RIGIDITY FOR ORIENTABLE FUNCTORS

I.PANIN STEKLOV MATHEMATICAL INSTITUTE (ST.PETERSBURG)

S.YAGUNOV MAX-PLANCK-INSTITUT FÜR MATHEMATIK STEKLOV MATHEMATICAL INSTITUTE (ST.PETERSBURG)

Introduction

Seventeen years ago Andrei Suslin published the following remarkable result concerning the algebraic K-theory of algebraically closed fields. [12]

Theorem 0.1. (Suslin) Let F_{0} ⊂ F be an extension of algebraically closed fields. Then for (p, char F) = 1 the natural map

K_{n}(F_{0}, ℤ/p) → K_{n}(F, ℤ/p)

is an isomorphism.

Later, in papers of H.Gillet, R.Thomason [4] and O.Gabber [3] similar results were established for the case of strict Henselization at a smooth point of an algebraic variety and for Henselian pairs.

Both the proofs are based on the concept of “rigidity” for K-theory, which appeared in [12].

Further developments in this field were made in a paper of Suslin-Voevodsky [13] in 1996. There the K-functor was replaced by a “homotopy invariant functor with transfers”. It was shown that for such a class of functors similar rigidity property also holds. Unfortunately, it is sometimes very non-trivial question whether a functor is a “functor with transfers”.

The aim of the present work is to prove the rigidity property (and, therefore, an analog of the Suslin’s theorem as well) for “orientable” functors. Roughly speaking, we call a multiplicative cohomology theory on schemes orientable if it has Chern classes which satisfy the standard list of properties (including the “projective bundle” theorem). We actually prove that an orientable functor can be endowed with transfers (in a sense different from the one in [13]). In order to construct the desired transfers, we replant the topological construction [1, 11] into the soil of algebraic geometry.

The orientability seems to be a very natural and easy-to-check property, which holds for a rather wide class of theories. Besides the K-functor, one can see that étale cohomology [5], motivic cohomology [16], and algebraic cobordism [17, 18] are orientable in our sense. The latter example seems to be the most interesting one.

We learned the concept of orientable spectrum in the algebraic context from lectures given by Morel during the workshop in Münster in June 1999. We are very grateful to Fabian Morel and Andrei Suslin for inspiring discussions during the work.

1. Functors with weak transfers

We are starting with some basic definitions. Let Sch/k be a category of schemes of finite type over a field k. We denote by Sm/k a full subcategory of Sch/k, whose objects are smooth algebraic varieties over k.

Definition 1.1. Consider the Cartesian square

\xymatrix{X′ \ar@{^{(}->}[r] \ar[d] & Y′ \ar[d]^{f} \\ X \ar@{^{(}->}[r] & Y}

of objects and morphisms in the category Sm/k with regular closed embeddings X ↪ Y and X′ ↪ Y′. We call this square transversal if the natural map of the corresponding normal bundles

f^{∗}(N_{Y/X}) → N_{Y′/X′}

is an isomorphism.

Example 1.2. Flatness of the morphism f in the square above implies the transversality of the square. (See [2] B 7.4.)

Example 1.3. Let

∐_{y_{i}/x}y_{i} ⟶ Y
↓  ↓^{f}
x ⟶ X

be a Cartesian square, where f: Y → X is a morphism of smooth curves, étale over the point x. Then the square above is transversal.

Let F: (Sm/k)^{∘} → Ab be a contravariant functor on the category Sm/k. Suppose that for every finite projective morphism f: X → Y and its decomposition X ↪^{τ} ℙ^{n} × Y →^{p_{Y}} Y there exists corresponding map f_{∗}^{τ}: F(X) → F(Y) called the transfer map and the family of the transfer maps satisfies three properties below.

Remark 1.4. From now on we accept the following convention. Let X ↪^{τ} Y ×ℙ^{n} → Y be a projective morphism with fixed decomposition. Let f′ be a base-change of f with respect to a morphism g. We shall consider only the decompositions of the map f′ which make all squares in the diagram below Cartesian.

\xymatrix{X′ \ar[r]^{τ′} \ar[rr]^{f′} \ar[d] & Y′ × ℙ^{n} \ar[d] \ar[r] & Y′ \ar[d]^{g} \\ X \ar[r]^{τ} \ar[rr]_{f} & Y × ℙ^{n} \ar[r] & Y}

This actually means we should choose the embedding τ′: X′ ↪ Y × ℙ^{n} canonically with respect to morphisms τ and g. (Namely, τ′ = id ×_{Y} τ.)

Under this convention we usually omit mentioning the embedding maps in the notation for transfers.

Property 1.5. (Base change) For any projective morphism f: X → Y endowed with a decomposition X ↪^{τ} ℙ^{n} × Y → Y and a morphism g: Y′ → Y such that the square

\xymatrix{X′ = Y′ ×_{Y} X \ar@{^{(}->}[r]^{τ′} \ar[d]_{g′} & Y′ × ℙ^{n} \ar[d]^{g×id} \\ X \ar@{^{(}->}[r]^{τ} & Y × ℙ^{n}}

is transversal, the diagram:

\begin{matrix}F(X′) & ⟶^{f′_{∗}} & F(Y′) \\ ↑ & & ^{g^{∗}}↑ \\ F(X) & ⟶^{f_{∗}} & F(Y) \end{matrix} (1.1)

commutes.

Property 1.6. (Finite additivity) Let X = X_{0} ∐ X_{1}, j_{m}: X_{m} ↪ X (m = 0, 1) be the embedding maps, and f: X → Y be a projective morphism. Setting f_{m} = fj_{m}, we have:

f_{0,∗}j_{0}^{∗} + f_{1,∗}j_{1}^{∗} = f_{∗}. (1.2)

Property 1.7. (Normalization) If X = Y = Spec(k) and f is the identity map, then for any decomposition X ↪^{τ} ℙ^{n} → Y the map f_{∗}: F(X) → F(Y) is the identity.

Definition 1.8. A functor F: (Sm/k)^{∘} → Ab called a functor with weak transfers if it can be endowed with the family of transfer maps {f_{∗}}, satisfying properties 1.5-1.7.

Definition 1.9. A functor F: (Sm/k)^{∘} → Ab called a homotopy invariant functor if for every X ∈ Sm/k the map p_{X}^{∗}: F(X) → F(X × 𝔸^{1}) is an isomorphism.

