1 and k finite, one sees from the Kummer sequence that T^{1}(V/k, ℓ) implies that the map (3) is surjective.

6For such a curve, even CH^{1}(V̄)^{Γ} ⟶^{degree} ℤ fails to be surjective (Lang and Tate 1958).

7For this, one needs to use that the odd-dimensional Steenrod operations with values in H^{∗}(V̄ , ℤ/ℓℤ) vanish on algebraic cycles (Brosnan 1999).

8A statement is said to hold for a very general hypersurface of degree d if it holds for all hypersurfaces in the complement of countably many proper subvarieties of the space of hypersurfaces of degree d.

4. The Adèlic Fibre Functor

Isomotives. Fix a field k of characteristic exponent p, and let S be a class of smooth projective varieties over k satisfying the following condition:

4.1. S is closed under passage to a connected component and under the formation of products and disjoint unions, and it contains the zero-dimensional varieties and the projective spaces.

4.2. Fix an adequate equivalence relation ∼. Recall that, for smooth projective varieties V and W over k, the group of correspondences of degree i from V to W is

Corr^{i}(V, W) = ⊕_{j}Z_{∼}^{dim(V_{j})+i}(V_{j} × W)_{ℚ}

where the V_{j} are the equidimensional components of V.

4.3. The category CV_{∼}^{0}(k) of correspondences based on S, has one object hV for each V ∈ S, and

Hom_{CV_{∼}^{0}(k)}(hV, hW) = Corr^{0}(V, W)_{ℚ}.

With the definition

hV ⊗ hW = h(V × W)  (4)

and the obvious constraints, CV_{∼}^{0}(k) becomes a tensor category over ℚ (Saavedra 1972, VI 4.1.1).

4.4. The category of effective isomotives M_{∼}^{+}(k; ℚ; S) is the pseudo-abelian hull of CV_{∼}^{0}(k). Its objects are pairs (hV, e) with e an idempotent in End(hV). It is a pseudo-abelian tensor category over ℚ.

4.5. In M_{∼}^{+}(k; ℚ; S) there is a canonical decomposition

hℙ^{1} = 11 ⊕ 𝕃,

and the category of isomotives M_{∼}(k; ℚ; S) is obtained from M_{∼}^{+}(k; ℚ; S) by inverting 𝕃 (Saavedra 1972, VI 4.1.3.1). Its objects are pairs ((hV, e), m) with (hV, e) an effective motive and m ∈ ℤ. It is a pseudo-abelian rigid tensor category over ℚ (Saavedra 1972, VI 4.1.3.5). We sometimes write h(V, e, m) for ((hV, e), m).

4.6. We say that the Künneth projectors are algebraic for a smooth projective variety V if this is so for the l-adic étale cohomologies for all l ≠ p and the crystalline cohomology (characteristic p ≠ 0). There are then well-defined elements π^{i} ∈ Z_{num}^{i}(V)_{ℚ} (independent of the cohomology theory) that give these projectors for each cohomology theory (when k is finite, this follows from the Riemann hypothesis, and when k is arbitrary it can be proved by a specialization argument).

When the Künneth projectors are algebraic for all V in S, we use them to modify the commutativity constraints in M_{∼}^{+}(k; ℚ; S) and M_{∼}(k; ℚ; S) (Saavedra 1972, VI 4.2.1.5).

For the remainder of this section, we assume that ∼ is not coarser than ℓ-adic homological equivalence on Z^{r}(V)_{ℚ} for any ℓ ≠ p, so that the formula

ω_{ℓ}(h(V, e, m)) = e (H^{∗}(V̄ , ℚ_{ℓ}(m))) , V̄ = V_{k^{sep}},

defines a tensor functor ω_{ℓ}: M_{∼}(k; ℚ; S) → Vec(ℚ_{ℓ}), and we drop ∼ from the notation.

The adèlic fibre functor. With the notation of (2.6), define

ω^{p}(hV) = H^{∗}(V̄ , 𝔸_{f}^{p}).

Because of (2.7d) and our assumption on ∼, a correspondence f of degree 0 from V to W defines a homomorphism ω^{p}(f): ω^{p}(hV) → ω^{p}(hW), and one sees immediately from the definitions that ω^{p}(f ∘ g) = ω^{p}(f) ∘ ω^{p}(g). The isomorphisms

ω^{p}(hV ⊗ hW) =^{(4)} ω^{p}(h(V × W)) ≅^{2.7c} ω^{p}(hV) ⊗ ω^{p}(hW)

provide ω^{p} with the structure of a tensor functor CV^{0}(k) → Mod(𝔸_{f}^{p}). It is ℚ-linear (in particular, additive).

Theorem 4.7. The functor ω^{p}: CV^{0}(k) → Mod(𝔸_{f}^{p}) extends (essentially uniquely) to a ℚ-linear tensor functor ω^{p}: M(k; ℚ; S) → Mod(𝔸_{f}^{p}). For each ℓ ≠ p, ω^{p} ⊗_{𝔸_{f}^{p}} ℚ_{ℓ} ≅ ω_{ℓ}.

Proof. As Mod(𝔸_{f}^{p}) is abelian, ω^{p} extends (essentially uniquely) to M^{+}(k; ℚ; S) because of the universality of the pseudo-abelian hull. Because ω^{p}(𝕃) = 𝔸_{f}^{p}(−1) is invertible in Mod(𝔸_{f}^{p}), ω^{p} has an essentially unique extension from M^{+}(k; ℚ; S) to M(k; ℚ; S). For V ∈ S, ω^{p}(hV) ⊗_{𝔸_{f}^{p}} ℚ_{ℓ} ≅ ω_{ℓ}(hV), and this property is preserved under each extension. □

Remark 4.8. There is the following formula,

ω^{p}(h(V, e, m)) = e(H^{∗}(V̄ , 𝔸_{f}^{p}(m)))

where, on the right, e denotes the map defined in (2.7d). For a motive X = h(V, e, m), define λ_{ℓ}(X) to be the image of

H^{∗}(V̄ , ℤ_{ℓ}(m)) → H^{∗}(V̄ , ℚ_{ℓ}(m)) →^{e} H^{∗}(V̄ , ℚ_{ℓ}(m)).

Then, ω^{p}(X) is the restricted direct product

ω^{p}(X) = ∏_{ℓ≠p}(ω_{ℓ}(X), λ_{ℓ}(X)).

Proposition 4.9. If ∼ is numerical equivalence and the Künneth projectors of all varieties in S are algebraic, then for all isomotives X,

(a) ω^{p}(X) is a free 𝔸_{f}^{p}-module of finite rank;

(b) for all morphisms α of isomotives, ω^{p}(α) has constant rank.

Proof. The hypotheses imply that M(k; ℚ; S) is a Tannakian category over ℚ (Jannsen 1992). For any fibre functor ω on a Tannakian category with values in the vector spaces over a field, and any morphism α: X → Y in the category

dim(ω(X)) = rank(X)
rank(ω(α)) = rank X − rank Ker(α).

In particular, dim(ω(X)) and rank(ω(α)) are independent of ω. When we apply this statement to the functors ω_{ℓ}, we obtain the theorem. □

Remark 4.10. Because we are assuming ∼ is not coarser than ℓ-adic homological equivalence, the hypotheses of the proposition are that, for all ℓ ≠ p, numerical equivalence coincides with ℓ-adic homological equivalence on algebraic cycles with ℚcoefficients and that the Künneth projectors are algebraic. Without these hypotheses, the conclusion is not known (see the discussion Katz 1994, p. 28, (3)).

The functor ω^{p}: M(k; ℚ; S) → R(k; 𝔸_{f}^{p}).

Theorem 4.11. The functor ω_{l} has a canonical factorization through the forgetful functor R(k; ℚ_{ℓ}) → Vec(ℚ_{ℓ}). Under the hypotheses of 4.9, the functor ω^{p} has a canonical factorization through the forgetful functor R(k; 𝔸_{f}^{p}) → Mod(𝔸_{f}^{p}).

Proof. Let V ∈ S. When k is finitely generated over the prime field, there is a canonical continuous action of Gal(k^{sep}/k) on H^{i}(V̄ , ℚ_{ℓ}(m)), which therefore lies in R(k; ℚ_{ℓ}). For any correspondence f ∈ Corr^{r}(V, W), the map

f: H^{i}(V̄ , ℚ_{ℓ}(m)) → H^{i+r}(W̄ , ℚ_{ℓ}(m + r))

respects the actions of Gal(k^{sep}/k). Therefore, the functor

ω_{ℓ}: M(k; ℚ) → Vec(ℚ_{ℓ})

has a canonical factorization through the forgetful functor R(k; ℚ_{ℓ}) → Vec(ℚ_{ℓ}). The argument for ω^{p} is similar.

For an arbitrary field k, any varieties V, W and correspondence f will have models V_{1}, W_{1}, f_{1} over some subfield k_{1} of k finitely generated over the prime field. The image of ω_{l}(f_{1}): ω_{l}(V_{1}) → ω_{l}(W_{l}) under the functor R(k_{1}; ℚ_{l}) → R(k; ℚ_{l}) is independent of the choice of k_{1}, V_{1}, W_{1}, f_{1}. Thus, the statements hold also for k. □

5. The Category of Motives

We fix a field k of characteristic exponent p, and a class S of smooth projective varieties over k satisfying the condition (4.1) and such that

(a) for all V in S, the Künneth projectors are algebraic, and

(b) for all V in S and all l (including p), numerical equivalence coincides with l-adic homological equivalence for algebraic cycles with ℚ-coefficients.

Let M^{+}(k; ℚ) and M(k; ℚ) be the categories M_{num}^{+}(k; ℚ; S) and M_{num}(k; ℚ; S) defined in 4.6. Recall that they are semisimple abelian ℚ-categories (Jannsen 1992) and that M(k; ℚ) is Tannakian (Deligne 1990).

Because ω_{l} is a fibre functor on M(k; ℚ), the map

Hom_{M(k;ℚ)}(X, Y)_{ℚ_{l}} → Hom_{R(k;ℚ_{l})}(ω_{l}(X), ω_{l}(Y))  (5)

defined by ω_{l} is injective (Deligne 1990, 2.13). It follows that it is also injective when M and R are replaced by M^{+} and R^{+}. We say that the l-adic Tate conjecture holds for M(k; ℚ) or M^{+}(k; ℚ) if the corresponding map is an isomorphism for l.

While conditions (a) and (b) are conjectured to hold for all smooth projective varieties, they have been proved for very few varieties (see Kleiman 1994 for a list). In the final subsection of this section (Variants, p22), we list some alternative categories M^{+}(k; ℚ) and M(k; ℚ) to which our constructions apply, but which require no unproven conjectures.

Our construction of M(k; ℤ) is suggested by the following simple observation: for a finitely generated ℤ-module M, M ≅ M_{ℤ̂} ×_{M_{𝔸_{f}}} M_{ℚ}; therefore, any category whose Hom-sets are finitely generated ℤ-modules admits a fully faithful functor into the fibre product of a ℤ̂-category with a ℚ-category over a 𝔸_{f}-category.

The category of effective motives M^{+}(k; ℤ).

Definition 5.1. The category of effective motives M^{+}(k; ℤ) over k is the full subcategory of the fibre product category

R^{+}(k; ℤ̂) ×_{R^{+}(k;𝔸_{f})} M^{+}(k; ℚ)

whose objects (X_{f}, X_{0}, x_{f}) are those for which the torsion subgroup of X_{f} is finite. Thus, an effective motive X over k is a triple (X_{f}, X_{0}, x_{f}) consisting of

(a) an object X_{f} = (X_{l})_{l} of R^{+}(k; ℤ̂) such that X_{l} is torsion-free for almost all l,

(b) an effective isomotive X_{0}, and

(c) an isomorphism x_{f}: (X_{f})_{ℚ} → ω_{f}(X_{0}) in R^{+}(k; 𝔸_{f}). A morphism of effective motives α: X → Y is a pair of morphisms

(α_{f}: X_{f} → Y_{f}, α_{0} : X_{0} → Y_{0})

such that

X_{f}  ⟶^{α_{f}} Y_{f}
↓^{x_{f}}  ↓^{y_{f}}
ω_{f}(X_{0}) ⟶^{ω_{f} (α_{0})} ω_{f}(Y_{0})

commutes. We say that a motive X = (X_{f}, X_{0}, x_{f}) is finite if X_{0} = 0.

Note that to give (X_{f}, x_{f}) amounts to giving an X_{l} in R(k; ℤ_{l}) for each l together with homomorphisms x_{l}: X_{l} → ω_{l}(X_{0}) that induce isomorphisms X_{l} → λ_{l}(X_{0}) for almost all l and isomorphisms (X_{l})_{ℚ} → ω_{l}(X_{0}) for all l. We usually regard x_{f} as a map X_{f} → ω_{f}(X_{0}) with finite kernel. We use the notations: X_{f} = X^{p} × X_{p}, X^{p} = ∏_{l≠p} X_{l}, x_{f} = x^{p} × x_{p}, α_{f} = α^{p} × α_{p}.

Proposition 5.2. The category M^{+}(k; ℤ) is abelian. A sequence

X → Y → Z

in M^{+}(k; ℤ) is exact if and only if the sequences

X_{0} → Y_{0} → Z_{0} and X_{f} → Y_{f} → Z_{f}

are exact in M^{+}(k; ℚ) and R^{+}(k; ℤ̂) respectively.

Proof. Both statements are true for the fibre product category, and so it remains to show that M^{+}(k; ℤ) is closed under the formation of kernels and cokernels in the larger category. Any subobject of an object in M^{+}(k; ℤ) is also in M^{+}(k; ℤ). Therefore, M^{+}(k; ℤ) is closed under the formation of kernels, and, in proving that the cokernel of α is in M^{+}(k; ℤ), we may assume that α is injective. Thus, given a short exact sequence

0 → X →^{α} Y → Z → 0

in the fibre product category with X, Y in M^{+}(k; ℤ), we have to show that Z is in M^{+}(k; ℤ). As M^{+}(k; ℚ) is semisimple, the sequence of M^{+}(k; ℚ)-components splits. But ω^{p} is additive, and so

0 → ω^{p}(X_{0}) ⟶^{ω^{p}(α_{0})} ω^{p}(Y_{0}) → ω^{p}(Z_{0}) → 0

is split-exact, which implies that

0 → λ_{ℓ}(X_{0}) → λ_{ℓ}(Y_{0}) → λ_{ℓ}(Z_{0}) → 0

is split-exact for almost all ℓ. In turn, this implies that

0 → X_{ℓ} →^{α_{ℓ}} Y_{ℓ} → Z_{ℓ} → 0

is split-exact for almost all ℓ. Thus, the torsion subgroup of Z_{ℓ} is zero for almost all ℓ and finite for all ℓ, which shows that Z is in M^{+}(k; ℤ). □

Corollary 5.3. Let X and X′ be submotives of an effective motive Y, and let X ∩ X′ = X ×_{Y} X′. If X_{0} = Y_{0}, then the cokernel Z of X ∩ X′ → X′ is finite.

Proof. Because X ↦ X_{0} is exact, (X ×_{Y} X′ → X′)_{0} ≅ (X_{0} ×_{Y_{0}} X′_{0} → X′_{0}), which is an isomorphism if X_{0} = Y_{0}. □

Proposition 5.4. The category M^{+}(k; ℤ) is noetherian (i.e., its objects are noetherian).

Proof. Let

X(1) ⊂ X(2) ⊂ X(3) ⊂ ⋯

be a sequence of a submotives of an effective motive Y. Eventually, the sequence X(1)_{0} ⊂ X(2)_{0} ⊂ ⋯ becomes stationary, equal, say, to X(∞)_{0} ⊂ Y_{0}. Let X(∞)_{f} = Y_{f} ×_{ω_{f} (Y_{0})} ω_{f}(X(∞)_{0}). Then X(∞) = (X(∞)_{f}, X(∞)_{0}, ·) is an effective motive, and X(1) ⊂ X(2) ⊂ ⋯ is a sequence of submotives of X(∞) such that X(r)_{0} = X(∞)_{0} for r ≥ r_{1}, some r_{1}. Now X(∞)/X(r_{1}) is finite (5.3), and so the sequence X(1) ⊂ X(2) ⊂ ⋯ becomes stationary. □

Lemma 5.5. Given a morphism β: X_{0} → Y_{0} of effective isomotives, there exists an integer m such that mβ is in the image of Hom(X, Y) → Hom(X_{0}, Y_{0}).

Proof. We have to find an integer m and morphisms α_{l} such that the diagrams

X_{l}  ⟶^{α_{l}} Y_{l}
↓^{x_{l}}  ↓^{y_{l}}
ω_{l}(X_{0}) ⟶^{ω_{l}(mβ)} ω_{l}(Y_{0})

commute. Because ω_{l}(β) is the l-component of a homomorphism ω^{p}(X_{0}) → ω^{p}(Y_{0}), it maps x_{l}(X_{l}) into y_{l}(Y_{l}) for almost all l, and so there exists an integer m such that mω_{l}(β) maps x_{l}(X_{l}) into y_{l}(Y_{l}) for all l. Choose an integer n that kills the torsion in Y_{l} for all l. Then the map

X_{l} ⟶^{x_{l}} x_{l}X_{l} ⟶^{ω_{l}(mnβ)} y_{l}Y_{l} = Y_{l}/torsion.

lifts to a map X_{l} → Y_{l}. □

Proposition 5.6. For all X, Y in M^{+}(k; ℤ), Hom(X, Y) is a finitely generated ℤ-module (modulo p-torsion when k is infinite and p ≠ 1).

