Projective Bundle Theorem

in Homology Theories with Chern Structure

Alexander Nenashev

Received: December 17, 2003 Revised: December 5, 2004

Communicated by Ulf Rehmann

Abstract. Panin and Smirnov deduced the existence of push-forwards, along projective morphisms, in a cohomology theory with cup products, from the assumption that the theory is endowed with an extra structure called orientation. A part of their work is a proof of the Projective Bundle Theorem in cohomology based on the assumption that we have the first Chern class for line bundles. In some examples we have to consider a pair of theories, cohomology and homology, related by a cap product. It would be useful to construct transfer maps (pull-backs) along projective morphisms in homology in such a situation under similar assumptions. In this note we perform the projective bundle theorem part of this project in homology.

Keywords and Phrases: (Co)homology theory, Chern structure, projective bundle, algebraic variety

1. Introduction

Let k be a field and Sm be the category of smooth quasi-projective algebraic varieties over k. Let P denote the category of pairs (X, U), with X ∈ Sm and U a Zarisky open in X, where a morphism (X, U) → (X′, U′) is a morphism f : X → X′ in Sm such that f(U) ⊂ U′. Sm embeds into P by X ↦ (X, ∅). For any functor A defined on P, we can compose it with this embedding and write A(X) for A(X, ∅). For f : (X, U) → (X′, U′) we will denote by f_{A} (resp. f^{A}) the morphism A(f) : A(X, U) → A(X′, U′) (resp. A(f) : A(X′, U′) → A(X, U)) if A is covariant (respectively, contravariant). We will call such maps push-forwards or pullbacks respectively. Note that the rule (X, U) ↦ (U, ∅) defines an endofunctor on P.

Definition. A homology theory over k with values in an abelian category M is a covariant functor A. : P → M endowed with a natural transformation d : A.(X, U) → A.(U) called the boundary homomorphism, subject to the following requirements: (h1) (Homotopy invariance) The arrow p_{A} : A.(X × 𝔸^{1}) → A.(X) induced by the projection p : X × 𝔸^{1} → X is an isomorphism for any X ∈ Sm. (h2) (Localization sequence) For any (X, U) ∈ P, the sequence

… → A.(U) → A.(X) → A.(X, U) ⟶^{d} A.(U) → A(X) → …

is exact. (h3) (Nisnevich excision) Let (X, U), (X′, U′) ∈ P, Z = X − U, and Z′ = X′ − U′. Then for any étale morphism f : X′ → X such that f^{−1}(Z) = Z′ and f : Z′ → Z is an isomorphism, the map f_{A} : A.(X′, U′) → A.(X, U) must be an isomorphism.

These axioms are dual to the axioms of a cohomology theory given in [PS] and [PS1].^{1} The objective of [PS1] is to provide simple conditions under which one can construct transfer maps (push-forwards) along projective morphisms in a cohomology theory. This, in its turn, is a prerequisite for the proof of a very general version of the Riemann-Roch Theorem in [Pa]. All the assumptions made in [PS] and [Pa] are true for many particular cohomology theories such as, for instance, K-theory, étale cohomology, higher Chow groups, and the algebraic cobordism theory introduced by Voevodsky in [V]. We therefore get, in a very uniform way, the existence of push-forwards and the Riemann-Roch Theorem in all these theories. In some situations we have to consider a pair of theories (A^{·}, A.) consisting of a cohomology and a homology theory related by a cap-product. An important example of this is given by motivic cohomology and homology introduced by Suslin and Voevodsky in [SV]. An ultimate goal in such a situation is to obtain a Poincaré duality in the sense of [PY] for the pair (A^{·}, A.). Among the assumptions from which the Poincaré duality is deduced in [PY], there is the assumption of existence of transfer maps in both A^{·} and A.. However, the homology part of this, i.e. the verification of existence of transfers (pull-backs) in homology is still lacking. A general objective in this context is to construct transfers along projective morphisms in a homology theory starting from simple assumptions analogous to those made in [PS] for cohomology. The purpose of this note is to prove the Projective Bundle Theorem in homology (PBTH), which is a part of the whole program aimed towards the existence of transfer maps in homology. In Section 2 we provide definitions and state the main result (PBTH). Its proof is given in Sections 3 and 4. A similar result was obtained independently by K. Pimenov in a slightly different framework [Pi].