We prove here one property of homotopy invariant functors, which will be used in further sections.

Proposition 1.10. Let X ∈ Sm/k and E →^{p} X be a vector bundle over X of

dimension n. Then the map: F(X) →^{p^{∗}} F(E) is an isomorphism.

Proof. Let s: X → E be the zero-section of the considered bundle. We have: s^{∗}p^{∗} = (ps)^{∗} = id_{X}, which shows p^{∗} is a monomorphism.

Consider now two maps i_{0}, i_{1} : E → E × 𝔸^{1}, given by points 0, 1 of 𝔸^{1}. Let pr: E × 𝔸^{1} → E be the projection map. We get pr ∘ i_{m} = id_{E}, therefore, i_{0}^{∗} ∘ pr^{∗} = i_{1}^{∗} ∘ pr^{∗}. Since pr^{∗} is an isomorphism, i_{0}^{∗} = i_{1}^{∗}.

Let {X_{i} = Spec U_{i}} be an affine covering of the scheme X such that the bundle E is trivial over each chart X_{i}. Let us define a map φ̄_{i}: U_{i}[x_{1}, …, x_{n}] → U_{i}[x_{1}, …, x_{n}, t], setting φ̄_{i}(x_{k}) = tx_{k} (k = 1, …, n). These maps can be, obviously, patched together, forming the morphism of schemes φ: E × 𝔸^{1} → E.

One can easily check that φi_{1} = id and φi_{0} = sp. Therefore, we have: p^{∗}s^{∗} = (sp)^{∗} = (φi_{0})^{∗} = i_{0}^{∗}φ^{∗} = (φi_{1})^{∗} = id_{E}. Summarizing the observations above, we see that p^{∗} is an isomorphism inverse to s^{∗}.

From now on we consider the case of algebraically closed base field k. Let F: (Sm/k)^{∘} → Ab be a contravariant functor and X be a smooth affine curve over k. We can construct a map Φ: Div(X) → Hom(F(X), F(k)) defined on canonical generators by the formula: [x] ↦ x^{∗}, where x^{∗}: F(X) → F(k) is the pull-back map, corresponding to the point x ∈ X(k).

Below we will need a notion of the relative Picard group Pic(X̄, X_{∞}), where X̄ is the smooth compactification of X and X_{∞} is the complement of X in X̄. Let us recall that this group is generated by divisors D of the curve X and two divisors D and D′ are considered to be equivalent iff there exists a rational function f on X regular in a neighborhood of X_{∞} and such that f|_{X_{∞}} = 1, div_{0}(f) = D, and div_{∞}(f) = D′.

Theorem 1.11. Let F be a homotopy invariant functor with weak transfers. Then, in the above notation, the map Φ can be decomposed in the following way:

\xymatrix{Div(X) \ar[rr]^{Φ} \ar[dr]_{Ω} &  & Hom(F(X), F(k)) \\  & Pic(X̄, X_{∞}), \ar[ur]_{Ψ}}

where Pic(X̄, X_{∞}) is the relative Picard group of the smooth compactification X̄ of X, X_{∞} = X̄\X, and the map Ω is the canonical homomorphism.

Proof. Since the field k is algebraically closed, the group Pic(X̄, X_{∞}) is generated by unramified divisors on the curve X ( i.e. the divisors which have the form ∑_{i} ±[x_{i}], where x_{i} are pairwise different points on X ). All the relations between unramified divisors come from ones of the following type: “D ∼ D′ iff there exists a rational function on X regular in a neighborhood of X_{∞} and such that f|_{X_{∞}} = 1, div_{0}(f) = D, and div_{∞}(f) = D′ ”. Let us define Ψ(D) = Φ[Ω]^{−1}(D). In order to check this map is well-defined, we need to prove that Φ(D) = Φ(D′) for equivalent divisors D ∼ D′. One may assume that Supp(D) ∩ Supp(D′) = ∅. Let us take a function f ∈ k(X) such that f|_{X_{∞}} = 1, div_{0}(f) = D, and div_{∞}(f) = D′.

Denote by X^{0} the open locus of X̄ where f ≠ 1. By the choice of the function f, we have: X^{0} ⊂ X and the morphism f: X^{0} → 𝔸^{1} = ℙ^{1} − {1} = Y is finite. Moreover, since the corresponding divisors are unramified, the map f is étale over points 0 and ∞. Consider the diagram:

\xymatrix{⊕_{x/∞}F(x) \ar@{-->}[dr]_{∑_{x/∞}f_{x,∗}} & F(D′) \ar[l] \ar[d]^{f_{∞,∗}} & F(X^{0}) \ar[l] \ar[r] \ar[d]^{f_{∗}} & F(D) \ar[d]^{f_{0,∗}} \ar[r] & ⊕_{x/0}F(x) \ar[dl]^{∑_{x/0}f_{x,∗}} \\ & F(k) & F(Y) \ar[l]_{i_{∞}^{∗}} \ar[r]^{i_{0}^{∗}} & F(k)} (1.3)

In this diagram f_{0,∗}, f_{∞,∗} are transfer maps corresponding to the morphisms f_{0}, f_{∞}, which are restrictions of the morphism f to divisors D, D′, respectively. Maps i_{0}^{∗}, i_{∞}^{∗} are induced by points 0 and ∞ of ℙ^{1}.

Finally, all summations in the very right (left) part of the diagram are over the set (finite) of all points in the preimage of 0 (∞).

Example 1.3 and Property 1.5 of functors with weak transfers show that two central squares of the diagram above commute. Property 1.6 implies that the triangles commute as well. Finally, from Property 1.7 we can see that for each point x over ∞ or over 0 one has a relation: f_{x,∗} = id.

Since Y = 𝔸^{1}, both the maps i_{∞}^{∗} and i_{0}^{∗} are isomorphisms which are inverse to the isomorphism p^{∗}: F(k) → F(𝔸^{1}) (the homotopy invariance property). So that, we have:

∑_{x/∞} x^{∗} = ∑_{x/0} x^{∗}: F(X^{0}) → F(k). (1.4)

This clearly implies:

Φ(D′) = ∑_{x/∞} x^{∗} = ∑_{x/0} x^{∗} = Φ(D): F(X) → F(k). (1.5)

Corollary 1.12. Let us assume, in addition to the hypothesis of Theorem 1.11, that there exists an integer n relatively prime to char(k) such that nF(Y) = 0 for every Y ∈ Sm/k. Then the map Ψ can be passed through the degree map Pic(X̄, X_{∞}) →^{deg} ℤ. Namely, if D, D′ are two divisors of the same degree, then Ψ(D) = Ψ(D′): F(X) → F(k).