Proof. As Hom(X, Y) ⊂ End(X ⊕ Y), it suffices to show that all endomorphism rings are finitely generated ℤ-modules (modulo …). Because ω^{p} is faithful, X ↦ X_{f} is faithful. In particular, the map

α ↦ α_{f}: End(X) → End(X_{f})

is injective, and so the torsion subgroup of End(X) is contained in that of End(X_{f}), which is finite except possibly for the p-torsion when k is infinite and p ≠ 1. The kernel of the map

α ↦ α_{0} : End(X) → End(X_{0})

is obviously torsion, and so it remains to show that its image R is finitely generated over ℤ. If β ∈ End(X_{0}) is in R, then ω_{l}(β) stabilizes the lattice x_{l}(X_{l}) for all l. Thus, the characteristic polynomial of β has coefficients in ℤ ⊂ ℚ. Let α_{1}, …, α_{n} be a basis for End(X_{0}) as a ℚ-vector space. According to Lemma 5.5, after possibly replacing each α_{i} with mα_{i} for some integer m, we may suppose that each α_{i} ∈ R. Because M^{+}(k; ℚ) is semisimple, End(X_{0}) is a semisimple ℚ-algebra, and so the trace pairing α, β ↦ Tr_{End(X_{0})/ℚ}(αβ) is nondegenerate. Therefore, there is a ℚ-basis β_{1}, …, β_{n} for End(X_{0}) dual to α_{1}, …, α_{n}. Now the usual argument shows that

R ⊂ ℤβ_{1} + ⋯ + ℤβ_{n}.

Namely, let β ∈ R. Then β can be written uniquely as a linear combination β = ∑ b_{j}β_{j} of the β_{j} with coefficients b_{j} ∈ ℚ, and we have to show that each b_{j} ∈ ℤ. As α_{i} and β are in R, so also is β · α_{i}, and so Tr(β · α_{i}) ∈ ℤ. But

Tr(β · α_{i}) = Tr(∑_{j} b_{j}β_{j}α_{i}) = ∑_{j} b_{j} Tr(β_{j}α_{i}) = b_{i}. □

Remark 5.7. (a) If Y has no p-torsion (i.e., Y_{p} is torsion-free), then Hom(X, Y) has no p-torsion and hence is finitely generated over ℤ.

(b) Let p ≠ 1. The effective motive X with X_{0} = 0 and X_{f} = X_{p} = k (F acting as zero) has End(X) = k. The same phenomenon occurs with subgroup schemes of abelian varieties: a supersingular elliptic curve has α_{p} = Spec k[T]/(T^{p}) as a subgroup, and End(α_{p}) = k.

Proposition 5.8. The functor X ↦ X_{0} : M^{+}(𝔽; ℤ) → M^{+}(𝔽; ℚ) defines an equivalence of categories

M^{+}(𝔽; ℤ)_{ℚ} → M^{+}(𝔽; ℚ).

Proof. Because the kernel of

α ↦ α_{0} : Hom(X, Y) → Hom(X_{0}, Y_{0})

is torsion, the map

r ⊗ α ↦ rα_{0} : ℚ ⊗ Hom(X, Y) → Hom(X_{0}, Y_{0})

is injective, and Lemma 5.5 shows that it is surjective. Thus, the functor is faithful and full. Given an X in M^{+}(k; ℚ), clearly there exists an X^{p} in R^{+}(k; ℤ^{p}) such that (X^{p})_{ℚ} ≈ ω^{p}(X). Using this, one sees that X ↦ X_{0} is essentially surjective. □

Remark 5.9. An object X of M^{+}(k; ℤ) is said to be torsion if mX = 0 for some m (equivalently, X is finite). The torsion objects form a thick subcategory of M^{+}(k; ℤ) and Proposition 5.8 shows that the functor

X ↦ X_{0} : M^{+}(k; ℤ) → M^{+}(k; ℚ)

realizes M^{+}(k; ℚ) as the quotient of M^{+}(k; ℤ) by its subcategory of torsion objects. Proposition 5.10. The functor

M^{+}(k; ℤ)_{ℤ_{l}} → R^{+}(k; ℤ_{l})

defined by X ↦ X_{l} is faithful; it is full if and only if the l-adic Tate conjecture holds for M^{+}(k; ℚ_{l}).

Proof. We first show that

z ⊗ α ↦ zα_{l}: ℤ_{l} ⊗_{ℤ} Hom_{M^{+}(k;ℤ)}(X, Y) → Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l})

is injective. The torsion subgroup of ℤ_{l} ⊗ Hom_{M^{+}(k;ℤ)}(X, Y) is equal to the l-primary component of the torsion subgroup of Hom_{M^{+}(k;ℤ)}(X, Y), and this maps injectively to Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l}) (because the functor X ↦ X_{f} is faithful). Thus, the kernel of z ⊗ α ↦ zα_{l} is torsion-free. Consider the diagram

ℤ_{l} ⊗_{ℤ} Hom(X, Y) ⟶  Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l})
↓  ↓
ℚ_{l} ⊗_{ℚ} Hom(X_{0}, Y_{0}) ⟶ Hom_{R^{+}(k;ℚ_{l})}(ω_{l}(X_{0}), ω_{l}(Y_{0})).

The left vertical arrow is obtained from

Hom(X, Y) → Hom(X_{0}, Y_{0})

by tensoring with ℤ_{l}, and hence has torsion kernel (5.8). The bottom arrow is injective (5). Hence, the kernel of the top horizontal arrow is torsion, but we have already shown it to be torsion-free; it is therefore zero.

We next show that the cokernel of z ⊗ α ↦ zα_{l} is torsion-free. Suppose that β ∈ Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l}) is such that l^{m}β = zα_{l} for some m ∈ ℕ, α ∈ Hom(X, Y), and z ∈ ℤ_{l}. Because z ⊗ α ↦ zα_{l} is (obviously) surjective on torsion, we may suppose that z is not divisible by l. Then α_{l} is divisible by l^{m} in Hom(X_{l}, Y_{l}). For l′ ≠ l, l^{m} is a unit in ℤ_{l′}, and so α_{l′} is also divisible by l^{m}. Now α′, with

α′_{0} = l^{−m}α_{0}, α′_{l} = z^{−1}β, α′_{l′} = l^{−m}α_{l′} for l′ ≠ l

is a morphism X → Y such that zα′_{l} = β.

Finally, we assume the l-adic Tate conjecture for M^{+}(k; ℚ), and we prove that the cokernel of z ⊗ α ↦ zα_{l} is torsion. For this, it suffices to prove that

ℚ_{l} ⊗_{ℤ} Hom(X, Y) → ℚ ⊗_{ℤ} Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l})

is surjective. But this follows from the isomorphisms

ℚ ⊗_{ℤ} Hom(X, Y) ≅^{(5.8)} Hom(X_{0}, Y_{0}),

and

ℚ_{l} ⊗_{ℚ} Hom(X_{0}, Y_{0}) →^{Tate} Hom_{R^{+}(k;ℚ_{l})}(ω_{l}(X_{0}), ω_{l}(Y_{0})) ≅ ℚ ⊗_{ℤ} Hom_{R^{+}(k;ℤ_{l})}(X_{l}, Y_{l}).

□ Proposition 5.11. The category

M^{+}(k; ℤ) = \underrightarrow{lim}_{k′} M^{+}(k′; ℤ)

where k′ runs over the subfields of k finitely generated over the prime field.

Proof. The statement is true for each of the categories M^{+}(k; ℚ), R^{+}(k; ℤ̂), and R^{+}(k; 𝔸_{f}), and it follows easily for their fibre product and its subcategory M^{+}(k; ℤ).

□

The category of motives.

Definition 5.12. The category of motives M(k; ℤ) is the full subcategory of

R(k; ℤ̂) ×_{R(k;𝔸_{f})} M(k; ℚ)

whose objects (X_{f}, X_{0}, x_{f}) are those for which the torsion subgroup of X_{f} is finite.

The results in the previous subsection hold mutandis mutatis for M(k; ℤ) — in particular, M(k; ℤ) is a noetherian abelian category whose quotient by its torsion subcategory is M(k; ℚ).

Proposition 5.13. The evident functor M^{+}(k; ℤ) → M(k; ℤ) is faithful, and it is full on torsion-free objects (on all objects when k has characteristic zero). It realizes M(k; ℤ) as the category obtained from M^{+}(k; ℤ) by inverting 𝕃.

Proof. Straightforward. □

Remark 5.14. (a) When k = ℚ there is a canonical functor from M(k; ℤ) to the category of systems of realizations over ℚ with integer coefficients defined in Deligne 1989, 1.23.

(b) Let k be a subfield of ℂ, and let M(k; ℚ) be the category based on a class S satisfying (4.1) but using the “motivated” classes as correspondences (André 1996, 4.2). In this case, our definition of M(k; ℤ) is equivalent to that of André (ibid. 8.1).

Artin motives. Fix a separable algebraic closure k̄ of k, and let Γ = Gal(k̄/k). The category of Artin isomotives is the full subcategory M^{Artin}(k; ℚ) of M^{+}(k; ℚ) of objects of the form X = hV with V a variety of dimension 0. For such an X, define π_{0}(X) = π_{0}(V̄) — it is a finite set with a continuous action of Γ. The functor H: X ↦ ℚ^{π_{0}(X)} =_{df} Map(π_{0}(X), ℚ) defines an equivalence of M^{Artin}(k; ℚ) with the category R(Γ; ℚ) of continuous representations of Γ on finite-dimensional ℚ-vector spaces (cf. Deligne and Milne 1982, 6.17). Hence, M^{Artin}(k; ℚ) is a Tannakian category.

We define the category of Artin motives to be the full subcategory M^{Artin}(k; ℤ) of M^{+}(k; ℤ) of objects X such that X_{0} is an Artin isomotive and (when p ≠ 1) F_{X_{p}} is bijective.

Recall (Saavedra 1972, VI 3.1.2; Demazure 1972, p. 69) that there is a fully faithful functor

γ: R(Γ; ℤ_{p}) → Crys^{+}(k^{pf}) = R^{+}(k; ℤ_{p})  (6)

whose essential image consists of the crystals with bijective F. Let X be an Artin motive. When p ≠ 1, we choose X′_{p} ∈ R(Γ; ℤ_{p}) so that γ(X′_{p}) = X_{p}, and we let ω′_{p}(X_{0}) = (X′_{p})_{ℚ}; otherwise we set X′_{p} = 0 = ω′_{p}(X_{0}). Define H(X) so that the right hand square in the following diagram is cartesian:

0 ⟶ H(X)_{tors} ⟶ H(X) ⟶  H(X_{0})
↓≅  ↓  ↓
0 ⟶ X_{tors}^{p} × X′_{ptors} ⟶ X^{p} × X′_{p} ⟶ ω^{p}(X_{0}) × ω′_{p}(X_{0})

The diagram is exact and commutative, and H(X) is a finitely generated ℤ-module with a continuous action of Γ (discrete topology on H(X)).

Proposition 5.15. The functor H: M^{Artin}(k; ℤ) → R(Γ; ℤ) is an equivalence of categories.

Proof. The functor H: M^{Artin}(k; ℚ) → R(Γ; ℚ) defines an equivalence of categories

R(Γ; ℤ) ×_{R(Γ;ℚ)} M^{Artin}(k; ℚ) → R(Γ; ℤ) ×_{R(Γ;ℚ)} R(Γ; ℚ) = R(Γ; ℤ)

and γ defines a full faithful functor

R(Γ; ℤ) ×_{R(Γ;ℚ)} M^{Artin}(k; ℚ) → R(k; ℤ̂) ×_{R(k;𝔸_{f})} M^{Artin}(k; ℚ)

whose essential image is M^{Artin}(k; ℤ). □

Rationally decomposed one-motives. Recall that a one-motive is a triple (G, X, u) with G a semi-abelian variety, X a finitely generated free ℤ-module on which Γ acts continuously, and u a Γ-homomorphism X → G(k^{sep}). A one-motive X ⟶^{u} G is said to be decomposed if u = 0 and the extension

0 → T → G → A → 0 (T a torus, A an abelian variety)

splits, and it is rationally decomposed if it is isogenous to a decomposed one-motive. Over a finite field, every one-motive is rationally decomposed. The one-motives form an additive category M_{1}(k) with the decomposed and rationally decomposed onemotives as additive subcategories M_{1}^{d}(k) ⊂ M_{1}^{rd}(k) ⊂ M_{1}(k).

Choose a quasi-inverse F: R(Γ; ℤ) → M^{Artin}(k; ℤ) to the functor H of (5.15) .

Proposition 5.16. For L = (X ⟶^{0} T ×A) in M_{1}^{d}(k), define hL to be the effective motive F (X) ⊕ h_{ℤ}^{1}A ⊕ F (X_{∗}(T)(1)). Then h extends to a functor h: M_{1}^{rd}(k) → M^{+}(k; ℤ). (Here h_{ℤ}^{1}A denotes h^{1}A endowed with the ℤ-structure provided by the maps H^{1}(A, ℤ_{l}) → H^{1}(A, ℚ_{l}).) Proof. Let L be a rationally decomposed one-motive. There exists a decomposed one-motive L′ and an isogeny α: L → L′ such that (L′, α) is universal, and α induces isomorphisms ω^{p}(L) ≅ ω^{p}(L′) and ω_{p}(L) ≅ ω_{p}(L′). We define hL to be the isomotive (hL′)_{0} equipped with the structures

λ^{p}L → ω^{p}(L) ≅ ω^{p}(L′) ≅ ω^{p}((hL′)_{0})
λ_{p}L → ω_{p}(L) ≅ ω_{p}(L′) ≅ ω_{p}((hL′)_{0}). □

Proposition 5.17. The functor h: M_{1}^{rd}(k) → M^{+}(k; ℤ) is fully faithful (except possibly when k has characteristic 2 and is not algebraic over 𝔽_{2}).

Proof. It suffices to prove this in the case that k is finitely generated over the prime field, in which case it is a consequence of the following theorem of Tate (1966a), Zarhin (1975), Faltings (Faltings and Wüstholz 1984), and de Jong (1998): for abelian varieties A and B over k, the map Hom_{k}(A, B)_{ℤ_{l}} → Hom_{R(k;ℤ_{l})}(T_{l}A, T_{l}B) is bijective (except possibly when k has characteristic 2 …). □

Remark 5.18. Propositions 5.16 and 5.17 show that, for any two rationally decomposed one-motives L, M, there is an injective homomorphism

Ext_{M_{1}(k)}^{1}(L, M) → Ext_{M^{+}(k;ℤ)}^{1}(hL, hM).

This map will not in general be surjective because an extension of two W [F]-modules for which there exist maps V with F V = p = V F need not be similarly endowed. In fact, the formula Milne 1968, Theorem 3, for the order of the Ext^{1} of abelian varieties disagrees with that in Theorem 10.1 below.

The category M(𝔽_{q}; ℤ). For a motive X = h(V, e, m) over 𝔽_{q}, the Frobenius endomorphism of V/𝔽_{q} defines a canonical Frobenius element π_{X} ∈ End(X). For a motive X over 𝔽, each model X_{1}/𝔽_{p^{n}} of X defines a pair (π, n) with π equal to π_{X_{1}} regarded as an endomorphism of X. Any two pairs (π, n), (π′, n′) arising in this way are equivalent in the sense that π^{n′m} = π^{′nm} for some integer m > 0. The germ of a Frobenius endomorphism of X is the equivalence class of pairs containing the pair arising from one (hence, every) model of X over a finite field.

Proposition 5.19. Assume the Tate conjecture holds for M(𝔽; ℚ). Then

X ↦ X_{/𝔽}: M(𝔽_{p^{n}}; ℤ) → M(𝔽; ℤ)

defines an equivalence of M(𝔽_{p^{n}}; ℤ) with the category of pairs (X, π) consisting of a motive X in M(𝔽; ℤ) and an endomorphism π of X such that (π, n) represents π_{X}. A similar statement holds for effective motives.

Proof. Easy consequence of the similar statement for isomotives (Milne 1994, 3.5). □

Remarks on the definition of M(k; ℤ).

5.20. As André notes (1996, 8.1), in view of the counterexamples of Atiyah and Hirzebruch (1962) to the original Hodge conjecture, it would be “peu judicieux” to define a category of integral motives by using algebraic cycles with integer coefficients in Grothendieck’s construction, in other words, with

Hom(11, h_{ℤ}V) = Z_{num}^{dim V} (V × V).

There is also the difficulty of knowing what to replace the pseudo-abelian hull with.