1[PS] is a part of [PS1] which has been published already.

Projective Bundle Theorem in Homology Theories … 489

Acknowledgements. I wish to thank Ivan Panin for attracting my attention to the problem of existence of pull-backs in homology, for his continuous interest in my work, and for numerous useful remarks. Special thanks are due to the Alexander von Humboldt Foundation (Germany) for sponsoring my stay in Bielefeld in May 2003, where the very initial steps of the project were undertaken. I am grateful to the University of Regina (Canada) where the work was completed.

2. Definitions and the Main Result

Let A. be a homology theory satisfying (h1-h3) and let A^{·} be a cohomology theory in the sense of [PS, Def. 2.0.1]. The latter means that A^{·} is a contravariant functor P → M equiped with a natural transformation d : A^{·}(U) → A^{·}(X, U) and satisfying the dual set of axioms that we will call (c1-c3). All the general properties of a cohomology theory deduced from (c1-c3) in [PS, Sect. 2.2] have their duals for a homology theory, obtained by inverting the arrows. In particular, the Mayer-Vietoris exact sequence in homology and the localization sequence for a triple can be deduced from (h1-h3). We will use the “(co)homology with support” notation A_{Z}(X) = A(X, U), where Z = X − U, for both A. and A^{·}. For simplicity, we will assume that A. and A^{·} take their values in the category Ab of abelian groups. From now on we will often write just A for the homology groups, while keeping the upper dot in the cohomology notation.

2.1. Product structures. We will assume that A^{·} is a ring cohomology theory in the sense of [PS, Sect. 2.4]. This, in particular, means that A^{·} is equiped with cup-products

∪ : A_{Z}^{·}(X) × A_{Z′}^{·}(X) → A_{Z∩Z′}^{·}(X)

that are functorial with respect to pull-backs and satisfy the following properties: (cup1)(associativity) (a ∪ b) ∪ c = a ∪ (b ∪ c) in A_{Z_{1}∩Z_{2}∩Z_{3}}^{·}(X) for any a ∈ A_{Z_{1}}^{·}(X), b ∈ A_{Z_{2}}^{·}(X), c ∈ A_{Z_{3}}^{·}(X). (cup2) The absolute cohomology groups A^{·}(X) become associative unitary rings; the pull-back maps f^{A} : A^{·}(X) → A^{·}(Y) are homomorphisms of such rings for all f : Y → X. (cup3) The groups A_{Z}^{·}(X) become two-sided unitary modules over A^{·}(X) for all X and closed Z ⊂ X. We say that a ∈ A_{Z}^{·}(X) is a central element if a ∪ b = b ∪ a for any b ∈ A^{·}(X). We say that a is universally central if f^{A}(a) ∈ A_{Z′}^{·}(X′) is central for any f : (X′, X′−Z′) → (X, X −Z) in P. Note that the notion of a ring cohomology theory also requires compatibility of cup-products with boundary maps, which implies compatibility of cup-products with Mayer-Vietoris arguments, etc. We will also assume that A is a left unitary module over A^{·} in the sense that we have cap-products

∩ : A_{Z}^{·}(X) × A_{Z∩Z′}(X) → A_{Z′}(X) satisfying the properties: (cap1) (a ∪ b) ∩ c = a ∩ (b ∩ c) in A_{Z_{3}}(X) for any a ∈ A_{Z_{1}}^{·}(X), b ∈ A_{Z_{2}}^{·}(X), c ∈ A_{Z_{1}∩Z_{2}∩Z_{3}}(X). (cap2) 1 ∩ a = a whenever defined. (cap3) Let U and U′ (resp. V and V′) be Zarisky opens in X (resp. in Y). Let Z = X − U, Z′ = X − U′, T = Y − V, T′ = Y − V′. Then for any f : (Y, V, V′) → (X, U, U′) and any a ∈ A_{Z}^{·}(X), b ∈ A_{T ∩T′}(Y), we have f_{A}(f^{A}(a) ∩ b) = a ∩ f_{A}(b) in A_{Z′}(X). (cap4) (compatibility with boundary maps)