Proof. We have an exact sequence:

0 ⟶ Pic(X̄, X_{∞})^{∘} ⟶ Pic(X̄, X_{∞}) ⟶^{deg} ℤ ⟶ 0 (1.6)

Since the group Pic(X̄, X_{∞})^{∘} is n-divisible over an algebraically closed field of characteristic relatively prime to n, we have: Ψ|_{Pic(X̄,X_{∞})^{∘}} = 0. The corollary now follows.

Theorem 1.13 (The Rigidity Theorem). Let T be a smooth affine variety over the algebraically closed field k. Let t_{1}, t_{2} ∈ T (k) be two k-points. Then, under the hypothesis of the previous corollary, two maps t_{1}^{∗}, t_{2}^{∗}: F(T) → F(k) coincide.

Proof. There exists an irreducible curve C ⊂ T joining the points t_{1} and t_{2} (see [9] p.56). The statement follows from the application of Corollary 1.12 to the normalization of C.

Extend the functor F to pro-objects in the category Sm/k setting

F(\underleftarrow{lim}(X_{i})) = \underrightarrow{lim} F(X_{i}) (1.7)

Theorem 1.14. Let F: (Sm/k)^{∘} → Ab be a homotopy invariant functor with weak transfers. Let n be an integer relatively prime to char(k) such that for every Y ∈ Sm/k nF(Y ) = 0 . Let k ⊂ K be an extension of algebraically closed fields. Then the natural map Spec K →^{π} Spec k induces an isomorphism π^{∗}: F(Spec k) →^{≃} F(Spec K).

Remark 1.15. Spec K is a pro-object in the category Sm/k. Moreover, schemes in the pro-system defining Spec K can be chosen affine [4, 12].

Proof. Let us take α ∈ F(Spec k) such that π^{∗}(α) = 0. Since F(Spec K) = \underrightarrow{lim} F(U), we can choose a smooth affine variety U →^{φ} Spec k such that φ^{∗}(α) = 0. Since the field k is algebraically closed, there exists a k-point Spec k →^{x} U. Since the composition F(Spec k) →^{φ^{∗}} F(U) →^{x^{∗}} F(Spec k) is the identity map, we obtain: α = 0. This shows that π^{∗} is a monomorphism.

Let us now prove the surjectivity of π^{∗}. Suppose we are given β ∈ F(Spec K). Choose an affine smooth variety U together with a morphism Spec K →^{y} U such that β = y^{∗}(β′), where β′ ∈ F(U). As before, U has to have a k-point Spec k →^{x} U. Consider the commutative diagram:

\xymatrix{U_{K} \ar[r]^{π̄} \ar[d] & U \ar[d] \\ Spec K \ar[r]^{π} \ar[ur]^{y} \ar[u]^{ỹ} \ar[u]_{x̃} & Spec k, \ar[u]_{x}}

(1.8)

where U_{K} = Spec K ×_{Spec k} U. Two K-points x ∘ π and y of U determine two Kpoints x̃, ỹ of U_{K}. We have: β−π^{∗}∘x^{∗}(β′) = (y^{∗}−π^{∗}∘x^{∗})(β′) = (ỹ^{∗}−x̃^{∗})∘π̄^{∗}(β′) = 0, because ỹ^{∗} − x̃^{∗} = 0 due to the Rigidity Theorem 1.13. Thus, β ∈ Im π^{∗}.

2. Orientable functors

The purpose of the current section is to introduce a class of “Orientable Functors” for which a family of transfers will be constructed in further sections.

Let us consider a category Sm^{2} /k whose objects are pairs (X, U) such that X ∈ Sm/k and U is open in X. Morphisms in this category are usual morphisms of pairs.

Orientable functors are among contravariant functors on this category. Typical examples are the functors which are represented by commutative ring T -spectra and orientable in the sense of Morel [6]. Functors given by these examples are double-indexed and the product satisfies the usual skew-commutativity property. In order to get a notion of an orientable functor, we axiomatize these and some other properties of mentioned functors. In fact, we never really need double-indexation but we decided to keep it to be consistent with the typical examples.

Let E^{p,q}: (Sm^{2} /k)^{∘} → Ab (p, q ∈ ℤ) be a functor (family of functors).

Remark 2.1. Further, we often write E_{Z}^{∗,∗}(X) for E^{∗,∗}(X, X\Z) and E^{∗,∗}(X) for E^{∗,∗}(X, ∅).

Consider the following list of axioms.

Eilenberg – Steenrod Type Axioms

Axiom 2.2 (Vanishing property). For every X E_{∅}^{∗,∗}(X) = 0.

Axiom 2.3 (Additivity). E^{∗,∗}(X_{1} ∐ X_{2}) →^{≃} E^{∗,∗}(X_{1}) ⊕ E^{∗,∗}(X_{2}).

Axiom 2.4 (Excision). Let X ⊇^{i} X_{0} ⊇ Z, where X_{0} is open in X and Z is closed in X. Then the induced map i^{∗}: E_{Z}^{∗,∗}(X) →^{≃} E_{Z}^{∗,∗}(X_{0}) is an isomorphism.

Axiom 2.5 (Homotopy Invariance). A functor E^{∗,∗} is homotopy invariant (see 1.9).

Axiom 2.6 (Localization). Let f: U → X be an open embedding morphism. Then we have the following long exact sequence:

⋯ ⟶ E^{∗,∗}(X) ⟶^{f^{∗}} E^{∗,∗}(U) ⟶ E^{∗+1,∗}(X, U) ⟶ ⋯

Product structure

Axiom 2.7 (Multiplicativity). There exists a pairing

E^{p,q}(X) ⊗ E_{Z}^{p′,q′}(X) →^{⌣} E_{Z}^{p+p′,q+q′}(X),

which is:

a) functorial;

b) for every X there exists two sides unit 1_{X} ∈ E^{0,0}(X) and it is functorial;

c) associative;

d) skew-commutative: a ⌣ b = (−1)^{p·p′}b ⌣ a.