With our definitions, a smooth projective variety V defines a complex of motives, rather than a motive (see an article in preparation), but we can make the ad hoc definitions: h_{ℤ}^{r}V is the isomotive h^{r}V with the ℤ-structure provided the maps H^{r}(V, ℤ_{l}) → H^{r}(V, ℚ_{l}), and h_{ℤ}V = ⊕h_{ℤ}^{r}V. If V has torsion-free cohomology, then

Hom(11, hV) = Z_{num}^{dim V} (V × V) ∩ End(H^{∗}(V, ℤ_{l})).

There is a map from Z_{num}^{dim V} (V ×V) to this group, but it will not always be onto (p11).

5.21. J-M. Fontaine has pointed out to us that there is an incompatibility between our definitions in characteristic 0 and in characteristic p: in the first case, we use the étale cohomology to define the integral structure at every prime, whereas in characteristic p we use the crystalline cohomology at the prime p; but the crystalline cohomology in characteristic p corresponds to the de Rham cohomology in characteristic zero, and the integral structures on the de Rham cohomology and the p-adic étale cohomology do not agree. We do not know how to resolve this problem (or even whether it is resolvable).

Variants. We list some alternative categories M(k; ℚ) to which our construction applies.

5.22. Let S consist of the smooth projective varieties over k for which the Künneth projectors are algebraic. This class satisfies the condition 4.1, includes all abelian varieties and surfaces, and includes all smooth projective varieties when k is finite. Let M_{l}(k; ℚ) and M(k; ℚ) be the categories of motives based on S using respectively the algebraic classes modulo l-adic homomological equivalence and numerical equivalence as the correspondences. There is a tensor functor

M_{l}(k) → M(k)

which, according to André and Kahn 2001, has a tensor section s_{l}. Therefore, ω_{l} ∘ s_{l} is a fibre functor on M(k; ℚ). Moreover, s_{l} can be chosen so that (ω_{l} ∘ s_{l})(hV) = H^{∗}(V, ℚ_{l}). If, as seems likely, s_{l} can be chosen^{9} so that the action of Z_{num}^{dim V} (V × V) ⊂ End(hV) on (ω_{l} ∘ s_{l})(hV) preserves the image of H^{∗}(V, ℤ_{l}) in H^{∗}(V, ℚ_{l}) for all V with torsion-free cohomology, then the ω_{l} ∘ s_{l} for l ≠ p are the l-components of an 𝔸_{f}^{p}-valued fibre functor on M(k; ℚ), and we can construct M^{+}(k; ℤ) and M(k; ℤ) as above.

5.23. Let S consist of all smooth projective varieties over k. For any V in S, the Künneth projectors are almost algebraic (in the sense of Tate 1994, p76). Therefore, the discussion in 5.22 applies to the category M(k; ℚ) defined using numerical equivalence classes of almost algebraic classes rather than algebraic classes.

5.24. Recall that a Lefschetz class modulo ∼ on a variety V is an element of the ℚ-subalgebra of ⊕Z_{∼}^{r}(V) generated by divisor classes. Let S be the smallest class of varieties over k satisfying (4.1) and including the abelian varieties. Then it is possible to define a category M(k; ℚ) of isomotives based on S using the Lefschetz classes modulo numerical equivalence as correspondences; moreover, M(k; ℚ) is a semisimple Tannakian category over ℚ and there exist canonical functors ω_{l} (Milne 1999, §1).

5.25. For any field k of characteristic zero, define M(k; ℚ) to be the category of motives based on some class S satisfying (4.1) and using the absolute Hodge classes as correspondences (Deligne and Milne 1982, §6).

5.26. Let CM(ℚ^{al}) be the category of motives based on the abelian varieties of CM-type over ℚ^{al} using the absolute Hodge classes as the correspondences, and let M(𝔽; ℚ) be the quotient of CM(ℚ^{al}) corresponding to a fibre functor on CM(ℚ^{al})^{P} which is correct at every prime (see Milne 2002; here P is the Weil-number protorus). As above, we can define M^{+}(𝔽; ℤ) and M(𝔽; ℤ) . Almost by definition, the Tate conjecture holds for M(𝔽; ℚ).

6. Tannakian Properties

Throughout this section, R is a commutative noetherian ring.

9According to André (email 25.03.02) it does not appear possible to prove this by the methods of their paper.

Tannakian categories over rings. In Saavedra 1972, Tannakian categories are defined only over a base field. We adapt his definition (ibid. III 1.1.1) of an ind-Tannakian category to obtain the notion of a Tannakian category over an arbitrary noetherian ring.

Recall that a tensor category C is a category together with a bifunctor ⊗: C ×C → C and compatible associativity and commutativity constraints for which there exists an identity object 11. When End(11) = R, we call (C, ⊗) a tensor category over R or a tensor R-category. Let R′ be an R-algebra. An R′-valued fibre functor is an R-linear exact tensor functor C → Mod(R′).

Some linear algebra. Let C be an R-linear category, and let R′ be an R-algebra. When C has arbitrary direct limits, Saavedra (1972, I 1.5) constructs an R′-linear category C_{(R′)} by “extension of the base ring”. We adapt his construction to the case that R′ is a finite R-algebra and C has finite direct limits (for example, C is abelian).

Thus, let C be an abelian R-linear category. Let R′ be a finite R-algebra, and define C_{(R′)} to be the category whose objects are pairs (X, i_{X}) comprising an object X of C and a homomorphism i_{X}: R′ → End_{R}(X) of R-algebras (Saavedra 1972, II 1.5).

6.1. For any finitely generated R-module M, there is a functor

X ↦ M ⊗_{R} X: C → C

and an isomorphism (natural in X and Y)

Hom_{C}(M ⊗_{R} X, Y) → Hom_{R}(M, Hom_{C}(X, Y))

(Saavedra 1972, II 1.5.1.1).

6.2. The category C_{(R′)} is an abelian R′-linear category. A sequence

(X′, i_{X′}) → (X, i_{X}) → (X′′, i_{X′′})

in C_{(R′)} is exact if and only if

X′ → X → X′′

is exact in C. The functor j^{R′/R}: C → C_{(R′)} sending X to R′ ⊗_{R} X (with its canonical R′-structure) is left adjoint to the functor j_{R/R}: C_{(R′)} → C sending (X, i_{X}) to X (Saavedra 1972, II 1.5.2). For any X, Y ∈ ob(C),

Hom_{C_{(R′)}}(j^{R′/R}X, j^{R′/R}Y) ≅ Hom_{C}(X, Y) ⊗_{R} R′,  (7)

and so j^{R′/R} realizes C_{R′} as a full subcategory of C_{(R′)}.

6.3. When R′ is a flat R-algebra, the functor j^{R′/R}: C → C_{(R′)} is exact.

6.4. Let C be an abelian tensor category over R such that ⊗ is right exact. The category C_{(R′)} has a canonical structure of a tensor category over R′ for which j^{R′/R} and j_{R′/R} are tensor functors; moreover ⊗ on C_{(R′)} is right exact (Saavedra 1972, II 1.5.4). Let D be an abelian tensor category over R′ for which ⊗ is right exact. The map u ↦ u ∘ j^{R′/R} defines an equivalence of the category of right exact R′-linear tensor functors C_{(R′)} → D with the category of right exact R-linear tensor functors C → D. When R′ is a flat R-algebra, u ↦ u ∘ j^{R′/R} carries exact functors to exact functors. (Ibid. II 1.5.3, 1.5.4.) (But it is not known that u ∘ j^{R′/R} exact implies u exact, even when R′ is faithfully flat over R, except in the case that k and k′ are fields, when the proof requires the main theorem of Deligne 1990).

Definitions. We say that an abelian tensor R-category C is finitely generated if there exists a finite set of objects of C contained in no proper strictly full abelian tensor R-subcategory of C.

Definition 6.5. (a) A neutral Tannakian category over R is a tensor category over R that is tensor equivalent to Rep(G; R) for some flat affine group scheme G over R.

(b) An algebraic Tannakian category over R is a finitely generated abelian tensor category C over R such that C_{(R′)} becomes a neutral Tannakian category for some faithfully flat finite R-algebra R′.

(c) A Tannakian category over R is an abelian tensor category C over R that is a filtered union of algebraic Tannakian categories.

Note that, in a Tannakian category, the Homs are finitely generated R-modules, because

Hom_{C}(X, Y) ⊗_{R} R′ ≅^{(7)} Hom_{C_{(R′)}}(j^{R′/R}X, j^{R′/R}Y) ⊂ Hom_{R′}(ω(X), ω(Y)).

A criterion to be a Tannakian category over a Dedekind domain. Let C be an abelian tensor category over a noetherian ring R. When internal Homs exist in C (Saavedra 1972, I 3), we let X^{∨} = \underline{Hom}(X, 11). There is a canonical morphism X → X^{∨∨}, and X is said to be reflexive when this morphism is an isomorphism.

For example, when G is an affine group scheme over R, an object X of Rep(G; R) is reflexive if and only if its underlying R-module is projective. Serre (1968, 2.2) proves that, when R is a Dedekind domain, every object in Rep(G; R) is a quotient of a representation of G on a projective R-module. Thus every object in Rep(G; R) is a quotient of a reflexive object. This property can be used to characterize Tannakian categories over Dedekind domains.

Proposition 6.6. Let C be an abelian tensor category over a Dedekind domain R such that every object of C is a quotient of a reflexive object. If there exists an R′-valued fibre functor on C_{(R′)} for some Dedekind domain R′ finite and flat over R, then C is Tannakian.

Proof. Let ω: C_{(R′)} → Mod(R′) be such a fibre functor. To prove that C is Tannakian, we shall show that the functor \underline{Aut}^{⊗}(ω) of R′-algebras is representable by a flat affine group scheme G over R′, and that ω defines an equivalence of R′-tensor categories C_{(R′)} → Rep(G; R′).

When R′ = R, this is proved in Saavedra 1972, II 4.1. In detail: ibid. II 4.1.1 shows that \underline{End}^{⊗}(ω) is represented by a flat affine monoid scheme G over R and that ω defines an equivalence of R-linear tensor categories C → Rep(G; R); but \underline{End}^{⊗}(ω) ≅ \underline{End}^{⊗}(ω_{0}) where ω_{0} is the restriction of ω to the full subcategory of reflexive objects; this last category is rigid, and so every endomorphism of ω_{0} is an automorphism (ibid. I 5.2.3), which implies that G is a group scheme.

According to (6.2) and (6.4), C_{(R′)} is an abelian tensor category over R′. Thus, the general case will follow from the neutral case once we show that C_{(R′)} inherits the property that every object is a quotient of a reflexive object. Let (Y, i_{Y}) be an object of C_{(R′)}, and let q: X → Y express Y as a quotient of a reflexive object. The composite

j^{R′/R}X ⟶^{j^{R′/R}(q)} j^{R′/R}Y → (Y, i_{Y})

is again surjective, and j^{R′/R}X is reflexive because X is. □

Example 6.7. For ℓ ≠ p, the category R(k; ℤ_{ℓ}) is Tannakian over ℤ_{ℓ} (apply 1.4). Example 6.8. (a) Let k be a finite field, and let ω be the forgetful functor

R(k; ℤ_{p}) → Mod(W (k)). For (X, i_{X}) in R(k; ℤ_{p})_{(W (k))}, there are two actions of W (k) on ω(X), namely, that coming from the action i_{X} of W (k) on X and that coming from the action of W (k) on ω, and we define ω′(X, i_{X}) = W (k) ⊗_{W (k)⊗_{ℤp}W (k)} ω(X). Then ω′ is exact. Indeed, W (k) ⊗_{ℤ_{p}} W (k) is isomorphic to a product of copies of W (k) indexed by the elements of Gal(k/𝔽_{p}), and the map W (k) ⊗_{ℤ_{p}} W (k) → W (k) corresponds to the projection to the “id”-component. Therefore ω′ is a W (k)-valued fibre functor on R(k; ℤ_{p})_{(W (k))}, and R(k; ℤ_{p}) is Tannakian (apply 1.8(b)).

(b) When k has nonzero characteristic p and is infinite, then R(k; ℤ_{p}) is not Tannakian (at least according to our definition, which may be too strict), because there exist objects X for which End(X) is not a finitely generated ℤ_{p}-module.

Existence of fibre functors. We refer to Deligne 1989, §6, for the notion of the fundamental group of a Tannakian category T : it is an affine group scheme π(T ) in Ind T acting on the objects of T ; each fibre functor ω carries π(T ) to \underline{Aut}^{⊗}(ω) and the action of π(T ) on an object X to the action of \underline{Aut}^{⊗}(ω) to ω(X). When π(T ) is commutative, it lies in Ind T^{0} where T^{0} is the full subcategory of trivial objects. Since Hom(11, −): T^{0} → Vec(k) is an equivalence of categories, in this case π(T ) can be identified with an affine group scheme in the usual sense.

Proposition 6.9. Let M be a Tannakian category over ℚ whose fundamental group T is an algebraic group of multiplicative type, and let ω^{p} be a ℚ-linear tensor functor M → Mod(𝔸_{f}^{p}) such that ω_{ℓ} =_{df} ω^{p} ⊗_{𝔸_{f}^{p}} ℚ_{ℓ} is a ℚ_{ℓ}-valued fibre functor for all ℓ ≠ p. Then there exists a finite field extension L of ℚ, a fibre functor ω: M → Vec(L), and an isomorphism ω ⊗_{L} 𝔸_{f,L}^{p} → ω^{p} ⊗_{𝔸_{f}^{p}} 𝔸_{f,L}^{p} of tensor functors M → Mod(𝔸_{f,L}^{p}).

Proof. The hypotheses imply that ω^{p}(X) is a free 𝔸_{f}^{p}-module for all X (see the proof of 4.9).

Assume first that T is split over ℚ and that there exists a fibre functor ω: M → Vec(ℚ). We show in this case that there is an isomorphism ω ⊗_{ℚ} 𝔸_{f}^{p} → ω^{p}. We use ω to identify M with Rep(T; ℚ). Let Ξ be a basis for the group X^{∗}(T) of characters of T, and let X be the representation X = ⨁_{λ∈Ξ} X_{λ} where X_{λ} is a one-dimensional representation of T with character λ. Choose a graded lattice Λ in X, i.e., a ℤ-lattice such that Λ = ⨁_{λ∈Ξ} Λ∩X_{λ}. Likewise, choose a graded lattice Λ_{f} in ω^{p}(X). Then Λ_{ℓ} =_{df} Λ_{f} ⊗_{𝔸_{f}^{p}} ℤ_{ℓ} is a graded lattice in ω_{ℓ}(X). As T is a split torus, H^{1}(ℚ_{ℓ}, T) = 0, and so the theory of Tannakian categories shows that there exists an isomorphism of fibre functors α_{ℓ}: ω ⊗_{ℚ} ℚ_{ℓ} → ω_{ℓ} which is uniquely determined up to an element of T (ℚ_{ℓ}). We may choose α_{ℓ} to map Λ ⊗_{ℤ} ℤ_{ℓ} onto Λ_{ℓ}. Now the family (α_{ℓ}) defines an isomorphism ω ⊗_{ℚ} 𝔸_{f}^{p} → ω^{p}.

In the general case, there will exist a finite field extension L of ℚ such that T splits over L and such that there exists a fibre functor ω: M → Vec(L). Then ω defines a fibre functor M_{(L)} → Vec(L) (Deligne and Milne 1982, 3.11), and the previous argument applied to M_{(L)} proves the general case of the proposition. □

The category M(k; ℤ). Fix a field k, and let M(k; ℚ) and M(k; ℤ) be as in §5. Lemma 6.10. The category M(k; ℤ) has internal Homs.

Proof. Since each of the categories R(k; ℤ̂), R(k; 𝔸_{f}^{p}), M(k; ℚ) has internal Homs, it is straightforward to check this. □

Lemma 6.11. Every motive in M(k; ℤ) is a quotient of a reflexive motive.

Proof. The proof is similar to that of Proposition 1.8: a motive is reflexive if and only if it is torsion-free; every motive is a direct sum of a torsion motive and a torsion-free motive; every torsion motive is a quotient of 11(r)^{s} for some r and s. □

Theorem 6.12. The category M(𝔽_{q}; ℤ) is Tannakian.

Proof. We apply Proposition 6.6. It remains to prove the existence of fibre functors. We replace S with a subset generated by a finite set of varieties, and prove that M(k; ℤ; S)_{(R′)} admits an R′-valued fibre functor for some Dedekind domain R′ finite and flat over R. For this, we can apply Proposition 6.9 because the fundamental group of M =_{df} M(k; ℚ; S) is of multiplicative type: it is reductive because M is semisimple, and it is commutative because its extension to ℚ_{ℓ} is contained in the algebraic group fixing the Tate classes, which is commutative.

According to Proposition 6.9, there exists a finite field extension L of ℚ splitting T, a fibre functor ω_{0} : M → Vec(L), and an isomorphism ξ^{p}: L⊗_{ℚ}ω^{p} → 𝔸_{f}^{p} ⊗_{ℚ}ω_{0}. After possibly extending L, we may suppose that there is also an isomorphism ξ_{p}: L⊗_{ℚ}ω_{p} → ℚ_{p} ⊗_{ℚ} ω_{0} where ω_{p} is X ↦ X_{p} (regarded as a B(𝔽_{q})-module). Let R be the integral closure of ℤ in L.