Chern structure. We will assume that A^{·} is equiped with a Chern structure in the sense of [PS, Def. 3.2.1], i.e., to any X ∈ Sm and any line bundle L over X there is assigned a universally central element c(L) ∈ A^{·}(X) called the (first) Chern class of L, subject to the requirements: (ch1) Functoriality with respect to pull-backs; c(L) = c(L′) if L ≅ L′ over X. (ch2) c(1_{X}) = 0 ∈ A^{·}(X), where 1_{X} denotes the trivial line bundle X × 𝔸^{1} over X, for any X. (ch3) For any X ∈ Sm, let ξ = c(O_{X×ℙ^{1}}(−1)) ∈ A^{·}(X × ℙ^{1}), where O_{X×ℙ^{1}}(−1) = p^{∗}(O_{ℙ^{1}}(−1)), O_{ℙ^{1}}(−1) denotes the tautological line bundle over ℙ^{1}, and p : X × ℙ^{1} → ℙ^{1} is the projection. Define the maps f^{(0)}, f^{(1)} : A^{·}(X) → A^{·}(X × ℙ^{1}) by f^{(0)} = p^{A} and f^{(1)} = (ξ ∪ −) ∘ p^{A}. Then the map (f^{(0)}, f^{(1)}) : A^{·}(X) ⊕ A^{·}(X) → A^{·}(X × ℙ^{1}) is an isomorphism. In the homology, define the maps f_{0}, f_{1} : A.(X × ℙ^{1}) → A.(X) by f_{0} = p_{A} and f_{1} = p_{A} ∘ (ξ ∩ −). We will say that we have an extended Chern structure (extended to homology) if (ch4) The map (f_{0}, f_{1}) : A.(X × ℙ^{1}) → A.(X) ⊕ A.(X) is an isomorphism for any X ∈ Sm. The axioms (ch3) and (ch4) can be considered as a dim = 1 case of the PBTC and PBTH accordingly. Our goal is to show that the extended Chern structure on (A^{·}, A.) implies the following general version of PBTH for A.

Projective Bundle Theorem. Let X be a smooth quasiprojective variety over k and E a vector bundle over X of rank n + 1. Assume that the pair of theories (A^{·}, A.) is endowed with a product structure and an extended Chern structure. Denote ℙ(E) the projectivisation of E, O(−1) the tautological line bundle over ℙ(E), and let ξ = c(O(−1)) ∈ A^{·}(ℙ(E)) be its Chern class. For

0 ≤ i ≤ n, denote f_{i} = f_{n,i} the composite map A.(ℙ(E)) ⟶^{ξ^{i}∩−} A.(ℙ(E)) ⟶^{p_{A}} A.(X), where p : ℙ(E) → X is the natural projection. Then the map

F_{n} := (f_{0}, f_{1}, …, f_{n}) : A.(ℙ(E)) ⟶ A.(X) ⊕ A.(X) ⊕ … ⊕ A.(X)

is an isomorphism.

A crucial reason for which we cannot consider the theory A. separately and must rather work with the pair (A^{·}, A.) is that ξ lives in the cohomology. However, everything works smoothly along the same guidelines as in [PS, Sect. 3.3].

3. Proof: Part I

Localizing and applying the Mayer-Vietoris, we reduce the situation to the case of a trivial bundle E ≅ X × 𝔸^{n+1}, ℙ(E) ≅ X × ℙ^{n}. Next we can reduce it to the case X = pt. We leave it to the reader to check that X × − can be inserted throughout the proof. Thus we want to prove that the map

(f_{0}, …, f_{n}) : A.(ℙ^{n}) → A.(pt) ⊕ … ⊕ A.(pt)

is an isomorphism. We proceed by induction on n. Choose homogeneous coordinates [x_{0} : …: x_{n}] in ℙ^{n} and introduce the following notation:

(i) 0 = [1 : 0 : …: 0] the distinguished point; (ii) for 0 ≤ i ≤ n, ℙ_{i}^{n−1} is the projective hyperplane x_{i} = 0; (iii) for 1 ≤ i ≤ n, ℙ_{i}^{1} is the projective axis on which all x_{j} = 0 for j ≠ 0, i; (iv) 𝔸_{i}^{n} = ℙ^{n} − ℙ_{i}^{n−1} for 0 ≤ i ≤ n; we will often write just 𝔸^{n} for 𝔸_{0}^{n};