Chern classes

Axiom 2.8 (Chern classes). For a linear bundle L over a smooth variety X we can choose a class c_{1}(L) ∈ E^{2,1}(X) ( called the first Chern class of the bundle L ) such that the family c_{1}(L) satisfies the following properties:

a) If L ≃ L′ then c_{1}(L) = c_{1}(L′);

b) For a morphism f: X → Y we have: f^{∗}(c_{1}(L)) = c_{1}(f^{∗}(L));

c) c_{1}(1) = 0;

and Properties 2.9,2.11 below.

Axiom 2.9 (Projective Bundle Theorem). Let E be a vector bundle of rank r over X.

Denote by ℙ(E) the projective bundle over X associated to E. Its fiber over a point x in X is the projective space of lines in the fiber E_{x}. Let O(−1) be the tautological line bundle over ℙ(E).

The map: ⊕_{i=0}^{r−1}E^{∗−2i,∗−i}(X) \underrightarrow{∑(⌣ξ^{i})} E^{∗,∗}(ℙ(E)) is an isomorphism. In this formula ξ = c_{1}(O(−1)) is the first Chern class of the tautological line bundle and ⌣ ξ^{i} means the multiplication by ξ^{i};

The latter axiom implies the following relation:

ξ^{r} − c_{1}(E)ξ^{r−1} + ⋯ + (−1)^{r−1}c_{r−1}(E)ξ + (−1)^{r}c_{r}(E) = 0 (2.1)

Definition 2.10. The coefficients c_{i}(E) ∈ E^{2i,i}(X) called the Chern classes of the vector bundle E. (We follow here the construction proposed by Morel [6]).

Axiom 2.11. If E is a trivial vector bundle then c_{i}(E) = 0 for i > 0.

Blow-up property

Axiom 2.12. If B(X, Z) = (𝔸^{1} × X)_{{0}×Z} is the deformation to the normal cone, N is the normal bundle over Z corresponding to the embedding Z ⊂ X, and i_{0} : N ↪ B(X, Z) and i_{1} : X ↪ B(X, Z) are canonical embeddings over 0 and 1, respectively, then the induced maps:

E_{Z}^{∗,∗}(N) ←^{i_{0}^{∗}} E_{𝔸^{1}×Z}^{∗,∗}(B(X, Z)) →^{i_{1}^{∗}} E_{Z}^{∗,∗}(X) (2.2)

are isomorphisms. (For more details about the meaning of this property see [8],section 3.2.3).

Definition 2.13. We call a functor (family of functors) E^{p,q}: (Sm^{2} /k)^{∘} → Ab, satisfying axioms 2.2 – 2.12 an oriented functor.

Consider a functor 𝔊: Sm/k → Sm^{2} /k given as follows: 𝔊(X) = (X, ∅). This functor embeds Sm/k into Sm^{2} /k as a full subcategory. Given a functor E^{∗,∗}: (Sm^{2} /k)^{∘} → Ab we can consider the composite: E^{∗,∗}𝔊^{∘}: (Sm/k)^{∘} → Ab.

Theorem 2.14. Let E^{∗,∗}: (Sm^{2}/k)^{∘} → Ab be an oriented functor. Then we can correspond to every projective morphism f: X → Y and its decomposition X ↪^{τ}

Y × ℙ^{n} →^{p^{Y}} Y the push-forward map f_{∗}^{τ}: E^{∗,∗}(X) → E^{∗+2r,∗+r}(Y) (r = dim Y −

dim X) such that the functor E^{∗,∗}𝔊^{∘} becomes a homotopy invariant functor with weak transfers.

The rest of the paper is devoted to the proof of Theorem 2.14.

Example 2.15. The following functors supply us with examples of oriented functors:

a) Motivic cohomology.

Most of the orientability axioms for motivic cohomology are checked in [14].

b) Étale cohomology.

c) Algebraic K-theory.

We put K^{p,q}(X, X − Z) = K_{2q−p}(X on Z), where the groups on the right-hand side are the Thomason-Trobaugh K-groups (see [15]). These groups coincide with the Quillen’s groups K′ [10], but they are obviously functorial with respect to morphisms of pairs.

d) algebraic cobordism MGL^{p,q}.

Algebraic cobordism admits a multiplicative structure and a theory of Chern classes (see [7]).

3. Construction of the Thom class

From now on E^{∗,∗} denotes an orientable contravariant functor on the category Sm^{2} /k.

Let X be a smooth variety over k and let E be a vector bundle over X of a constant rank r. Let 1 be the trivial vector bundle over X, ℙ(E) and ℙ(E ⊕ 1) be projectivizations of bundles E and E ⊕ 1, respectively. Consider the following diagram:

[figure: X; E; ℙ(1); s_{1}; s̄_{1}; ℙ(E ⊕ 1)\ℙ(E); j_{1}; ℙ(E ⊕ 1); j_{E}; ℙ(E ⊕ 1)\ℙ(1); s_{E}; ℙ(E); s̄_{E}] (3.1)

where s_{1}, s_{E}, s̄_{1}, and s̄_{E} are corresponding canonical embeddings.

Lemma 3.1. The following induced maps are isomorphisms.

a) s_{E}^{∗}: E^{∗,∗}(ℙ(E ⊕ 1)\ℙ(1)) → E^{∗,∗}(ℙ(E)).

b) j_{1}^{∗}: E_{ℙ(1)}^{∗,∗}(ℙ(E ⊕ 1)) → E_{ℙ(1)}^{∗,∗}(ℙ(E ⊕ 1)\ℙ(E)) = E_{X}^{∗,∗}(E).

Proof. a) Let us mention that ℙ(E ⊕ 1)\ℙ(1) is the total space of the tautological line bundle O(−1) over ℙ(E) and the map s_{E} is its zero-section. The claim now follows from this observation and Proposition 1.10.

b) It follows from the excision property.

Proposition 3.2. The sequence:

0 → E_{X}^{∗,∗}(E) → E^{∗,∗}(ℙ(E ⊕ 1)) →^{s̄_{E}^{∗}} E^{∗,∗}(ℙ(E)) → 0

is exact.