For X_{0} in M_{(L)}, there are two actions of L on ω_{0}(X_{0}), namely, that coming from the action of L on ω_{0} and that coming from the action of L on X_{0}. Define

ω_{1}(X_{0}) = L ⊗_{L⊗_{ℚ}L} ω_{0}(X_{0}).

Then ω_{1} is an L-valued fibre functor on M_{(L)} (Deligne and Milne 1982, 3.11). For (X, i_{X}) in M_{(R)}, set

ω_{2}(X) = (X^{p}, ω′(X_{p}))

where ω′ is the functor defined (6.8a). Then ω_{2} is an R̂-valued fibre functor on M_{(R)}, and ξ defines an isomorphism L ⊗_{ℚ} ω_{2} → 𝔸_{f} ⊗_{ℚ} ω_{1}.

Thus, we have defined an exact tensor functor

X ↦ (ω_{2}(X), ω_{1}(X_{0}), ξ): M_{(R)} → Modf(R̂) ×_{Modf(𝔸_{f} ⊗R)} Modf(L).

Its image lies in the full subcategory of triples whose Modf(R̂)-component has finite torsion, which can be identified with the category Modf(R). It is therefore an Rvalued fibre functor on M(𝔽_{q}; ℤ)_{(R)}. □

Remark 6.13. If M(k; ℚ) is as in 5.24 or 5.26 and k is finite, or as in 5.25, then M(k; ℤ) is Tannakian. In the first two cases, this can be proved as in Theorem 6.12, and in the third case, there is a Betti fibre functor on M(k; ℤ).

Stacks of categories of motives. We refer the reader to Saavedra 1972, I 4.5, for the notion of tensor fibred category p: M → E. Briefly, it is a fibred category p: M → E endowed with an E-bifunctor ⊗: M ×_{E} M → M and compatible associativity and commutativity constraints for which there exists a cartesian section to p giving identity objects in each fibre; thus, for each E in E, the fibre M_{E} is a tensor category. When E is endowed with a topology (in the sense of Grothendieck) and p: M → E is a stack, we call it a stack of tensor categories. When the fibres are abelian categories and the inverse image functors are exact, we call p: M → E a stack of abelian tensor categories. Finally, when End(11) = R for each identity object 11 in each fibre, we call p: M → E a stack of abelian tensor R-categories. In particular, then each fibre is an abelian tensor R-category.

For a scheme S = ∐ Spec k_{i}, finite and étale over 𝔽_{q}, let M(S; ℤ) = ∏ M(k_{i}; ℤ). Theorem 6.14. Let Et_{𝔽_{q}} be the category of all schemes finite and étale over 𝔽_{q},

and endow Et_{𝔽_{q}} with the étale topology. For k = 𝔽_{q} or 𝔽, let M(k; ℚ) be as at the start of §5 and assume the Tate conjecture holds for M(k; ℚ). The categories M(−; ℤ) form (in a natural way) the fibres of a stack of abelian tensor ℤ-categories over Et_{𝔽_{q}}.

Proof. Immediate consequence of Proposition 5.19. □

7. Some Spectral Sequences

Abstract spectral sequences. We prove an abstract version of Tate’s spectral sequence (Milne 1986b, I 0.3). We fix a field k, and let Et_{k} be the category of all schemes finite and étale over k. Also, we fix a ring R, and let M → Et_{k} be a stack of noetherian abelian tensor R-categories with internal Homs over Et_{k}.

7.1. When X and Y are objects of an abelian category A, Ext^{r}(X, Y) denotes the Yoneda extension group (Mitchell 1965, VII). It agrees with the group defined in terms of the derived category (Verdier 1996, III 3.2.12). When A is noetherian, the Yoneda extension group also agrees with that defined using injective resolutions in the ind-category Ind(A) (Oort 1964, p. 235; see also Huber 1993).

7.2. An object of a tensor category is trivial if it is a quotient of a finite direct sum of copies of 11, and an object Y of an abelian tensor category is flat if the functor X ↦ X ⊗ Y is exact. When X and Y are objects of an abelian tensor category over R, we write

Tor_{i}^{R}(X, Y) = 0 for all i > 0

to mean Y has a resolution

⋯ → Y^{1} → Y^{0} → Y → 0

by flat objects which remains exact when it has been tensored by X.

7.3. Let k′ be a finite Galois extension of k with Galois group Γ, and write X ↦ X′ for an inverse image functor M(k) → M(k′). Because M → Et_{k} is a stack, the functor

X ↦ N(X) =^{df} Hom(11, X′)

defines an equivalence from the full subcategory of M(k) of objects becoming trivial in M(k′) to the category Modf(R[Γ]) with quasi-inverse N ↦ N ⊗ 11.

7.4. Let k′ be a finite étale k-algebra. The theory of “Weil restriction of scalars” (Weil 1982, 1.3) applies in the present situation and gives an exact functor M(k′) → M(k) that is both left and right adjoint to the inverse image functor M(k) → M(k′).

Proposition 7.5. Let k′ be a finite Galois extension of k with Galois group Γ. Write T′ for the inverse image in M(k′) of an object T of M(k). For all X, Y, Z in M(k) such that X′ is trivial and Tor_{i}^{R}(X, Y) = 0 for all i > 0, there is a spectral sequence

Ext_{R[Γ]}^{r}(N(X), Ext_{M(k′)}^{s}(Y′, Z′)) ⟹ Ext_{M(k)}^{r+s}(X ⊗ Y, Z).

Consider the functors (for fixed X, Y)

α: M(k) → Mod(R[Γ]),  α(Z) = Hom_{M(k′)}(Y′, Z′)
β: Mod(R[Γ]) → Ab  β(M) = Hom_{R[Γ]}(N(X), M)
γ: M(k) → Ab  γ(Z) = Hom_{M(k)}(X ⊗ Y, Z).

To prove the proposition, we shall show:

(a) β ∘ α ≅ γ;

(b) the functor α takes injective objects in Ind(M(k)) to β-acyclic R[Γ]-modules;

(c) the functor Z ↦ Z′: M(k) → M(k′) takes injective objects in Ind(M(k)) to injective objects in Ind(M(k′)).

Statements (a) and (b) imply that there is a spectral sequence

(R^{r}β ∘ R^{s}α)(Z) ⟹ (R^{r+s}γ)(Z);

hence,

Ext_{R[Γ]}^{r}(N(X), (R^{s}α)(Z)) ⟹ Ext_{M(k)}^{r+s}(X ⊗ Y, Z).

Statement (c), together with the exactness of Z ↦ Z′, shows that (R^{s}α)(Z) = Ext_{M(k′)}^{s}(Y′, Z′). Proof of (c). The functor Y ↦ Y′ has an exact left adjoint (7.4). □ Proof of (a). We have to prove

Hom_{R[Γ]}(N(X), Hom_{M(k′)}(Y′, Z′)) ≅ Hom_{M(k)}(X ⊗ Y, Z)

functorially in Z. From the definition of \underline{Hom} (Saavedra 1972, I 3.1.1)

Hom_{M(k′)}(X′ ⊗ Y′, Z′) ≅ Hom_{M(k′)}(X′, \underline{Hom}_{M(k′)}(Y′, Z′)).

But X′ ≅ N(X) ⊗ 11, and so

Hom_{M(k′)}(X′ ⊗ Y′, Z′) ≅^{6.1} Hom_{R}(N(X), Hom(11, \underline{Hom}_{M(k′)}(Y′, Z′)))
≅ Hom_{R}(N(X), Hom_{M(k′)}(Y′, Z′)).

On taking Γ-invariants we obtain (a). □ Proof of (b). We write Hom_{k} and Hom_{k′} for Hom_{Ind(M(k))} and Hom_{Ind(M(k′))} respectively. We have to show that if I is injective in Ind(M(k)), then

Ext_{R[Γ]}^{r}(N(X), Hom_{k′}(Y′, I′)) = 0 for r > 0.

We first show that, if Y is flat and I is injective, then Hom_{k′}(Y′, I′) is an injective R[Γ]-module. For this, we must show that the functor

Hom_{R[Γ]}(−, Hom_{k′}(Y′, I′)): Mod(R[Γ]) → Ab

is exact, at least on finitely generated R[Γ]-modules, but (a) shows that the composite of this functor with the equivalence X ↦ N(X) is the composite of the exact functors − ⊗ Y and Hom_{k}(−, I).

We now prove (b). Let

Y^{•} → Y

be a flat resolution of Y such that X ⊗ (Y^{•} → Y) is exact, and let I be an injective object of Ind(M(k)). Then, by (c), I′ is injective, and so

Hom_{k′}(Y′, I′) → Hom_{k′}(Y^{•′}, I′)

is an exact complex; in fact, it is an injective resolution of Hom_{k′}(Y′, I′). We shall use it to compute the groups Ext_{R[Γ]}^{r}(X, Hom_{k′}(Y′, I′)). Statement (a) shows that the complex

Hom_{R[Γ]}(N(X), Hom_{k′}(Y′, I′)) → Hom_{R[Γ]}(N(X), Hom_{k′}(Y^{•′}, I′))

is isomorphic to the complex

Hom_{k}(X ⊗ Y, I) → Hom_{k}(X ⊗ Y^{•}, I)

which is exact because X ⊗(Y → Y^{•}) is exact and I is injective. □

Corollary 7.6. Let k̄ be a Galois extension of k (possibly infinite), and let M(k̄) = \underrightarrow{lim}_{k′⊂k̄} M(k′) where k′ runs over the finite Galois extensions of k. Denote the inverse image functor M(k) → M(k̄) by X ↦ X̄, and let Γ = Gal(k̄/k). Let Y be an object of M(k) that admits a resolution by flat objects. For all Z in M(k), there is a spectral sequence

H^{r}(Γ, Ext_{M(k̄)}^{s}(Ȳ , Z̄)) ⟹ Ext_{M(k)}^{r+s}(Y, Z).

Proof. The condition on Y is that Tor_{i}^{R}(11, Y) = 0 for all i > 0. When k̄ is finite over k, the two definitions of M(k̄) coincide, and we obtain the spectral sequence by taking M = 11 in Proposition 7.5. When k̄ is of infinite degree over k, we pass to the limit over subfields k′ ⊂ k̄ that are finite and Galois over k. □

Spectral sequences at ℓ.

Proposition 7.7. Let Γ = Gal(𝔽/𝔽_{q}). For M, N in R(𝔽_{q}; ℤ_{ℓ}), there is a spectral sequence

H^{r}(Γ, Ext_{R(𝔽;ℤ_{ℓ})}^{s}(M, N)) ⟹ Ext_{R(𝔽_{q};ℤ_{ℓ})}^{r+s}(M, N).  (8)

Proof. Proposition 1.4 shows M has a resolution by flat objects, i > 0, and so this is a special case of Corollary 7.6. □

Denote Exts in Mod(ℤ_{ℓ}) by Ext_{ℤ_{ℓ}}.

Proposition 7.8. Let Γ = Gal(𝔽/𝔽_{q}). For M, N in R(𝔽_{q}; ℤ_{ℓ}), there is a spectral sequence

H_{cts}^{r}(Γ, Ext_{ℤ_{ℓ}}^{s}(M, N)) ⟹ Ext_{R(𝔽_{q};ℤ_{ℓ})}^{r+s}(M, N).  (9)

Here H_{cts}^{r} is computed using continuous cochains relative to the ℓ-adic topology on Ext_{ℤ_{ℓ}}^{s}(M, N).

Proof. When N is finite, (9) will become (8) once we have shown that the natural map

Ext_{R(𝔽;ℤ_{ℓ})}^{j}(M, N)) → Ext_{ℤ_{ℓ}}^{j}(M, N)

is an isomorphism. When M is finite, this is obvious (with no condition on N), because it is true for j = 0 and the forgetful functor Ind R(𝔽; ℤ_{ℓ}) → Mod(ℤ_{ℓ}) is exact and preserves injectives. The Ext(M, −) sequences of

0 → M ⟶^{ℓ^{n}} M → M/ℓ^{n}M → 0 , ℓ^{n}N = 0,  (10)
0 → M_{tors} → M → M/M_{tors} → 0

allow us to deduce it, first for all torsion-free M, and then for all M.

There are spectral sequences

E_{2}^{ij} = Ext_{R(𝔽_{q};ℤ_{ℓ})}^{i}(M, \underleftarrow{lim}^{j}N^{(ℓ^{n})})
E_{2}^{i,j} = \underleftarrow{lim}^{i} Ext_{R(𝔽_{q};ℤ_{ℓ})}^{j}(M, N^{(ℓ^{n})}),

with isomorphic final terms (cf. Jensen 1972, 4.3). The higher inverse limits vanish because the groups Ext_{R(𝔽_{q};ℤ_{ℓ})}^{j}(M, N^{(ℓ^{n})}) and N^{(ℓ^{n})} are finite. Moreover, \underleftarrow{lim} N^{(ℓ^{n})} ≅ N, and so this shows that

Ext_{R(𝔽_{q};ℤ_{ℓ})}^{i}(M, N) ≅ \underleftarrow{lim}_{n} Ext_{R(𝔽_{q};ℤ_{ℓ})}^{i}(M, N^{(ℓ^{n})}).  (11)

Similarly,

Ext_{ℤ_{ℓ}}^{j}(M, N) ≅ \underleftarrow{lim}_{n} Ext_{ℤ_{ℓ}}^{j}(M, N^{(ℓ^{n})}),

and so (Tate 1976, 2.2)

H_{cts}^{i}(Γ, Ext_{ℤ_{ℓ}}^{j}(M, N)) ≅ \underleftarrow{lim}_{n} H_{cts}^{i}(Γ, Ext_{ℤ_{ℓ}}^{j}(M, N^{(ℓ^{n})})).

Therefore, (9) can be obtained by passing to the limit in the spectral sequences

H_{cts}^{r}(Γ, Ext_{ℤ_{ℓ}}^{s}(M, N^{(ℓ^{n})})) ⟹ Ext_{R(𝔽_{q};ℤ_{ℓ})}^{r+s}(M, N^{(ℓ^{n})}).
□

Spectral sequences at p.

Lemma 7.9. For torsion-free objects M, N in R^{+}(𝔽; ℤ_{p}), Ext_{R^{+}(𝔽;ℤ_{p})}^{2}(M, N) is torsion-free.

Proof. From the exact sequences

0 → Ext_{R^{+}}^{1}(M, N)^{(p^{n})} → Ext_{R^{+}}^{1}(M, N^{(p^{n})}) → Ext_{R^{+}}^{2}(M, N)_{p^{n}} → 0

we obtain a surjection

\underrightarrow{lim}_{n} Ext_{R^{+}}^{1}(M, N^{(p^{n})}) → Ext_{R^{+}}^{2}(M, N)_{tors}.

Let R_{p^{n}}^{+} denote the full subcategory of R^{+}(𝔽; ℤ_{p}) of objects killed by p^{n}. The same argument as in the proof pp34–34 shows that the map

s: Ext_{R^{+}}^{1}(M, N^{(p^{n})}) → Ext_{R_{p^{n}}^{+}}^{1}(M^{(p^{n})}, N^{(p^{n})})

that replaces each term E in a short exact sequence with E^{(p^{n})} is an isomorphism. Therefore, it remains to show that

\underrightarrow{lim}_{n} Ext_{R_{p^{n}}^{+}}^{1}(M^{(p^{n})}, N^{(p^{n})}) = 0.  (12)

We claim (i) that the map

Ext_{Crys^{+}(𝔽)}^{1}(M ⊗ W (𝔽), N ⊗ W (𝔽)) → Ext_{R_{p^{n}}^{+}}^{1}(M^{(p^{n})}, N^{(p^{n})})

that replaces each term E in a short exact sequence by E^{(p^{n})} is surjective, and (ii) that Ext_{Crys^{+}(𝔽)}^{1}(M, N) is a torsion group. Together, (i) and (ii) imply (12) , because (i) gives a surjection

Ext_{Crys^{+}(𝔽)}^{1}(M ⊗ W (𝔽), M ⊗ W (𝔽)) ⊗ ℚ_{p}/ℤ_{p} → \underrightarrow{lim}_{n} Ext_{R_{p^{n}}^{+}}^{1}(M^{(p^{n})}, N^{(p^{n})})

and (ii) implies that the first group is zero.

To prove (i), note that an element of either group splits when regarded as an extension of W -modules, and is therefore determined by the σ-linear map describing the action of the Frobenius. Since every σ-linear map M^{(p^{n})} → N^{(p^{n})} lifts to a σ-linear map M → N, (i) is clear. According to a theorem of Manin (Demazure 1972, p85), the category Isoc^{+}(k^{al}) is semisimple, which implies (ii). □

Lemma 7.10. When k = 𝔽 or 𝔽_{q}, every object in R^{+}(k; ℤ_{p}) is a quotient of a torsion-free object.