Proof. Let us look at the following fragment of the localization long exact sequence corresponding to the open embedding ℙ(E ⊕ 1)\ℙ(1) ⊂ ℙ(E ⊕ 1):

⋯ → E_{ℙ(1)}^{∗,∗}(ℙ(E ⊕ 1)) →^{i} E^{∗,∗}(ℙ(E ⊕ 1)) → E^{∗,∗}(ℙ(E ⊕ 1)\ℙ(1)) → ⋯ (3.2)

Applying the statements of Lemma 3.1 to this sequence, we get:

⋯ → E_{X}^{∗,∗}(E) → E^{∗,∗}(ℙ(E ⊕ 1)) →^{s̄^{∗}} E^{∗,∗}(ℙ(E)) → ⋯ (3.3)

To complete the proof, it is sufficient to show that the map

E^{∗,∗}(ℙ(E ⊕ 1)) →^{s̄^{∗}} E^{∗,∗}(ℙ(E)) (3.4)

is an epimorphism. The Projective Bundle Theorem tells us that both E^{∗,∗}- algebras E^{∗,∗}(ℙ(E ⊕ 1)) and E^{∗,∗}(ℙ(E)) are generated by the first Chern classes of the corresponding tautological bundles. The restriction of the tautological line bundle on ℙ(E ⊕ 1) to ℙ(E) coincides with the tautological line bundle on ℙ(E). Thus, Property 2.8.b of the first Chern class completes the proof of surjectivity. The proposition follows.

Let ζ and ξ ∈ E^{2,1}(X) be the first Chern classes of the tautological line bundles over projectivizations ℙ(E ⊕ 1) and ℙ(E), respectively. As before, the functoriality of the first Chern class implies that s̄_{E}^{∗}(ζ) = ξ. Consider now the Chern polynomial:

c_{t}(E) = ∑_{k=0}^{r}(−1)^{r−k}c_{r−k}(E)t^{k}. (3.5)

We have: c_{ζ}(E) ∈ E^{2r,r}(ℙ(E ⊕ 1)). It follows from the Projective Bundle Theorem, that s̄_{E}^{∗}(c_{ζ}(E)) = c_{s̄_{E}^{∗}ζ}(E) = c_{ξ}(E) = 0. So that, from 3.2 we have: c_{ζ}(E) ∈ Im i(E_{X}^{2r,r}(E)). Let us mention that the excision axiom implies: E_{ℙ(1)}^{2r,r}(ℙ(E ⊕ 1)) = E_{X}^{2r,r}(E).

Definition 3.3. (The Thom class) We set th(E) = j_{1}^{∗}(i^{−1}c_{ζ}(E)) ∈ E_{X}^{2r,r}(E) and call th(E) the Thom class of the vector bundle E.

Proposition 3.4. Let f: Y → X be a morphism of smooth varieties. Let E be a vector bundle of dimension r over X. Then, we have:

th(f^{∗}E) = f^{∗}(th(E))

Proof. Let ξ_{X} (resp. ξ_{Y}) be the first Chern class of the bundle O(−1) over ℙ_{X}(E) (resp. ℙ_{Y} (f^{∗}E)). Since ξ_{Y} = f^{∗}ξ_{X}, we have a relation:

ξ_{Y}^{r} + f^{∗}(−c_{1}(E))ξ_{Y}^{r−1} + ⋯ + f^{∗}((−1)^{r}c_{r}(E)) = 0 (3.6)

which holds in E^{∗,∗}(ℙ_{Y} (f^{∗}E)).

On the other hand, one has a similar relation for the class ξ_{Y}^{r} with coefficients (−1)^{i}c_{i}(f^{∗}(E)). Since E^{∗,∗}(ℙ_{Y} (f^{∗}E)) is a free E^{∗,∗}(Y)-module with a base 1, ξ_{Y} , …, ξ_{Y}^{r−1}, this implies the functoriality of the higher Chern classes: f^{∗}c_{i}(E) = c_{i}(f^{∗}E).

Therefore, f^{∗}(c_{t}(E)) = c_{t}(f^{∗}E). Since the inverse image of the tautological bundle O_{ℙ_{X}(E⊕1)}(−1) equals to O_{ℙ_{Y} (f^{∗}E⊕1)}(−1), we have the desired result.

4. Gysin map for a closed embedding

Let τ: X ↪ Y be a closed embedding of codimension d of smooth k-varieties and let N = N_{Y/X} be the corresponding normal bundle. We need to recall a wellknown construction of the deformation to the normal cone. Let B(Y, X) denote the blow-up of Y × 𝔸^{1} with center at X × {0}. Considering the fibers of the map B(Y, X) → Y ×𝔸^{1} → 𝔸^{1} over the points 0 and 1 one easily obtains two embeddings: i_{0} : ℙ(N ⊕ 1) ↪ B(Y, X) and i_{1} : Y ↪ B(Y, X). The subvariety X × 𝔸^{1} contains the center X × {0} of the blow-up as a divisor.Therefore, it lifts canonically to a subvariety in B(Y, X). Since X ×𝔸^{1} crosses ℙ(N ⊕1) along ℙ(1) and crosses i_{1}(Y) along i_{1}(X), one has following embeddings of the pairs:

(N , N − X) ↪^{i_{0}} (B, B − X × 𝔸^{1}) ↩^{i_{1}} (Y, Y − X) (4.1)

From Property 2.12, we have:

\xymatrix{E_{X}^{∗,∗}(N) & E_{X×𝔸^{1}}^{∗,∗}(B) \ar[l]_{i_{0}^{∗}}^{≃} \ar[r]^{i_{1}^{∗}}_{≃} & E_{X}^{∗,∗}(Y)} (4.2)

Construction-Definition 4.1. We define the Gysin map τ_{∗} as the following chain of maps:

E^{∗−2d,∗−d}(X) ⟶^{⌣th(N)} E_{X}^{∗,∗}(N) ⟶^{i_{1}^{∗}∘(i_{0}^{∗})^{−1}} E_{X}^{∗,∗}(Y) ⟶^{j^{∗}} E^{∗,∗}(Y),

where j: (Y, ∅) ↪ (Y, Y − X) and we mean under ⌣ th(N) the composition of the canonical isomorphism (s^{∗})^{−1} : E^{∗,∗}(X) →^{≃} E^{∗,∗}(N) induced by the zero-section of the bundle N and the multiplication with the Thom class th(N).

Let us prove some properties of the constructed Gysin map.

Proposition 4.2. (Base change property) For a transversal square:

\xymatrix{X′ \ar[d]^{g} \ar@{^{(}->}[r]^{τ′} & Y′ \ar[d]^{f} \\ X \ar@{^{(}->}[r]^{τ} & Y}

the square

\xymatrix{E^{∗,∗}(X′) \ar[r]^{τ′_{∗}} & E^{∗,∗}(Y′) \\ E^{∗,∗}(X) \ar[r]^{τ_{∗}} \ar[u]_{g^{∗}} & E^{∗,∗}(Y) \ar[u]_{f^{∗}}}

commutes.

Proof. Consider the diagram:

\begin{matrix}E^{∗−2d,∗−d}(X′) & ⟶^{⌣th(N′)} & E_{X′}^{∗,∗}(N′) & ⟵^{i′_{0}^{∗}} & E_{𝔸^{1}×X′}^{∗,∗}(B′) & ⟶^{j′^{∗}i′_{1}^{∗}} & E^{∗,∗}(Y′) \\ ↑^{g^{∗}} & & ↑^{N(g)^{∗}} & & ↑^{B(f)^{∗}} & & ↑^{f^{∗}} \\ E^{∗−2d,∗−d}(X) & ⟶^{⌣th(N)} & E_{X}^{∗,∗}(N) & ⟵^{i_{0}^{∗}} & E_{𝔸^{1}×X}^{∗,∗}(B) & ⟶^{j^{∗}i_{1}^{∗}} & E^{∗,∗}(Y). \end{matrix} (4.3)

All the squares but the very left one are obviously commutative. Therefore, to obtain the relation τ′_{∗}g^{∗} = f^{∗}τ_{∗}, we just need to check the relation f^{∗}(th(N)) = th(f^{∗}(N)) = th(N′), which holds due to Proposition 3.4, since the initial square is transversal.

Proposition 4.3. (Additivity) Let us consider the commutative diagram:

\xymatrix{Z_{0} \ar[dr]^{j_{0}} \ar[ddr]_{τ^{0}} &  & Z_{1} \ar[dl]_{j_{1}} \ar[ddl]^{τ_{1}} \\  & Z = Z_{0} ∐ Z_{1} \ar@{^{(}->}[d]^{τ} \\  & X}

with canonical maps j_{0}, j_{1} and the map τ to be a closed embedding. Then:

τ_{∗} = τ_{0,∗}j_{0}^{∗} + τ_{1,∗}j_{1}^{∗}

Proof. Consider commutative diagrams of natural inclusions (for m = 0, 1):

\xymatrix{(Z_{m}, ∅) \ar@{^{(}->}[r]^{j_{m}} \ar@{^{(}->}[rr]_{ψ_{m}} & (Z, ∅) \ar@{^{(}->}[r]^{φ_{m}} & (Z, Z_{1−m})} (4.4)

Applying the functor E^{∗,∗} to both diagrams and adding the results, we obtain the following commutative diagram:

\xymatrix{E^{∗,∗}(Z) \ar[r]^{j_{0}^{∗}⊕j_{1}^{∗}}_{≃} & E^{∗,∗}(Z_{0}) ⊕ E^{∗,∗}(Z_{1}) \\ E_{Z_{0}}^{∗,∗}(Z) ⊕ E_{Z_{1}}^{∗,∗}(Z) \ar[u]^{φ_{0}^{∗}+φ_{1}^{∗}} \ar[ur]_{(ψ_{0}^{∗},ψ_{1}^{∗})}^{≃}} (4.5)

In the diagram above the horizontal arrow is an isomorphism since the functor E^{∗,∗} is additive 2.3. The excision property 2.4 makes the diagonal arrow an isomorphism as well. Since two arrows in the considered diagram are isomorphisms, the last one φ_{0}^{∗} + φ_{1}^{∗} is also an isomorphism. One can easily see that the composition: φ_{0}^{∗}[ψ_{0}^{∗}]^{−1}j_{0}^{∗} + φ_{1}^{∗}[ψ_{1}^{∗}]^{−1}j_{1}^{∗} is the identity map on E^{∗,∗}(Z). Let us now rewrite what we want to prove: τ_{0,∗}j_{0}^{∗} +τ_{1,∗}j_{1}^{∗} = τ_{∗} = τ_{∗} ∘[φ_{0}^{∗}[ψ_{0}^{∗}]^{−1}j_{0}^{∗} + φ_{1}^{∗}[ψ_{1}^{∗}]^{−1}j_{1}^{∗}]. Therefore, in order to prove the proposition, it is sufficient to check that τ_{m,∗} = τ_{∗}φ_{m}^{∗}[ψ_{m}^{∗}]^{−1} for m = 0, 1. To verify this relation for m = 1 (the case of m = 0 is exactly the same), consider the diagram:

[figure: (N , N − Z); (B, B − Z × 𝔸^{1}); (X, X − Z); (N , N − Z_{1}); (B, B − Z_{1} × 𝔸^{1}); σ; (X, X − Z_{1}); (N_{1}, N_{1} − Z_{1}); (B_{1}, B_{1} − Z_{1} × 𝔸^{1}); (X, X − Z_{1})] (4.6)

The top (bottom) row of this diagram corresponds to the morphism Z ↪^{τ} X (Z_{1} ↪^{τ_{1}} X), respectively. We also set: B = B(X, Z) (B_{1} = B(X, Z_{1})) The morphism σ is the blow-down morphism corresponding to blowing up the component Z_{0}.

Since σ^{−1}(Z_{1} ×𝔸^{1}) = Z_{1} ×𝔸^{1}, the morphism σ gives us a well-defined morphism

of pairs: (B, B − Z_{1} × 𝔸^{1}) →^{σ} (B_{1}, B_{1} − Z_{1} × 𝔸^{1}).

Applying the functor E^{∗,∗} to Diagram 4.6 and combining the result with the definition of Gysin map, we obtain the following commutative diagram (here and below we sometimes omit the indexes for the shortness):

[figure: E(Z); ⌣th(N); τ_{∗}; E_{Z}(N); E_{Z×𝔸^{1}}(B); ≃; ≃; E_{Z}(X); E(X); E_{Z_{1}}(Z); ⌣th(N); φ_{1}^{∗}; ≃; ψ_{1}^{∗}; E_{Z_{1}}(N); N(ψ_{1})^{∗}; ≃; E_{Z_{1}×𝔸^{1}}(B); E_{Z_{1}}(X); E(X); E(Z_{1}); ⌣th(N_{1}); τ_{1,∗}; E_{Z_{1}}(N_{1}); E_{Z_{1}×𝔸^{1}}(B_{1}); ≃; σ^{∗}; ≃; E_{Z_{1}}(X); E(X)] (4.7)

This diagram proves the desired relation and the proposition as well.