Proof. Suppose that the object M is an extension of objects M′ and M′′ each of which is a quotient of a torsion-free objects, say, N′ ↠ M′, N′′ ↠ M′′. Pull back the original extension by N′′ → M′′:

0 ⟶ M′ ⟶ E ⟶ N′′ ⟶ 0
∥  ↓  ↓
0 ⟶ M′ ⟶ M ⟶ M′′ ⟶ 0

If the class of E in Ext^{1}(N′′, M′) is the image of an element of Ext^{1}(N′′, N′), then M also is a quotient of an torsion-free object. The obstruction to this is an element of Ext^{2}(N′′, K) where K is the kernel of N′ → M′. Certainly, when k = 𝔽_{q}, this group will be zero if there exists a resolution

0 → A ⟶^{·λ} A → N′′ → 0

(because in this case Exts in R_{p}^{+} agree with those in Mod(A) — see 7.1).

It suffices to prove the lemma in the case k = 𝔽_{q}, and, for a given M, an argument using restriction of scalars allows us to replace 𝔽_{q} by a finite extension. After such an extension, a torsion M will have a composition series whose quotients are isomorphic to one of (k, σ) or (k, 0). Each of these is a quotient

\begin{aligned}0 & → & (W, σ) & ⟶^{p} & (W, σ) & → & (k, σ) & → & 0 \\ 0 & → & (W, pσ) & ⟶^{p} & (W, pσ) & → & (k, 0) & → & 0. \end{aligned}

Since (W, σ) ≅ A/A(F − σ) and (W, pσ) ≅ A/A(F − pσ) the above remarks prove the lemma for M. Let M be an arbitrary object, and let M_{1} = M/M_{tors}. The obstruction to extending the statement from M_{tors} to M is a torsion element of Ext^{2}(M_{1}, K), which (see 7.9) becomes zero after a finite extension of 𝔽_{q}. □

Proposition 7.11. Let Γ = Gal(𝔽/𝔽_{q}). For M, N in R^{+}(𝔽_{q}; ℤ_{p}), there is a spectral sequence

H^{r}(Γ, Ext_{R^{+}(𝔽;ℤ_{p})}^{s}(M̄, N̄)) ⟹ Ext_{R^{+}(𝔽_{q};ℤ_{p})}^{r+s}(M, N).  (13)

Proof. Lemma 7.10 shows that M has a flat resolution. We can not apply Corollary 7.6 directly, because R^{+}(𝔽_{q′}; ℤ_{p}) does not have internal Homs. However, this is only used in the proof of (a) in 7.5, namely, in showing that

Hom_{ℤ_{p}[Γ]}(P, Hom_{R^{+}(𝔽_{q′};ℤ_{p})}(M′, N′)) ≅ Hom_{R^{+}(𝔽_{q};ℤ_{p})}(P ⊗ M, N)

where Γ = Gal(𝔽_{q′}/𝔽_{q}) and P is an object of R^{+}(𝔽_{q}; ℤ_{p}) that becomes isomorphic to a direct sum of copies of (W, σ) in R^{+}(𝔽_{q′}; ℤ_{p}). This can be shown directly. □

Proposition 7.12. Let Γ = Gal(𝔽/𝔽_{q}). For M, N in R^{+}(𝔽_{q}; ℤ_{p}), there is a spectral sequence

H_{cts}^{r}(Γ, Ext_{Crys^{+}(𝔽)}^{s}(M_{W (𝔽)}, N_{W (𝔽)})) ⟹ Ext_{R^{+}(𝔽_{q};ℤ_{p})}^{r+s}(M, N).  (14)

Here H_{cts}^{r} is computed using continuous cochains relative to the p-adic topology on Ext_{W (𝔽)}^{s}(M, N).

Proof. When N (or M) is finite, (14) becomes the (13) — see the proof of Proposition 7.8. The remainder of the proof of Proposition 7.8 applies to this case, except that N^{(p^{n})} and Ext_{Crys^{+}(𝔽)}^{j}(M, N^{(p^{n})}) need no longer be finite. However, N^{(p^{n})} is of finite length over W, and so we can apply Jensen 1972, 7.2, to see that \underleftarrow{lim}^{i}N^{(p^{n})} = 0 for i > 0. Moreover, there exists an N such that p^{N} · Ext_{Crys^{+}(𝔽)}^{j}(M, N^{(p^{n})}) is finite for all n, and so the inverse system Ext_{Crys^{+}(𝔽)}^{j}(M, N^{(p^{n})}) satisfies the Mittag-Leffler condition. □

8. Homological Algebra in the Category of Motives

In this section, we assume that the l-adic Tate conjectures hold for M^{+}(k; ℚ) (see p14).

Homological algebra in M^{+}(k; ℤ).

Theorem 8.1. Let X and Y be effective motives in M^{+}(k; ℤ).

(a) The map (α, β) ↦ α − β,

Hom_{M^{+}(k;ℚ)}(X_{0}, Y_{0}) × Hom_{R^{+}(k;ℤ̂)}(X_{f}, Y_{f}) → Hom_{R^{+}(k;𝔸_{f})}(ω_{f}(X_{0}), ω_{f}(Y_{0})),

is surjective with kernel Hom_{M^{+}(k;ℤ)}(X, Y).

(b) For i > 0, Ext_{M^{+}(k;ℤ)}^{i}(X, Y) is torsion and the sequence

0 → Ext_{M^{+}(k;ℤ)}^{i}(X, Y)_{ℤ_{l}} → Ext_{R(k;ℤ_{l})}^{i}(X_{l}, Y_{l}) → T_{l} Ext_{M^{+}(k;ℤ)}^{i+1}(X, Y) → 0 (15)

is exact (all l); hence,

Ext_{M^{+}(k;ℤ)}^{i}(X, Y) ≅ Ext_{R^{+}(k;ℤ̂)}^{i}(X_{f}, Y_{f})_{tors}.

(c) If k is separably closed, then Ext_{M^{+}(k;ℤ)}^{i}(X, Y) = 0 for all i > 2, and for all i > 1 when X and Y are torsion-free.

Proof. For any ℤ-module M of finite rank, the sequence

0 → M → M_{ℤ̂} × M_{ℚ} → M_{𝔸_{f}} → 0

is exact. We obtain (a) by taking M = Hom(X, Y) in this sequence and applying (5.6, 5.8, 5.10).

The remainder of the proof will require several steps.

Ext_{M^{+}(k;ℤ)}^{i}(X, Y) is torsion for i > 0: It suffices to prove that Ext_{M^{+}(k;ℤ)}^{1}(X, Y) is torsion. Moreover, we may assume that X and Y are torsion-free. Consider an exact sequence

0 → Y → E →^{π} X → 0

in M^{+}(k; ℤ). Because M^{+}(k; ℚ) is semisimple, there exists a morphism α_{0} : X_{0} → E_{0} such that π_{0} ∘ α_{0} = id_{X_{0}}. For some m ∈ ℤ, mα_{0} ∈ Hom_{M^{+}(k;ℤ)}(X, E) (see 5.5). Now the equation π ∘ (mα_{0}) = m id_{X} implies that the class of the sequence is killed by m.

Write f^{i}(X, Y) for the natural map

Ext_{M^{+}(k;ℤ)}^{i}(X, Y)_{ℤ_{l}} → Ext_{R^{+}(k;ℤ_{l})}^{i}(X_{l}, Y_{l}).

Recall (5.10) that f^{0}(X, Y) is an isomorphism.

The maps f^{i}(X, Y) are isomorphisms when X and Y are finite. The functor X ↦ X_{f} defines an equivalence of the category of torsion effective motives with the category of torsion objects in R^{+}(k; ℤ̂), and so it suffices to prove that the Exts in the torsion subcategories coincide with the Exts in the full categories. We do this for M^{+}(k; ℤ) since R^{+}(k; ℤ_{l}) is similar. Because M^{+}(k; ℤ) is noetherian, Ind(M^{+}(k; ℤ)) has enough injectives, and the Exts computed using injective resolutions coincide with the Yoneda Exts (7.1). Because n: 11 → 11 is injective, every injective object in Ind(M^{+}(k; ℤ)) is divisible, and so an injective resolution Y → I^{•} in Ind(M^{+}(k; ℤ)) gives an injective resolution Y → I_{tors}^{•} in the subcategory of torsion objects. As Hom(X, I_{tors}^{•}) ≅ Hom(X, I^{•}), this proves that the Exts in the two categories agree.

The maps f^{i}(X, Y) are isomorphisms when Y is finite. It suffices to prove this when X is torsion-free and Y is l-torsion, say l^{n}Y = 0. The Ext sequences of

0 → X ⟶^{l^{n}} X → X/l^{n}X → 0

give a diagram

0 ⟶ Ext_{M^{+}}^{i−1}(X, Y) ⟶ Ext_{M^{+}}^{i}(X/l^{n}X, Y) ⟶ Ext_{M^{+}}^{i}(X, Y) ⟶ 0
↓^{f^{i−1}(X,Y)}  ↓≅  ↓^{f^{i}(X,Y)}
0 ⟶ Ext_{R^{+}}^{i−1}(X_{l}, Y_{l}) ⟶ Ext_{R^{+}}^{i}(X_{l}/l^{n}X_{l}, Y_{l}) ⟶ Ext_{R^{+}}^{i}(X_{l}, Y_{l}) ⟶ 0

from which it follows that f^{i−1}(X, Y) is injective and f^{i}(X, Y) is surjective. Since this holds for all i, f^{i}(X, Y) is an isomorphism.

The maps f^{i}(X, Y) are isomorphisms when X is finite. Similar to the preceding case.

The sequence (15) is exact. Because f^{i}(X, Y) is an isomorphism when Y is finite, f^{i}(X, Y^{•}) (map of hyper-Exts) is an isomorphism when Y^{•} is a bounded complex with finite cohomology, for example, when Y^{•} = Y^{{l^{n}}} =_{df} Y ⟶^{l^{n}} Y:

Ext_{M^{+}}^{i}(X, Y^{{l^{n}}}) ≅ Ext_{R^{+}}^{i}(X_{l}, Y_{l}^{{l^{n}}}).

There are exact sequences

0 → Ext_{M^{+}}^{i}(X, Y)^{(l^{n})} → Ext_{M^{+}}^{i}(X, Y^{{l^{n}}}) → Ext_{M^{+}}^{i}(X, Y)_{l^{n}} → 0,  (16)

and

\underleftarrow{lim} Ext_{M^{+}}^{i}(X, Y^{\{l^{n}\}}) ≅ \underleftarrow{lim} Ext_{R^{+}}^{i}(X_{l}, Y_{l}^{\{l^{n}\}}) ≅ \underleftarrow{lim} Ext_{R^{+}}^{i}(X_{l}, Y_{l}^{(l^{n})}) ≅ Ext_{R^{+}}^{i}(X_{l}, Y_{l})

(for the third isomorphism, see the proofs of 7.8 and 7.12). Thus, on passing to the inverse limit in (16), we obtain (15).

This completes the proof of (b). We now assume that k = k^{sep}.

Ext_{M^{+}}^{2}(X, Y) = 0 if X and Y are torsion-free. For an integer m, let M_{m}^{+} (resp. R_{m}^{+}) denote the subcategory of M^{+}(k; ℤ) (resp. R^{+}(k; ℤ̂)) of objects killed by m. On passing to the direct limit in the sequence

0 → Ext_{M^{+}}^{1}(X, Y)^{(m)} → Ext_{M^{+}}^{1}(X, Y^{(m)}) → Ext_{M^{+}}^{2}(X, Y)_{m} → 0

and using that Ext_{M^{+}}^{1}(X, Y) and Ext_{M^{+}}^{2}(X, Y) are torsion, we find that

\underrightarrow{lim}_{m} Ext_{M^{+}}^{1}(X, Y^{(m)}) ≅ Ext_{M^{+}}^{2}(X, Y).

Consider the diagram

Ext_{M^{+}}^{1}(X^{(m)}, Y^{(m)})  \overline{π  ^{}}Ext_{M^{+}}^{1}(X, Y^{(m)})
[figure: i; s]
Ext_{M_{m}^{+}}^{1}(X^{(m)}, Y^{(m)})

in which π is the surjection induced by X → X^{(m)}, i is the obvious injection, and s applied to a short exact sequence replaces each motive E in the sequence with E^{(m)}. One checks easily that s ∘ (π ∘ i) = id and (π ∘ i) ∘ s = id, and so s is an isomorphism. Thus

Ext_{M^{+}}^{2}(X, Y) ≅ \underrightarrow{lim}_{m} Ext_{M_{m}^{+}}^{1}(X^{(m)}, Y^{(m)}) ≅ \underrightarrow{lim}_{m} Ext_{R_{m}^{+}}^{1}(X_{f}^{(m)}, Y_{f}^{(m)}).

For l ≠ p, Ext_{R_{l^{n}}^{+}}^{1}(X_{f}^{(l^{n})}, Y_{f}^{(l^{n})}) is obviously zero, and so it remains to show that

\underrightarrow{lim}_{p^{n}} Ext_{R_{p^{n}}^{+}}^{1}(X_{p}^{(p^{n})}, Y_{p}^{(p^{n})}) = 0.  (17)

But this is shown in the proof of (7.9) (for k = 𝔽, but the argument works generally).

Ext_{M^{+}}^{i}(X, Y) = 0 for all i > 2. This follows from (b) and the fact that Ext_{R^{+}(k;ℤ_{l})}^{i}(M, N) = 0 for i > 2 (for l ≠ p, this is obvious, and for l = p it can be proved by the arguments used to prove the p-case of Theorem 9.1 below). □

Remark 8.2. The group Ext_{R^{+}(𝔽;ℤ_{ℓ})}^{1}(M, N) need not be torsion. Consider, for example, the extension

0 → ℤ_{ℓ} → ℤ_{ℓ} ⊕ ℤ_{ℓ} → ℤ_{ℓ} → 0

on which the Frobenius element in Γ = Gal(𝔽/𝔽_{p}) acts on the middle term as the matrix ( \begin{matrix}1 & a \\ 0 & 1 \end{matrix} ) with a a nonzero element of ℤ_{ℓ}. Multiplying the class of the extension by m corresponds to multiplying a by m, and so this represents a nontorsion element of Ext_{R^{+}(𝔽_{p};ℤ_{ℓ})}^{1}(M, N), which remains nontorsion in Ext_{R^{+}(𝔽;ℤ_{ℓ})}^{1}(M, N). Thus, in general,

Ext_{M^{+}(𝔽;ℤ)}^{1}(X, Y)(ℓ) ≉ Ext_{R(𝔽;ℤ_{ℓ})}^{1}(X_{ℓ}, Y_{ℓ}).

Similarly, the group Ext_{R^{+}(𝔽;ℤ_{p})}^{1}(M, N) need not be torsion (because the category R^{+}(𝔽; ℤ_{p}), in contrast to Isoc^{+}(𝔽), is not semisimple).

Homological algebra in M^{+}(𝔽_{q}; ℤ).

Lemma 8.3. Every effective motive over 𝔽_{q} or 𝔽 is a quotient of a torsion-free effective motive.

Proof. Straightforward adaptation of the proof of Lemma 7.10. □

Theorem 8.4. Let Γ = Gal(𝔽/𝔽_{q}). For all motives X and Y over 𝔽_{q}, there is a spectral sequence:

H^{r}(Γ, Ext_{M^{+}(𝔽;ℤ)}^{s}(X_{/𝔽}, Y_{/𝔽})) ⟹ Ext_{M^{+}(𝔽_{q};ℤ)}^{r+s}(X, Y).

Proof. Lemma 8.3 shows that X has a flat resolution. We can not apply Corollary 7.6 directly because M^{+}(𝔽_{q}; ℤ) does not have internal Homs, but its proof can be adapted as in the proof of 7.11. □

Homological algebra in M(k; ℤ). The results of the last two subsections hold mutatis mutandis for motives.

Examples.

Example 8.5. We apply (8.1) to compute Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, ℤ(r)) (we are writing ℤ for 11 and ℤ(r) for 𝕋^{⊗r}).

For r = 0, Hom_{M^{+}(𝔽;ℤ)}(ℤ, ℤ(0)) ≅ ℤ and Ext_{M^{+}(𝔽;ℤ)}^{i}(ℤ, ℤ(0)) = 0 for i = 1, 2.. For r ≠ 0, Hom_{M^{+}(𝔽;ℤ)}(ℤ, ℤ(r)) = 0, and so

Hom_{M^{+}(𝔽;ℤ)}(ℤ, ℤ(r)^{(m)}) →^{≅} Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, ℤ(r))_{m}.

For m = p^{n}, the left hand side is zero, and for m = ℓ^{n} it is (ℤ^{(ℓ^{n})})(r). Therefore, on passing to the direct limit over m, we find that

Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, ℤ(r)) ≅ (⨁_{ℓ≠p} ℚ_{ℓ}/ℤ_{ℓ}) (r)

where (ℚ_{ℓ}/ℤ_{ℓ})(r) is ℚ_{ℓ}/ℤ_{ℓ} with the Frobenius element of Gal(𝔽/𝔽_{q}) acting as q^{r}. Moreover, Ext_{M^{+}(𝔽,ℤ)}^{i}(ℤ, ℤ(r)) = 0 for i ≠ 0, 2. (Without the +, these statement holds only modulo p-torsion.)

Example 8.6. We compute Ext_{M^{+}(𝔽_{q},ℤ)}^{1}(ℤ, ℤ(r)) for r > 0. The spectral sequence in (8.4) shows that Ext_{M^{+}(𝔽_{q},ℤ)}^{1}(ℤ, ℤ(r)) and Ext_{M^{+}(𝔽_{q},ℤ)}^{2}(ℤ, ℤ(r)) are the kernel and cokernel respectively of

⊕_{ℓ≠p}ℚ_{ℓ}/ℤ_{ℓ} ⟶^{x↦(q^{r}−1)x} ⊕_{ℓ≠p}ℚ_{ℓ}/ℤ_{ℓ}.

Therefore Ext_{M^{+}(𝔽_{q};ℤ)}^{1}(ℤ, ℤ(r)) is cyclic of order q^{r} − 1, and Ext_{M^{+}(𝔽_{q};ℤ)}^{i}(ℤ, ℤ(r)) = 0 for i ≠ 0, 2.

Example 8.7. Let A be an abelian variety over 𝔽, and let h_{1}A be the isomotive h_{1}(A)_{0} endowed with the ℤ-structure provided by the maps H_{1}(A, ℤ_{l}) → H_{1}(A, ℚ_{l}). Since Hom(ℤ, h_{1}A) = 0,

Hom_{M^{+}(𝔽;ℤ)}(ℤ, (h_{1}A)^{(m)}) ≅ Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, h_{1}A)_{m}

for all m, and so

Hom_{M^{+}(𝔽;ℤ)}(ℤ, h_{1}A ⊗ ℚ/ℤ) ≅ Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, h_{1}A).

But

Hom_{M^{+}(𝔽;ℤ)}(ℤ, h_{1}A ⊗ ℚ/ℤ) ≅ Hom_{R^{+}(𝔽;ℤ̂)}(ℤ, (h_{1}A)_{fℚ}/(h_{1}A)_{f}) ≅ A(𝔽).

Thus,

Ext_{M^{+}(𝔽;ℤ)}^{1}(ℤ, h_{1}A) ≅ A(𝔽).

When A is defined over 𝔽_{q}, Gal(𝔽/𝔽_{q}) acts, and, on taking invariants, we find that

Ext_{M^{+}(𝔽_{q};ℤ)}^{1}(ℤ, h_{1}A) ≅ A(𝔽_{q}).

A similar argument using that Hom_{M^{+}(𝔽;ℤ)}((h_{1}A)(−1), ℤ) = 0 shows that

Hom_{M^{+}(𝔽;ℤ)}((h_{1}A)(−1), ℚ/ℤ) ≅ Ext_{M^{+}(𝔽;ℤ)}^{1}((h_{1}A)(−1), ℤ).

But

Hom_{M^{+}(𝔽;ℤ)}((h_{1}A)(−1), ℚ/ℤ) ≅ Hom_{R^{+}(𝔽;ℤ̂)}((h_{1}A)_{f}(−1), ℚ/ℤ) ≅ A^{∨}(𝔽)

where A^{∨} is the dual abelian variety. For an abelian variety A over 𝔽_{q}, this becomes

Ext_{M^{+}(𝔽_{q};ℤ)}^{1}((h_{1}A)(−1), ℤ) ≅ A^{∨}(𝔽_{q}).

Remark 8.8. It is generally conjectured, that for a smooth variety V over a field F,

H_{mot}^{i}(V, ℚ(r)) ≅ (K_{2r−i}V)_{ℚ}^{(r)}  (18)

and

H_{mot}^{i}(V, ℚ(r)) ≅ Ext_{MM(F;ℚ)}^{i}(11, (hV)(r));  (19)

see, for example, Deligne 1994, 3.7, 3.3. Here Ext_{MM(F;ℚ)}^{i}(11, (hV)(r)) is computed in the conjectural category of mixed isomotives over F and (K_{2r−i}V)_{ℚ}^{(r)} is the subspace of (K_{2r−1}V)_{ℚ} on which the Adams operators ψ^{k} act as k^{r}.

In particular, for F itself, it is conjectured that

Ext_{MM(F;ℚ)}^{1}(ℚ, ℚ(r)) ≅ (K_{2r−1}F)_{ℚ}^{(r)}  (20)

For F = 𝔽_{q} or 𝔽, the statement (20) is true because the K-groups are torsion and, under any definition, the category of mixed isomotives will be semisimple, so both sides are 0.

With ℤ-coefficients, statement (20) is expected to be false in general, and (18) is definitely false^{10}. However, Quillen (1972) shows that K_{2r−1}𝔽 ≈ ⊕_{ℓ≠p}ℚ_{ℓ}/ℤ_{ℓ} with the Frobenius automorphism acting as p^{r} and K_{2r−1}𝔽_{q} = (K_{2r−1}𝔽)^{Gal(𝔽/𝔽_{q})}, and Hiller (1981) and Kratzer (1980) show that the Adams operator ψ^{k} acts on K_{2r−1}𝔽 as multiplication by k^{r}, and so an integral version of (20) does hold when F = 𝔽_{q} or 𝔽:

Ext_{M^{+}(F;ℤ)}^{1}(ℤ, ℤ(r)) ≈ (K_{2r−1}F)^{(r)}.  (21)

The isomorphism (21) can be made canonical: Soulé (1979, IV.2, p. 284) proves that the Chern class map c_{i,1} : K_{2i−1}(𝔽_{q})_{ℤ_{l}} → H^{1}(𝔽_{q}, ℤ_{l}(i)) is an isomorphism if i < l; later Dyer and Friedlander (1985) defined a Chern character ch_{i,1} : K_{2i−1}(𝔽_{q})_{ℤ_{l}} → H^{1}(𝔽_{q}, ℤ_{l}(i)) that is related the Chern class map by (−1)^{i−1}(i − 1)!ch_{i,1} = c_{i,1} (Soulé 1999, 3.2.2, p. 267).

9. Extensions and Zeta Functions: Local Case

We use the notations: Γ = Gal(𝔽/𝔽_{q}), q = p^{a}, W = W (𝔽_{q}), B = B(𝔽_{q}), A = W [F, σ], R_{ℓ} = R(𝔽_{q}; ℤ_{ℓ}) =_{df} R(Γ; ℤ_{ℓ}), R̄_{ℓ} = Mod(ℤ_{ℓ}), R_{p} = R^{+}(𝔽_{q}, ℤ_{p}) =_{df} Crys^{+}(𝔽_{q}), R̄_{p} = Crys^{+}(𝔽).

Statement of the theorem. Let M be an object of R_{l}. The Frobenius endomorphism π_{M} of M is the map induced by the element x ↦ x^{q} of Γ when l ≠ p and by F^{a} when l = p. We denote the characteristic and minimum polynomials of π_{M} acting on M_{ℚ_{l}} by m_{M}(t) and P_{M}(t) respectively. They are monic polynomials with coefficients in ℤ_{l} (see Demazure 1972, p89, for the case l = p). For M in R_{p}, let r(M) denote the rank of M and s(M) the sum of the slopes of M; thus if

P_{M}(T) = t^{h} + ⋯ + c

then r(M) = h and s(M) = ord_{p}(c)/a (cf. Demazure 1972, p90). For M in R_{ℓ} we set s(M) = 0.

Let M, N be objects of R_{l}. From the spectral sequences (7.8, 7.12) we obtain a map

HomR̄_{l}(M̄, N̄)_{Γ} → Ext_{R_{l}}^{1}(M, N).

Define

f = f(M, N): Hom_{R_{l}}(M, N) → Ext_{R_{l}}^{1}(M, N)

to be the composite of this with the obvious map Hom_{R_{l}}(M, N) → HomR̄_{l}(M̄, N̄)_{Γ}.

Theorem 9.1. If m_{M}(T) and m_{N}(T) have no multiple root in common, then z(f) is defined and satisfies

z(f) · [Ext_{R_{l}}^{2}(M, N)] = |q^{s(M)·r(N)} · ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} )|_{l}.  (22)

where (a_{i})_{1≤i≤r(M)} and (b_{j})_{1≤j≤r(N)} are the roots of P_{M}(t) and P_{N}(t) respectively.

Remark 9.2. (a) When l ≠ p, the term q^{s(M)·r(N)} = 1.

H_{mot}^{3}(V, ℤ(1)) = H^{2}(V, 𝔾_{m}),

10For a projective smooth variety over a field k, which is a nonzero torsion group in general whereas K_{−1}V = 0. See also Bloch and Esnault 1996, p. 304, which explains why a closely related conjecture should not be true integrally.

(b) In the course of the proof, we shall show that Ext_{R_{l}}^{i}(M, N) = 0 for i > 2 (without any condition on M and N) and that Ext_{R_{l}}^{2}(M, N) is finite.

(c) When one of M or N is finite, the theorem simply states that the groups Ext_{R_{l}}^{i}(M, N) are finite and the alternating product of their orders is 1.

(d) Let P_{M^{∨}⊗N}(t) = ∏ (1 − \frac{b_{j}}{a_{i}} t), and let ρ(M, N) be the rank of Hom_{R_{l}}(M, N). Then ρ(M, N) is the number of pairs (i, j) with a_{i} = b_{j}, and (22) becomes the equation

z(f) · [Ext^{2}(M, N)] = |q^{s(M)·r(N)} · lim_{t→1} \frac{P_{M^{∨}⊗N}(t)}{(1 − t)^{ρ(M,N)}} |_{l} .

(e) When the groups Ker(f), Coker(f), Ext_{R_{l}}^{2}(M, N), Ext_{R_{l}}^{3}(M, N), … are finite and almost all zero, then we say that χ^{×}(M, N) is defined, and we set it equal to the alternating product of the orders of the groups. Granted (b), Theorem 9.1 is equivalent to the statement: if m_{M}(T) and m_{N}(T) have no multiple root in common, then χ^{×}(M, N) is defined and equals

|q^{s(M)·r(N)} · ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} )|_{l}  (23)

(f) Let

0 → N′ → N → N′′ → 0

be an exact sequence in R_{l}. If χ^{×}(M, N′) and χ^{×}(M, N′′) are defined, then so also is χ^{×}(M, N), and

χ^{×}(M, N) = χ^{×}(M, N′) · χ^{×}(M, N′′).

Since a similar statement holds for exact sequences in M and the expression (23) is multiplicative, we see that it suffices to prove Theorem 9.1 for M and N running through a set of generators for the Grothendieck group K(R_{l}) of R_{l}.

Even more is true: both χ^{×} and the expression in (23) take values in ℚ_{+}^{×}, the group of positive rational numbers, which is torsion-free. Therefore, in order to prove Theorem 9.1, it suffices to verify it for M and N running through a set of generators for K(R_{l}) ⊗ ℚ.

Proof of Theorem 9.1 in the case l ≠ p. From Proposition 7.8, we obtain an exact sequence and isomorphisms

Hom_{R_{ℓ}}(M, N) ≅ Hom_{ℤ_{ℓ}}(M, N)^{Γ}
0 → Hom_{ℤ_{ℓ}}(M, N)_{Γ} ⟶^{c} Ext_{R_{ℓ}}^{1}(M, N) → Ext_{ℤ_{ℓ}}^{1}(M, N)^{Γ} → 0
Ext_{ℤ_{ℓ}}^{1}(M, N)_{Γ} ≅ Ext_{R_{l}}^{2}(M, N)
Ext_{R_{l}}^{i}(M, N) = 0 for i > 2.

Let f_{0} be the map

Hom_{ℤ_{ℓ}}(M, N)^{Γ} → Hom_{ℤ_{ℓ}}(M, N)_{Γ}

induced by the identity map. Then f = c ∘ f_{0}, and so (see 9.8)

z(f) = z(f_{0})/[Ext_{ℤ_{ℓ}}^{1}(M, N)^{Γ}].  (24)

Let γ be the generator x ↦ x^{q} of Γ. The family of eigenvalues of γ on Hom_{ℤ_{ℓ}}(M, N) is (\frac{b_{i}}{a_{j}})_{i,j}. Because of our hypothesis on m_{M}(t) and m_{N}(t), (9.9) applies, and shows that

z(f_{0}) = |∏_{a_{i}≠b_{j}} (1 − \frac{b_{i}}{a_{j}} )|_{ℓ} .

Because Ext_{ℤ_{ℓ}}^{1}(M, N) is finite,

[Ext_{ℤ_{ℓ}}^{1}(M, N)^{Γ}] = [Ext_{ℤ_{ℓ}}^{1}(M, N)_{Γ}]

and so

[Ext_{R_{l}}^{2}(M, N)] = [Ext_{ℤ_{ℓ}}^{1}(M, N)^{Γ}].

On substituting for z(f_{0}) and [Ext_{ℤ_{ℓ}}^{1}(M, N)^{Γ}] in (24), we obtain (22).

Proof of Theorem 9.1 in the case l = p. An A-module M will be said to be special if it is cyclic, M = A/A · λ, and λ lies in the centre ℤ_{p}[F^{a}] of A, in which case λ(F) = m_{M}(F^{a}) and P_{M}(t) = m_{M}(t)^{a}. We let k denote the A-module with underlying W -module 𝔽_{q} and with F acting as 0.

Let R_{p,f} be the full subcategory of R_{p} of finite objects. Then Exts computed in R_{p,f} coincide with those in R_{p} (see the argument p33) and with those in Pro-R_{p,f} (R_{p,f} is Artinian and so we can apply the opposite of 7.1). There is a canonical functor R_{p} →Pro-R_{p,f} and by using the usual exact sequences (10) one deduces that Exts in R_{p} coincide with those in Pro-R_{p,f}. This allows us to use resolutions by free finitely generated A-modules to compute Exts in R_{p}.

The case M or N is finite with F acting invertibly. Because of (6), this case is essentially the same as the non-p case. From the spectral sequence (7.12), we obtain exact sequences

0 → Exti−1 ; R̄_{p}(M_{W (𝔽)}, N_{W (𝔽)})_{Γ} → Ext_{R_{p}}^{i}(M, N) → Exti ; R̄_{p}(M_{W (𝔽)}, N_{W (𝔽)})^{Γ} → 0.

In this case, the groups Exti ; R̄_{p}(M_{W (𝔽)}, N_{W (𝔽)}) are finite, and it follows immediately that the alternating product of the orders of the Ext_{R_{p}}^{i}(M, N) is 1.

The case of M = k and N finite. We have the following Koszul-type resolution of k

0 → A ⟶^{t} A ⊕ A ⟶^{s} A → k → 0  (25)

where t(α) = (αF, pα) and s(x, y) = px − yF. For any N, the groups Ext^{i}(k, N) are the cohomology groups of the complex

Hom(A, N) ⟶^{∘s} Hom(A ⊕ A, N) ⟶^{∘t} Hom(A, N) → 0 → ⋯  (26)

Therefore, Ext^{i}(k, N) = 0 for i > 2, and if N is finite, then

Ext^{0}(k, N) · Ext^{2}(k, N) = Ext^{1}(k, N)

because the groups in (26) are finite.

The case M cyclic and N = k. The resolution,

0 → A ⟶^{·λ} A → M → 0,

gives an exact sequence

0 → Hom(M, k) → Hom(A, k) → Hom(A, k) → Ext^{1}(M, k) → 0

and equalities Ext^{i}(M, k) = 0 (i > 1). As Hom(A, k) ≅ k is finite, this implies that

[Ext^{0}(M, k)] = [Ext^{1}(M, k)].

The case M = k and N cyclic. From (25) and (26), we find that Ext^{i}(k, A) = 0 unless i = 2 in which case Ext^{2}(k, A) = k. Now the Ext(k, −)-sequence of

0 → A ⟶^{·λ} A → M → 0

gives an exact sequence

0 → Ext^{1}(k, N) → Ext^{2}(k, A) → Ext^{2}(k, A) → Ext^{2}(k, N) → 0

and equalities Ext^{i}(k, N) = 0 (i ≠ 1, 2). As Ext^{2}(k, A) is finite, the sequence gives that

[Ext^{1}(k, N)] = [Ext^{2}(k, N)].

The case M, N special, m_{M} and m_{N} relatively prime. Let

M = A/Aλ_{1}, λ_{1} = m_{M}(F^{a})
N = A/Aλ_{2}, λ_{2} = m_{N}(F^{a})

with m_{M}(t) and m_{N}(t) relatively prime. From the resolution

0 → A ⟶^{·λ_{1}} A → M → 0

we obtain an exact sequence

0 → Hom(M, N) → N ⟶^{λ_{1}·} N → Ext^{1}(M, N) → 0 → . . ..

As ·λ_{1} is injective, we see that Ext^{i}(M, N) = 0 for i ≠ 1, and so

z(f) · [Ext^{2}(M, N)] = z(λ_{1}·) =^{9.7} |det(λ_{1}·)|_{p}^{a} .

Here det(λ_{1}·) can be computed as the determinant of m_{M}(F^{a}) acting on the B-vector space N_{ℚ}. But P_{M}(t) = m_{M}(t)^{a}, and so

det(m_{M}(F^{a})|N_{ℚ})^{a} = ∏_{\begin{matrix}1≤i≤r(M) \\ 1≤j≤r(N) \end{matrix}} (a_{i} − b_{j}).

Thus,

z(f) · [Ext^{2}(M, N)] = |∏(a_{i} − b_{j})|_{p} = |q^{s(M)·r(N)} ∏ (1 − \frac{b_{j}}{a_{i}} )|_{p}

as required.