5. Transfer map and the proof of Theorem 2.14

We are now starting to construct the family of transfer maps. Let f: X → Y be a projective morphism of codimension d, with a decomposition:

\xymatrix{X \ar@{^{(}->}[r]^{τ} \ar[rr]_{f} & Y × ℙ^{n} \ar[r]^{p^{Y}} & Y,} (5.1)

where τ is a closed embedding and p^{Y} denotes the standard projection.

Definition 5.1. Define a map p_{∗}^{Y} : E^{∗,∗}(Y × ℙ^{n}) → E^{∗−2n,∗−n}(Y) as follows. By the Projective Bundle Theorem 2.9 one has a direct sum decomposition:

E^{∗,∗}(Y × ℙ^{n}) = ⨁_{i=0}^{n} E^{∗−2i,∗−i}(Y) × ξ^{i}, (5.2)

where ξ = c_{1}(O(−1)). Set p_{∗}^{Y} be a projection on the summand E^{∗−2n,∗−n}(Y).

Proposition 5.2. The following properties hold for the map p_{∗}^{Y}:

a) base change property for an arbitrary morphism;

b) p_{∗}^{Y} (ξ^{n}) = 1.

Proof. Both properties are direct corollaries of the definition above.

Definition 5.3. For each diagram of form 5.1 we can define the map f_{∗}^{τ}: E^{∗,∗}(X) → E^{∗+2d,∗+d}(Y), setting f_{∗}^{i} = p_{∗}^{Y} ∘ τ_{∗}. (d = dim Y − dim X).

To complete the proof of Theorem 2.14, we need to check that Properties 1.5-1.7 hold for the constructed transfer maps. We do it in Propositions 5.4-5.6.

Proposition 5.4. (Compare 1.5). Let

\xymatrix{X′ \ar[d]^{g′} \ar[r]_{f′} & Y′ \ar[d]^{g} \\ X \ar[r]^{f} & Y}

be a Cartesian square where the projective morphism X →^{f} Y is decomposed like in 5.1 and the square

\xymatrix{X′ = Y′ ×_{Y} X \ar[d]_{g′} \ar@{^{(}->}[r]^{τ′} & Y′ × ℙ^{n} \ar[d]^{G} \\ X \ar@{^{(}->}[r]^{τ} & Y × ℙ^{n}}

is transversal. Then the diagram:

\xymatrix{E^{∗,∗}(X′) \ar[r]^{f′_{∗}} & E^{∗,∗}(Y′) \\ E^{∗,∗}(X) \ar[r]^{f_{∗}} \ar[u]^{_{g′_{∗}}} & E^{∗,∗}(Y). \ar[u]_{g^{∗}}}

commutes.

Proof. Consider the diagram:

[figure: Y′ × ℙ^{n}; G; p^{Y ′}; Y × ℙ^{n}; p^{Y}; X′; g′; f′; τ′; X; f; τ; Y′; g; Y] (5.3)

where G = g × id. Since Square 5.4 is transversal, Proposition 4.2 implies that: G^{∗}τ_{∗} = τ′_{∗}g′^{∗}. On the other hand, from Proposition 5.2, we have: g^{∗}p_{∗}^{Y} = p_{∗}^{Y′}G^{∗}. Therefore, one gets: g^{∗}f_{∗} = g^{∗}p_{∗}^{Y} τ_{∗} = p_{∗}^{Y′}G^{∗}τ_{∗} = p_{∗}^{Y′}τ′_{∗}g′^{∗} = (f′)_{∗}g′^{∗}. The base change property is proven.

Proposition 5.5. (Compare 1.6) Let X = X_{0} ∐ X_{1}, j_{m}: X_{m} ↪ X (m = 0, 1) be embedding maps, and f: X → Y. Setting fj_{m} = f_{m} one has:

f_{0,∗}j_{0}^{∗} + f_{1,∗}j_{1}^{∗} = f_{∗} (5.4)

Proof. The additivity property 1.6 gives us the relation: τ_{∗} = τ_{0,∗}j_{0}^{∗} + τ_{1,∗}j_{1}^{∗}. So, we have the equality: f_{∗} = p_{∗}^{Y} τ_{∗} = p_{∗}^{Y} (τ_{0,∗}j_{0}^{∗} + τ_{1,∗}j_{1}^{∗}) = f_{0,∗}j_{0}^{∗} + f_{1,∗}j_{1}^{∗}, which proves the additivity property.

The last step is to check the normalization property.

Proposition 5.6. (Compare 1.7) Let X = Y = Spec(k) and f: X → Y be the identity morphism. Then for every decomposition pt ↪^{τ} ℙ^{n} →^{p} pt the map f_{∗}^{τ}: E^{∗,∗}(Spec k) → E^{∗,∗}(Spec k) is the identity.

Proof. Without loosing of generality we may assume τ(Spec(k)) = [0: 0: ⋯: 1]. Granting for a moment the equality τ_{∗}(1) = ξ^{n}, we easily see that the desired property follows from a chain of equalities: f_{∗}(1) = p_{∗}τ_{∗}(1) = p_{∗}(ξ^{n}) = 1. The last one follows from Proposition 5.2. Since both maps τ_{∗} and p_{∗} are E^{∗,∗}(Spec k)-linear, the normalization property follows.

Lemma 5.7. τ_{∗}(1) = ξ^{n}, where ξ = c_{1}(O(−1)) ∈ E^{2,1}(ℙ^{n}).

Proof. Let N denote now the normal bundle corresponding to the embedding pt ↪^{τ} ℙ^{n}. We have: N = 1^{n} and ℙ^{n} = ℙ(N ⊕ 1). Consider the transversal square:

[figure: pt; 𝔸^{n}; j; pt; ℙ^{n}] (5.5)

where j is the open embedding of 𝔸^{n} into ℙ^{n} given by the formula: j(x_{1}, …, x_{n}) = (x_{1} : ⋯: x_{n}: 1). The corresponding base-change diagram for the Gysin map looks now as follows:

[figure: E(Spec k); ⌣th(N); E_{{0}}(N); E_{{0}×𝔸^{1}}(B(ℙ^{n}, {0})); ī_{0}^{∗}; ī_{1}^{∗}; E_{{0}}(ℙ^{n}); ≃; j^{∗}; E(ℙ^{n}); E(Spec k); ⌣th(N); E_{{0}}(N); E_{{0}×𝔸^{1}}(B(𝔸^{n}, {0})); i_{0}^{∗}; i_{1}^{∗}; E_{{0}}(𝔸^{n}).] (5.6)

The map j^{∗} in the diagram above is an isomorphism due to the excision property 2.4. Since the Chern classes of a trivial vector bundle vanish in positive dimensions (see 2.11), we have an equality: c_{t}(N) = t^{n}.