The case M and N are special and equal. We first give an explicit description of the map

f: Hom(M, N) → Ext^{1}(M, N)

in the case that M = A/A · λ_{1} and N = A/A · λ_{2} are special but not necessarily equal. We use ^{−} to denote − ⊗_{W (𝔽_{q})} W (𝔽), and we let σ_{k} be the canonical generator of Gal(𝔽/𝔽_{q}). The Ext(−, N) sequence of

0 → A →^{·λ_{1}} A → M → 0

is

0 → Hom(M, N) → N →^{λ_{1}·} N → Ext^{1}(M, N) → 0

and the Ext^{r}(−, N̄) sequence of

0 → Ā →^{·λ_{1}} Ā → M̄ → 0

is

0 → Hom(M̄, N̄) → N̄ →^{λ_{1}·} N̄ → Ext^{1}(M̄, N̄) → 0.

The map f can be described as follows: let u ∈ Hom(M, N) and regard u as an element of N such that λ_{1}u = 0; then u can be written u = (σ_{k} − 1)v for some v ∈ N̄; now

(σ_{k} − 1)(λ_{1}v) = λ_{1}(σ_{k} − 1)v = λ_{1}u = 0,

and so λ_{1}v ∈ N̄^{Γ} = N; the image of λ_{1}v under N → Ext^{1}(M, N) is f(u). Now take N = M, so that multiplication by λ = λ_{1} = λ_{2} is zero on M. Thus

Hom(M, N) = A/Aλ = Ext^{1}(M, N)

and f is an endomorphism of A/A/λ. Since A/Aλ is torsion-free, z(f) is defined if and only if

f_{ℚ}: A_{ℚ}/A_{ℚ}λ → A_{ℚ}/A_{ℚ}λ

has nonzero determinant (9.7), in which case

z(f) = | det(f)|_{p}^{a}.

Let u ∈ A_{ℚ}/A_{ℚ}λ and choose v ∈ Ā_{ℚ}/Ā_{ℚ}λ such that u = σ_{k}v−v. Then σ_{k}^{i}v = iu+v for all i. Let

λ(F) = F^{ma} + b_{ma−a}F^{ma−a} + ⋯ + b_{0} = m(F^{a}).

Then

f(u) = λ(F)v
= muF^{ma} + (m − 1)b_{ma−a}uF^{ma−a} + ⋯ + 0 (as vλ = 0)
= uF^{a} \frac{d}{dF^{a}} (m(F^{a}))
= F^{a} \frac{d}{dF^{a}} (m(F^{a}))u.

From (9.7) we find that

z(f) = |det (\frac{d}{dF^{a}} (m(F^{a})))|_{p}^{a} · |det(F^{a})|_{p}^{a} .

But

| det(F^{a})|_{p}^{a} = |q^{a·s(N)}|_{p}

and

|det (\frac{d}{dF^{a}} (m(F^{a})))|_{p}^{a} = | ∏_{a_{i}≠b_{j}}(a_{i} − b_{j})|_{p} = |q^{s(N)(r(N)−a)} ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} )|_{p} .

Thus,

z(f) = |q^{r(N)·s(N)} ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} )|_{p} .

As Ext^{2}(M, N) = 0 this proves Theorem 9.1 in this case.

The general case. An isogeny (denoted ∼) of objects in R_{p} is a homomorphism whose kernel and cokernel have finite length over W. We say an object in R_{p} is indecomposable if it is not isogenous to a direct sum of two nonzero objects.

Lemma 9.3. Every object of R_{p} is isogenous to a direct sum of indecomposable objects, and the decomposition is unique up to isogeny.

Proof. Apply the Krull-Schmidt theorem in the category of A_{ℚ}-modules. □

Lemma 9.4. If M in R_{p} is indecomposable, then M_{ℚ} ≈ A_{ℚ}/A_{ℚ}λ for some λ ∈ A_{ℚ}; moreover, the minimum polynomial m_{M}(t) of the Frobenius endomorphism of M is a power of a ℤ_{p}-irreducible polynomial, and there exists an integer e such that ⊕^{e}M ∼ A/A · m_{M}(t).

Proof. The ring A_{ℚ} is the skew polynomial ring B[F, σ]. It is therefore a principal ideal domain (Jacobson 1943, III 1, p30), and every finitely generated module over such a ring is isomorphic a direct sum of cyclic modules (ibid. III, Theorem 19, p44). This proves the first part.

For the second, we use that the two-sided ideals in A_{ℚ} are precisely those generated by a polynomial in the centre ℚ_{p}[F^{a}] of A_{ℚ} (ibid. III 5, p38). It follows that A_{ℚ}·m(F^{a}) is the largest two-sided ideal contained in A_{ℚ}·λ, i.e., it is the bound of the ideal A_{ℚ}·λ in the sense of ibid. III 6, p38. Because A_{ℚ}/A_{ℚ}λ is indecomposable, this implies that m(t) is a power of a ℤ_{p}-irreducible polynomial (ibid. III Theorem 13, p40). Finally, when A_{ℚ}/A_{ℚ} · m(F^{a}) is decomposed into a direct sum of indecomposable modules A_{ℚ}/A_{ℚ}λ′, each of the ideals A_{ℚ}λ′ has A_{ℚ} · m(F^{a}) as its bound, and is therefore isomorphic to M (ibid. III Theorem 20, p45). □

Proposition 9.5. Objects of the following types generate K(R_{p})_{ℚ}:

(a) k = (𝔽_{q}, 0);

(b) (M, F_{M}) with M finite and F_{M} invertible;

(c) M special and m_{M}(t) is a power of an irreducible polynomial.

Proof. For every finite module M, there exists an n such that M = Im(F^{n}) ⊕ Ker(F^{n}) (Fitting’s Lemma). It follows that such an M has a composition series with quotients of type (a) or (b). The general statement now follows from Lemma 9.4. □

Therefore (see 9.2(f)), it suffices to prove Theorem 9.1 when each of M and N is of type (a), (b), or (c). Moreover, because of the condition on the minimum polynomials, in case (c) we can assume that either m_{M} and m_{N} have no common factor or they are irreducible and equal. This we have done.

Restatement of the theorem for complexes. We sketch a generalization of Theorem 9.1 to complexes. Let M^{•} and N^{•} be bounded complexes of A-modules such that H^{i}(M^{•}) lies in R_{p} for all i. Define

P_{M^{•∨}⊗N^{•}}(t) = ∏ P_{H^{i}(M^{•})^{∨}⊗H^{j}(N^{•})}(t)^{(−1)^{i+j}}.

Let f^{i}: Ext^{i}(M^{•}, N^{•}) → Ext^{i+1}(M^{•}, N^{•}) be the map rendering

Ext^{i}(M^{•}, N^{•}) ⟶^{f^{i}} Ext^{i+1}(M^{•}, N^{•})
↓  ↓
Ext^{i}(M̄^{•}, N̄^{•})^{Γ} ⟶^{∪θ_{p}} Ext^{i}(M̄^{•}, N̄^{•})_{Γ}

commutative. Here the vertical maps arise from a spectral sequence (cf. 7.12) and the lower map is cup-product with the canonical generator θ_{p} ∈ H^{1}(Γ, W) (equal to the map induced by the identity map on Ext^{i}(M̄^{•}, N̄^{•})). Because θ_{p}^{2} = 0,

Ext^{•}(M^{•}, N^{•}): ⋯ → Ext^{i}(M^{•}, N^{•}) →^{f^{i}} Ext^{i+1}(M^{•}, N^{•}) → ⋯

is a complex. Define

χ^{×}(M^{•}, N^{•}) = ∏[H^{i}(Ext^{•}(M^{•}, N^{•}))]^{(−1)^{i}}

when these numbers are finite.

Theorem 9.6. Let M^{•} and N^{•} be bounded complexes of A-modules such that H^{i}(M^{•}) and H^{i}(N^{•}) are semisimple and finitely generated over W for all i.

(a) The groups Ext^{i}(M^{•}, N^{•}) are finitely generated ℤ_{p}-modules for all i.

(b) The alternating sum ∑(−1)r_{i} of the ranks of the Ext^{i}(M^{•}, N^{•}) is zero.

(c) The order of zero of P (M^{•∨} ⊗ N^{•}, t) at t = 1 is equal to the secondary Euler characteristic ∑(−1)^{i+1} · i · r_{i}.

(d) The groups H^{i}(Ext^{•}(M^{•}, N^{•})) are finite, and

|q^{χ(M^{•},N^{•})} · lim_{s→1} \frac{P_{M^{•∨}⊗N^{•}}(t)}{(1 − t)^{ρ(M^{•},N^{•})}} |_{l} = χ^{×}(M^{•}, N^{•})

where χ(M^{•}, N^{•}) = ∑(−1)^{i+j}s(H^{i}(M^{•})) · r(H^{j}(N^{•})).

Proof. Consider first the case that M^{•} and N^{•} are modules M and N regarded as complexes concentrated in degree 0.

(a) Because M and N are semisimple, m(M, t) and m(N, t) have only simply roots, and therefore the hypothesis of Theorem 9.1 hold. Hence, the groups Ext^{i}(M, N) are finitely generated ℤ_{p}-modules for all i, finite for i = 2, and zero for i > 2.

(b) Moreover, f: Hom(M, N) → Ext^{1}(M, N) has finite kernel and cokernel, and so Hom(M, N) and Ext^{1}(M, N) have the same rank.

(c) In fact, one sees easily that the rank of Hom(M, N) is the number of pairs (i, j) such that a_{i} = b_{j} (see 9.2d).

(d) The map f of Theorem 9.1 coincides with f^{0}, and so (d) is a restatement of the theorem.

The general case case now follows from the usual induction argument on the number of nonzero terms in the complexes M^{•} and N^{•} (cf. 9.2f). □

There is a similar Theorem for l ≠ p, whose formulation and proof we leave to the reader.

Why R^{+}(𝔽_{q}; ℤ_{p}) rather than R(𝔽_{q}; ℤ_{p})? Write R^{+} and R for R^{+}(𝔽_{q}; ℤ_{p}) and R(𝔽_{q}; ℤ_{p}) respectively. Even the torsion subgroup of Ext_{R}^{1}(M, N) can be infinite. Suppose, for example, that M and N are torsion-free, and consider the diagram

0 ⟶ Hom_{R^{+}}(M, N)^{(p^{n})} ⟶ Hom_{R^{+}}(M, N^{(p^{n})}) ⟶ Ext_{R^{+}}^{1}(M, N)_{p^{n}} ⟶ 0
↓≅  ↓  ↓
0 ⟶ Hom_{R}(M, N)^{(p^{n})} ⟶ Hom_{R}(M, N^{(p^{n})}) ⟶ Ext_{R}^{1}(M, N)_{p^{n}} ⟶ 0.

Since [Hom_{R}(M, N^{(p^{n})})] = (q^{n})^{r(M)r(N)} is unbounded, whereas [Hom_{R^{+}}(M, N^{(p^{n})})] is bounded if Hom_{R^{+}}(M, N) = 0, we see that Ext_{R}^{1}(M, N)(p) is frequently infinite. From a different perspective, it follows from Theorem 9.1 that Ext_{R^{+}}^{1}(M, N) grows when M and N are replaced by M(1) and N(1) because s(M(1)) > s(M), but the functor M ↦ M(1) is an equivalence on the category R(𝔽_{q}; ℤ_{p}).

Appendix: Lemmas on abelian groups. Recall that, for a homomorphism f: M → N of abelian groups, we define z(f) = \frac{[Ker(f)]}{[Coker(f)]} when both the top and bottom are finite. There are the following elementary statements (Tate 1966b, §5). We let W = W (𝔽_{q}), q = p^{a}.

9.7. Let M and N be finitely generated ℤ-modules (resp. W -modules) of the same rank and let (x_{i}) and (y_{i}) be bases for M and N respectively modulo torsion. Suppose f(x_{i}) ≡ ∑ a_{ij}y_{j} modulo torsion. Then z(f) is defined if and only if det(a_{ij}) ≠ 0, in which case

z(f) = \frac{[M_{tors}]}{| det(a_{ij})|[N_{tors}]}

(resp.

z(f) = \frac{[M_{tors}]| det(a_{ij})|_{p}^{a}}{[N_{tors}]} ).

Note that, when M = N, det(a_{ij}) is the determinant of f_{ℚ}: M_{ℚ} → M_{ℚ}.

9.8. Consider maps f: M → N and g: N → P. If any two of the three numbers z(f), z(g), z(g ∘ f) are defined, then so is the third, and

z(g ∘ f) = z(g) · z(f).

9.9. Let Γ = ℤ̂, and let γ be its canonical topological generator. Let M be a Γ -module that is finitely generated as a ℤ-module, and let f: M^{Γ} → M_{Γ} be the map induced by the identity map on M. Then z(f) is defined if and only if the minimum polynomial of γ on M_{ℚ} does not have 1 as a multiple root, in which case

z(f) | ∏_{a_{i}≠1}(1 − a_{i})| = 1

where a_{1}, a_{2}, … is the family of eigenvalues of γ acting on M_{ℚ}.

10. Extensions and Zeta Functions: Global Case

In this section, M(𝔽; ℚ) will be one of the following categories:

(a) the category M_{num}(𝔽; ℚ; S) where S consists of all smooth projective varieties over 𝔽 and we assume that the Tate conjecture holds for all V ∈ S;

(b) the category M(𝔽; ℚ) defined in (5.26). In fact, if the category in (a) exists and the Hodge conjecture holds for complex abelian varieties of CM-type, then these categories are essentially the same: the reduction functor CM(ℚ^{al}) → M_{num}(𝔽; ℚ; S) defines a fibre functor ω on CM(ℚ^{al})^{P}, and the quotient of CM(ℚ^{al}) corresponding to ω is canonically equivalent with M_{num}(𝔽; ℚ; S).

The main theorem for extensions.

Theorem 10.1. For all effective motives X and Y in M^{+}(𝔽_{q}; ℤ), the groups Ext^{1}(X, Y) and Ext^{2}(X, Y)_{cotors} are finite, and

q^{χ(X,Y)}·ζ(X^{∨}⊗Y, s) ∼ ±\frac{[Ext^{1}(X, Y)] · D(X, Y)}{[Hom(X, Y)_{tors}] · [Ext^{2}(X, Y)_{cotors}]}·(1−q^{−s})^{ρ(X,Y)} as s → 0

where χ(X, Y) = r(X_{p})s(Y_{p}) (notations as on p37), D = D(X, Y) is the discriminant of the pairing

Hom(Y, X) × Hom(X, Y) ⟶^{∘} End(Y) ⟶^{trace} ℤ,  (27)

and ρ(X, Y) is the rank of Hom(X, Y).

Remark 10.2. Let X and Y be effective motives over 𝔽_{q}.

(a) Because M^{+}(𝔽_{q}; ℚ) is a semisimple category, End(X_{0} × Y_{0}) is a semisimple ring. As it is also finite dimensional over ℚ, the trace pairing

End(X_{0} × Y_{0}) × End(X_{0} × Y_{0}) → ℚ

is nondegenerate: D(X, Y) ≠ 0.

(b) Below we prove that [Ext^{2}(X, Y)_{cotors}] = [Hom(Y, X)_{tors}] . This makes the formula more symmetric — note that D(X, Y) = D(Y, X).

(c) When we replace X by X(r) or Y by Y (r) (r ∈ ℕ) in Theorem 10.1, we obtain a description of the behaviour of ζ(X^{∨} ⊗ Y, s) near −r or +r.

Proof of Theorem 10.1 with z(ε) for D(X, Y). Recall (8.1) that for all l, there is an exact sequence

0 → Ext_{M^{+}(𝔽_{q};ℤ)}^{1}(X, Y)_{ℤ_{l}} ⟶^{c_{l}} Ext_{R^{+}(𝔽_{q};ℤ_{l})}^{1}(X_{l}, Y_{l}) → T_{l} Ext_{M^{+}(𝔽_{q};ℤ_{l})}^{2}(X, Y) → 0.

Define

ε = (ε_{l})_{l}: Hom_{M^{+}(𝔽_{q};ℤ)}(X, Y)_{ℤ̂} → T Ext_{M^{+}(𝔽_{q};ℤ)}^{2}(X, Y)

to be the map making

Hom_{M^{+}(𝔽_{q};ℤ)}(X, Y)_{ℤ_{l}} ⟶^{ε_{l}} T_{l} Ext_{M^{+}(𝔽_{q};ℤ)}^{2}(X, Y)
↓≅  ↑^{c_{l}}
Hom_{R^{+}(𝔽_{q};ℤ_{l})}(X_{l}, Y_{l}) ⟶^{f_{l}} Ext_{R^{+}(𝔽_{q};ℤ_{l})}^{1}(X_{l}, Y_{l})

commute. Here f_{l} = f(X_{l}, Y_{l}) is the map defined on p37. According to 9.8 and 9.1

z(ε_{l}) = z(c_{l}) · z(f_{l}) = \frac{[Ext^{1}(X, Y)(l)]}{[Ext^{2}(X_{l}, Y_{l})]} · |q^{s(X_{l})·r(Y_{l})} · ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} )|_{l}. (28)

The a_{i}, b_{j} are algebraic numbers independent of l. Because Ext^{i}(X_{l}, Y_{l}) = 0 for i ≥ 3, (8.1b) shows that Ext^{i}(X, Y) = 0 for i ≥ 3, and it follows (from 8.1b again) that

Ext^{2}(X, Y)_{cotors}(l) ≅ Ext^{2}(X_{l}, Y_{l}).

On multiplying the equations (28) for the different l and applying the product formula, we find that

q^{χ(X,Y)} · ∏_{a_{i}≠b_{j}} (1 − \frac{b_{j}}{a_{i}} ) · z(ε) = ±\frac{[Ext^{1}(X, Y)]}{[Ext^{2}(X, Y)_{cotors}]}.

This will become Theorem 10.1 once we have shown that

z(ε) = ±\frac{[Hom(X, Y)_{tors}]}{D(X, Y)}.

Comparison of z(ε) with D(X, Y). Let N be a finitely generated torsion-free discrete Γ-module, and let N^{∨} = Hom(N, ℤ). The pairing

N^{Γ} × H^{2}(Γ, N^{∨}) → H^{2}(Γ, ℤ) ≅ ℚ/ℤ

realizes the compact group (N^{Γ})ℤ̂ as the Pontryagin dual of the discrete group H^{2}(Γ, N^{∨}) (e.g., Milne 1986b, I 1.10). There is therefore a canonical isomorphism

Hom(N^{Γ}, ℤ̂) → Hom(ℚ/ℤ, H^{2}(Γ, N^{∨})) =^{df} T H^{2}(Γ, N^{∨}).

Suppose we are also given a nondegenerate pairing

ψ: N × M → ℤ

where M is a discrete Γ-module such that M_{tors} is killed by some integer and M/M_{tors} is finitely generated. This defines a map M → N^{∨} with torsion kernel and cokernel, which induces isomorphisms H^{2}(Γ, M) → H^{2}(Γ, N^{∨}) and T H^{2}(Γ, M) → T H^{2}(Γ, N^{∨}). Hence, there is a canonical isomorphism

Hom(N^{Γ}, ℤ̂) → T H^{2}(Γ, M).

When combined with the map

M^{Γ} → Hom(N^{Γ}, ℤ)

defined by ψ, this gives a homomorphism

ε(ψ): (M^{Γ})_{ℤ̂} → T H^{2}(Γ, M).

Note that z(ε(ψ)) is defined if and only if M_{tors}^{Γ} is finite and the restriction ψ^{Γ} to a pairing N^{Γ} × M^{Γ} → ℤ is nondegenerate, in which case

z(ε(ψ)) = ±\frac{[M_{tors}^{Γ}]}{det(ψ^{Γ})}.

When we apply this remark to the pairing

Hom(Ȳ , X̄)/{torsion} × Hom(X̄, Ȳ) → ℚ,

we obtain a homomorphism

ε: Hom(X, Y)_{ℤ̂} → T H^{2}(Γ, Hom(X, Y)),

and

z(ε) = ±\frac{[Hom(X, Y)_{tors}]}{D(X, Y)}.

Since ε coincides with the map defined in the preceding subsubsection, this completes the proof of Theorem 10.1.

Comparison of Hom(Y, X)_{tors} with Ext^{2}(X, Y)_{cotors}. In this subsection, we prove that

[Hom(Y, X)_{tors}] = [Ext^{2}(X, Y)_{cotors}].

It follows from Theorem 8.1 that

Hom(Y, X)(l) ≅ Hom(Y_{l}, X_{l})_{tors}, Ext^{2}(X, Y)_{cotors}(l) ≅ Ext^{2}(X_{l}, Y_{l})_{cotors},

and so it suffices to show that, for M and N in R^{+}(𝔽_{q}; ℤ_{l}),

[Hom(N, M)_{tors}] = [Ext^{2}(M, N)_{cotors}].

Because Ext^{3} = 0 in R^{+}(𝔽_{q}; ℤ_{l}), Ext^{2}(M/M_{tors}, N) is divisible and

Ext^{2}(M, N)_{cotors} ≅ Ext^{2}(M_{tors}, N).

As Hom(N, M_{tors}) ≅ Hom(N, M)_{tors}, this shows that it suffices to prove that

[Hom(N, M)] = [Ext^{2}(M, N)]  (29)

when M is torsion. In fact, we shall show that

Hom(N, M) ≅ Ext^{2}(M, N)^{∗}  (30)

when N is torsion-free and M is torsion. Here ∗ denotes Hom(−, ℚ/ℤ). This implies (29) because an arbitrary N has a resolution by torsion-free objects (1.4b, 7.10).

Define

Ext^{i}(M, N ⊗ ℚ/ℤ) = \underrightarrow{lim}_{n} Ext^{i}(M, N^{(l^{n})}).

Then

Ext^{1}(M, N ⊗ ℚ/ℤ) ≅ Ext^{2}(M, N)

and so we have to show that

Hom(N, M) ≅ Ext^{1}(M, N ⊗ ℚ/ℤ)^{∗}  (31)

(M torsion, N torsion-free).

Proof of (31) in the case l ≠ p. We let M̄ and N̄ denote M and N regarded as ℤ_{l}-modules. There is a perfect pairing of finite ℤ_{l}-modules

Hom(N̄, M̄)×Hom(M̄, N̄ ⊗ℚ/ℤ) →^{∘} Hom(N̄, N̄ ⊗ℚ/ℤ) ≅ End(N̄)⊗ℚ/ℤ ⟶^{Tr ⊗1} ℚ/ℤ.

This gives a perfect pairing of Γ-cohomology groups

Hom(N̄, M̄)^{Γ} × Hom(M̄, N̄ ⊗ ℚ/ℤ)_{Γ} → ℚ/ℤ,

which can be identified with (31).

Proof of (31) in the case l = p. If F acts as an isomorphism on M, this can be proved as in the case l ≠ p. Thus, we may assume that F act nilpotently on M, which therefore has a composition series whose quotients are isomorphic to k =_{df} (𝔽_{q}, 0).

Any extension of N^{(p^{n})} by N^{(p^{n})} splits as an extension of W -modules, and therefore is described by a σ-linear endomorphism α of N^{(p^{n})}. The trace of α acting on N^{(p^{n})} is an element of W/p^{n}W, and the trace of this element is a well-defined element of ℤ_{p}/p^{n}ℤ_{p}. Thus, we have a homomorphism

Ext^{1}(N^{(p^{n})}, N^{(p^{n})}) → ℤ_{p}/p^{n}ℤ_{p} ≅ p^{−n}ℤ_{p}/ℤ_{p}.

On passing to the limit, we obtain a homomorphism

t: Ext^{1}(N, N ⊗ ℚ/ℤ) → ℚ/ℤ.

We claim that the pairings

Ext^{i}(N, M) × Ext^{1−i}(M, N ⊗ ℚ/ℤ) → Ext^{1}(N, N ⊗ ℚ/ℤ) →^{t} ℚ/ℤ  (32)

are perfect for i ≥ 0. When N is cyclic, N = A/A · λ, and M = k, this can be proved directly: either both groups are zero, or the pairing is the trace pairing

𝔽_{q} × 𝔽_{q} → 𝔽_{p}.

Consider an exact sequence

0 → N′ → N → k → 0.

If the pairings (32) are perfect for N, k, and i ≥ 0, then the diagram

0 ⟶ Hom(k, k) ⟶  Hom(N, k)  ⟶  Hom(N′, k)  ⟶ ⋯
↓≅  ↓≅  ↓  ↓≅
0 ⟶ Ext^{2}(k, k)^{∗} ⟶ Ext^{1}(k, N ⊗ ℚ/ℤ)^{∗} ⟶ Ext^{1}(k, N′ ⊗ ℚ/ℤ)^{∗} ⟶ ⋯

shows that they are perfect for N′, k, and i ≥ 0 (note that (24) shows that Ext^{i}(k, k) equals k, k ⊕ k, k respectively for i = 0, 1, 2). A similar argument applies when N/N′ is a finite module on which F acts invertibly. Since an arbitrary torsion-free N can be realized as a submodule of a direct sum of cyclic modules (9.3), this proves that the pairings (32) are perfect when M = k. An induction argument on the length of M extends this to an arbitrary torsion M. The case i = 0 completes the proof of (31).

The main theorem for Weil extensions. Let Γ_{0} be the subgroup of Γ generated by the Frobenius element (so Γ_{0} ≅ ℤ). Inspired by Lichtenbaum 2002, we define “Weil” extension groups satisfying an analogue of the spectral sequence (1).

As we noted in 5.19, to give a motive over 𝔽_{q} amounts to giving a motive X over 𝔽 together with an endomorphism π of X that represents its Frobenius germ. The π can be considered a descent datum on X, and the condition on it a continuity requirement. We now define the category of Weil motives over 𝔽_{q}, M(𝔽_{q}; ℤ)_{W} to be the category of pairs (X, π) consisting of a motive X over 𝔽 and an endomorphism π (“noncontinuous descent datum”). Equivalently, M(𝔽_{q}; ℤ)_{W} is the category of motives over 𝔽 with a ℤ-action. It contains M(𝔽_{q}; ℤ) as a full subcategory.

Similarly, we define the category M^{+}(𝔽_{q}; ℤ)_{W} of effective Weil motives. For effective motives X and Y over 𝔽_{q}, let

Ext^{i}(X, Y)_{0} = Ext_{M^{+}(𝔽_{q};ℤ)_{W}}^{i}(X, Y).

Proposition 10.3. For effective motives X, Y over 𝔽_{q}, there is a spectral sequence

H^{i}(Γ_{0}, Ext_{M^{+}(𝔽;ℤ)}^{j}(X̄, Ȳ)) ⟹ Ext^{i+j}(X, Y)_{0}

Proof. The proof of Theorem 8.4 can be adapted to the present situation. □

Lemma 10.4. The groups Ext^{i}(X, Y)_{0} are finitely generated for all i, torsion for i ≥ 2, and zero for i ≥ 3.

Proof. The groups Ext_{M^{+}(𝔽;ℤ)}^{j}(X̄, Ȳ) have composition series whose quotients are finitely generated ℤ-modules or 𝔽 with its natural Γ_{0}-action. Moreover, they are torsion for i ≥ 1 and zero for i ≥ 2. Thus the statement follows from 10.3. □

Let f: Hom(M, N)_{0} → Ext^{1}(M, N)_{0} be the map rendering

Hom(M, N)_{0}  ⟶^{f}  Ext^{1}(M, N)_{0}
↓  ↑
Hom_{M^{+}(𝔽;ℤ)}(M̄, N̄)^{Γ_{0}} ⟶ Hom_{M^{+}(𝔽;ℤ)}(M̄, N̄)_{Γ_{0}}

commutative. Here the vertical maps arise from a spectral sequence (10.3) and the lower map is induced by the identity map on Hom(M̄, N̄) (equal to cup-product with the canonical generator of H^{1}(Γ_{0}, ℤ) ≅ ℤ).

Theorem 10.5. For effective motives X and Y over 𝔽_{q}, z(f) is defined and

q^{χ(X,Y)} · ζ(X^{∨} ⊗ Y, s) ∼ ±z(f) · [Ext^{2}(X, Y)_{0}] · (1 − q^{−s})^{ρ(X,Y)} as s → 0.

(Notations as in 10.1.)

Proof. If either X or Y is finite, then the spectral sequence 10.3 coincides with 8.4 and so

Ext^{i}(X, Y) ≅ Ext^{i}(X, Y)_{0}.

Now the same argument as in §8 proves that, for arbitrary X, Y, there are exact sequences

0 → Ext^{i}(X, Y)_{0} ⊗ ℤ_{l} → Ext^{i}(X_{l}, Y_{l}) → T_{l} Ext^{i+1}(X, Y)_{0} → 0.

Lemma 10.4 shows that Ext^{i+1}(X, Y)_{0} has no l-divisible elements, and so these sequences reduce to isomorphisms

Ext^{i}(X, Y)_{0} ⊗ ℤ_{l} ≅ Ext^{i}(X_{l}, Y_{l}).

The spectral sequence (10.3) is compatible with the spectral sequences (7.8) and (7.12), and so the diagrams

Hom(X, Y)_{0} ⊗ ℤ_{l} ⟶^{f⊗1} Ext^{1}(X, Y)_{0} ⊗ ℤ_{l}
↓≅  ↓≅
Hom(X_{l}, Y_{l}) ⟶^{f_{l}} Ext^{1}(X_{l}, Y_{l})

commute. Here f_{l} is the map in Theorem 9.1. Therefore, Theorem 10.5 follows from Theorem 9.1 and the product formula. □

Remark 10.6. The advantage of statement 10.5 over 10.1 is that it has a natural extension to complexes of motives — see the sequel to this paper. Compare also Theorem 9.6.

Motivic cohomology. Let V be a smooth projective variety^{11} over 𝔽_{q}. In contrast to the pure case, V should define, not a mixed motive, but rather a complex which will not (in general) decompose when there is torsion, the torsion at p being particularly complicated. Nevertheless, for an r ≥ 0 such that the truncated de Rham-Witt cohomology groups H^{i}(V, W Ω^{≤r−1}) are finitely generated over W, we make the ad hoc definition: h^{i}V (r) is the isomotive h_{0}^{i}V (r) endowed with the ℤstructure provided by the maps H^{i}(V, ℤ_{l}(r)) → H^{i}(V, ℚ_{l}(r)) (étale cohomology for l ≠ p and crystalline cohomology l = p). Define the (Weil) motivic cohomology groups by^{12}

H_{mot}^{j}(V, ℤ(r)) = ⊕_{i} Ext^{i}(11, h^{j−i}V (r))_{0}.  (33)

Let

H_{mot}^{•}(V, ℤ(r)) = ⋯ → H_{mot}^{j}(V, ℤ(r)) ⟶^{f^{j}} H_{mot}^{j+1}(V, ℤ(r)) → ⋯

be the complex with

H_{mot}^{j}(V, Z(r)) Ext^{0}(11, h^{j}(V)(r))_{0} ⊕Ext^{1}(11, h^{j−1}(V)(r))_{0}⊕Ext^{2}(11, h^{j−2}(V)(r))_{0}
[figure: =; f; f]
H_{mot}^{j+1}(V, Z(r)) f^{j}_{}Ext^{0}(11, h^{j+1}(V)(r))_{0}⊕ Ext^{1}(11, h^{j}(V)(r))_{0} ⊕Ext^{2}(11, h^{j−1}(V)(r))_{0}

The maps f at right are as in Theorem 10.5.

Theorem 10.7. Let V be a smooth projective variety over 𝔽_{q}, and let r ≥ 0 be such that the groups H^{i}(V, W Ω^{≤r−1}) are finitely generated over W.

(a) The groups H_{mot}^{i}(V, ℤ(r)) are finitely generated abelian groups.

(b) The alternating sum ∑(−1)^{i}r_{i} of the ranks r_{i} of the groups H_{mot}^{i}(V, ℤ(r)) is zero.

(c) The order of the zero of ζ(V, s) at s = r is equal to the secondary Euler characteristic ρ(V, r) =_{df} ∑(−1)^{i}ir_{i}.

(d) The cohomology groups of the complex (H_{mot}^{•}(V, ℤ(r)), f) are finite, and the alternating product of their orders χ^{×}(V, ℤ(r)) satisfies

ζ(V, s) ∼ ±χ^{×}(V, ℤ(r)) · q^{χ(V,O,r)} · (1 − q^{r−s})^{ρ} as s → r

E_{2}^{ij} = Ext^{i}(11, h^{j}(V)) ⟹ H_{mot}^{i+j}(V, ℚ(r)). E_{2}^{ij} = Ext^{i}(11, h^{j}(V))_{0} ⟹ H_{mot}^{i+j}(V, ℤ(r)).

11If the category M(𝔽; ℚ) at the start of this section is taken to be as in 5.26 (case (b)), then V must be taken to be an abelian variety.

12According to Deligne 1994, 3.2.1, the motivic cohomology groups (ℚ-coefficients) should be the final term of a spectral sequence Over 𝔽_{q} it is natural to expect this also with ℤ-coefficients and (following Lichtenbaum’s ideas)

replace the Exts with the Weil Exts:

If this spectral sequence degenerates, then we arrive at (33). In the sequel to this paper, we shall take a less ad hoc approach.

where

χ(V, O, r) = ∑_{\begin{matrix}0≤i≤r \\ 0≤j≤dim V \end{matrix}} (−1)^{i+j}(r − i) · dim_{𝔽_{q}} H^{j}(X, Ω^{i}).

Proof. Immediate consequence of Theorem 10.5 and Milne 1986a, 4.1 (the condition on the de Rham-Witt groups implies that the numbers d^{i}(r − 1) in the second reference are zero). □

Remark 10.8. When r = 0, the condition on the de Rham-Witt groups is satisfied vacuously, and the groups H_{mot}^{i}(V, ℤ(0)) coincide with the Weil-étale groups H_{W}^{i} (V, ℤ) of Lichtenbaum 2002. Thus, in this case the theorem coincides with the smooth projective case of ibid., Theorem 8.2.

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