From the definition of the Thom class we can see that j^{∗}(c_{ζ}(N)) = th(N). Therefore, in order to verify the desired relation we just need to check that i_{1}^{∗}[i_{0}^{∗}]^{−1} = id.

Lemma 5.8. Let us consider the diagram:

E_{{0}}^{∗,∗}(𝔸^{n}) ⟵^{i_{0}^{∗}} E_{{0}×𝔸^{1}}^{∗,∗}(B(𝔸^{n}, {0})) ⟶^{i_{1}^{∗}} E_{{0}}^{∗,∗}(𝔸^{n}),

where maps i_{0}^{∗}, i_{1}^{∗} are isomorphisms (see 2.12) induced by embeddings over the points {0} and {1}, described above. Then i_{1}^{∗}(i_{0}^{∗})^{−1} = id.

Proof. Let us mention, first, that B(𝔸^{n}, {0}) = B = (𝔸^{n} ×𝔸^{1})_{{0}} is the total space of the tautological line bundle over ℙ^{n}. Consider the following Cartesian square:

\xymatrix{L \ar[r]^{J} \ar[d]_{q} & B \ar[d]^{q̄} \\ 𝔸^{n} \ar@{^{(}->}[r]^{j} & ℙ^{n}} (5.7)

In this square L = B ×_{ℙ^{n}} 𝔸^{n} is the restriction of the vector bundle B to an open subset of the base. Evidently, the map i_{0} : 𝔸^{n} → B can be decomposed as Jj_{0}, where j_{0} is the zero-section of the bundle L over 𝔸^{n}.

In the same way the embedding i_{1} comes from the section j_{1} : 𝔸^{n} → L given by x ↦ (x, 1) × x ∈ L.

The projection map q: L → 𝔸^{n} clearly induces a morphism of pairs (L, L − {0} × 𝔸^{1}) → (𝔸^{n}, 𝔸^{n} − {0}), which we also denote by q. Since L is a vector bundle over 𝔸^{n}, the map q^{∗} is an isomorphism (inverse to j_{0}^{∗}). Thus, we have: j_{1}^{∗}[j_{0}^{∗}]^{−1} = j_{1}^{∗}q^{∗} = (qj_{1})^{∗} = id. Since i_{k}^{∗} = j_{k}^{∗}J^{∗} (k = 0, 1), the relation i_{1}^{∗}[i_{0}^{∗}]^{−1} = id follows.

References

[1] Fomenko A.; Fuks D. Kurs gomotopicheskoj topologii. [A course in Homotopic Topology] (in Russian) “Nauka”, Moscow, 1989. 496 pp.

[2] Fulton, W. Intersection theory. Springer-Verlag, Berlin, 1998. xiv+470 pp.

[3] Gabber, O. K-theory of Henselian local rings and Henselian pairs. Contemp. Math., 126, Amer. Math. Soc., Providence, RI, 1992. 59–70.

[4] Gillet, H.; Thomason, R. The K-theory of strict Hensel local rings and a theorem of Suslin. J. Pure Appl. Algebra 34 (1984), no. 2-3, 241–254.

[5] Milne, J. Étale cohomology. Princeton University Press 1980. xiii+323 pp.

[6] Morel, F. Some basic properties of the stable homotopy category of schemes. (preprint)

www.math.jussieu.fr/ morel/listepublications.html

[7] Morel, F. Voevodsky’s proof of Milnor’s conjecture. Bull. Amer. Math. Soc. (N.S.) 35 (1998), no. 2, pp. 123–143.

[8] Morel, F.; Voevodsky, V. 𝔸^{1}-homotopy theory of schemes (preprint) www.math.uiuc.edu/

K-theory/0305/ 1998.

[9] Mumford, D. Abelian varieties. Oxford. University Press 1974.

[10] Quillen, D. Higher algebraic K-theory. I. Algebraic K-theory, I: Higher K-theories Lecture Notes in Math., Vol. 341, Springer, Berlin 1973, pp. 85–147.

[11] Rudyak, Yu. On Thom spectra, orientability, and cobordism. Springer-Verlag, Berlin, 1998. xii+587 pp.

[12] Suslin,A. On the K-theory of algebraically closed fields. Invent. Math. 73 (1983), no. 2, 241– 245.

[13] Suslin, A.; Voevodsky, V. Singular homology of abstract algebraic varieties. Invent. Math. 123 (1996), no. 1, 61–94.

[14] Suslin, A.; Voevodsky, V. Bloch-Kato conjecture and motivic cohomology with finite coeffi-

cients. (preprint) www.math.uiuc.edu/K-theory/0341/ 1999.

Bloch-Kato conjecture and motivic cohomology with finite coefficients,

[15] Thomason, R.; Trobaugh, T. Higher algebraic K-theory of schemes and of derived categories. The Grothendieck Festschrift, Vol. III, 247–435, Progr. Math., 88, Birkhuser Boston, Boston, MA, 1990.

[16] Voevodsky, V.; Friedlander,E.; and Suslin, A. Cycles, Transfers and Motivic Homology The-

ories. (book to appear) www.math.uiuc.edu/K-theory/0368/ 1999.

[17] Voevodsky, V. The Milnor Conjecture. Preprint MPI 97–8 51pp. 1997.

[18] Voevodsky, V. 𝔸^{1}-homotopy theory. Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998). Doc. Math. 1998, Extra Vol. I, pp. 579–604.

Ivan Panin Steklov Mathematical Institute (St.Petersburg) Fontanka 27, 191011 St.Petersburg, Russia.

Email address: panin@pdmi.ras.ru

Serge Yagunov Max Planck Institute für Mathematik, Vivatsgasse 7, 53111 Bonn, Germany.

Email address: yagunov@mpim-bonn.mpg.de

Steklov Mathematical Institute (St.Petersburg) Fontanka 27, 191011 St.Petersburg, Russia.

Email address: yagunov@pdmi.ras.ru