(v) 𝔸_{i}^{1} = ℙ_{i}^{1} ∩ 𝔸^{n} and 𝔸_{i}^{n−1} = ℙ_{i}^{n−1} ∩ 𝔸^{n} for 1 ≤ i ≤ n. Consider the localization sequence of the pair (ℙ^{n}, ℙ^{n} − 0):

(3.1)  … → A(ℙ^{n} − 0) ⟶^{u_{A}} A(ℙ^{n}) ⟶^{v_{A}} A_{0}(ℙ^{n}) → …,

where u : ℙ^{n}−0 → ℙ^{n} and v : (ℙ^{n}, ∅) → (ℙ^{n}, ℙ^{n}−0) are the natural maps. Note that ℙ^{n} − 0 can be considered as a line bundle over ℙ_{0}^{n−1}, with the projection map t : ℙ^{n} − 0 → ℙ_{0}^{n−1} given by [x_{0} : x_{1} : …: x_{n}] ↦ [0 : x_{1} : …: x_{n}]. Denote by s : ℙ_{0}^{n−1} → ℙ^{n}−0 the inclusion map, then by (h1), s_{A} : A(ℙ_{0}^{n−1}) → A(ℙ^{n}−0) and t_{A} : A(ℙ^{n} − 0) → A(ℙ_{0}^{n−1}) are inverse isomorphisms. Let u′ : ℙ_{0}^{n−1} → ℙ^{n} be the inclusion map, then u′ = us and u′_{A} = u_{A}s_{A}. Consider the diagram

(3.2)  \begin{matrix}A(ℙ_{0}^{n−1}) & ⟶^{u′_{A}} & A(ℙ^{n}) \\ ^{F_{n−1}}↓ & & ↓^{F_{n}} \\ ⨁_{i=0}^{n−1} A(pt) & ⟶^{a_{n−1,n}} & ⨁_{i=0}^{n} A(pt) \end{matrix}

where a_{n−1,n} maps each summand of ⨁_{i=0}^{n−1} A(pt) to the same summand in ⨁_{i=0}^{n} A(pt) as the identity map, the last summand in the latter group is therefore not being covered. We claim that the diagram commutes. For it suffices to prove that the diagram

A(ℙ_{0}^{n−1}) ⟶^{u′_{A}} A(ℙ^{n})
f_{n−1,i}↓  ↓^{f_{n,i}}
A(pt) ⟶^{1} A(pt)

commutes for every 0 ≤ i ≤ n − 1 and that f_{n,n}u′_{A} = 0. The first assertion follows from the commutativity of the diagram

A(ℙ_{0}^{n−1}) ⟶^{u′_{A}} A(ℙ^{n})
ξ_{n−1}^{i}∩−↓  ↓^{ξ_{n}^{i}∩−}
A(ℙ_{0}^{n−1}) ⟶^{u′_{A}} A(ℙ^{n})

which commutes by (cap3) since the restriction of O_{ℙ^{n}}(−1) to ℙ_{0}^{n−1} is isomorphic to O_{ℙ_{0}^{n−1}}(−1) and (u′)^{A}(ξ_{n}) = ξ_{n−1}. The same diagram with i = n implies that the composition f_{n,n}u′_{A} vanishes as ξ_{n−1}^{n} = 0. (See [PS, Sect. 3.3] for a standard argument that proves ξ_{n−1}^{n} = 0.) Now consider the map a_{n,n−1} : ⨁_{i=0}^{n} A(pt) → ⨁_{i=0}^{n−1} A(pt) that identically maps the ith summand to the ith summand for all 0 ≤ i ≤ n − 1 and vanishes on the nth summand. As a_{n,n−1}a_{n−1,n} = 1, the commutativity of (3.2) implies F_{n−1} = a_{n,n−1}a_{n−1,n}F_{n−1} = a_{n,n−1}F_{n}u′_{A}. By the inductional hypothesis F_{n−1} is an isomorphism, whence u′_{A} is a split monomorphism, and so is u_{A} as s_{A} is an isomorphism. This has two important consequences:

(i) (3.1) is in fact a split short exact sequence; (ii) the map f_{n,n} : A(ℙ^{n}) → A(pt) factors uniquely through v_{A}. Denote by g : A_{0}(ℙ^{n}) → A(pt) the factoring map: f_{n,n} = gv_{A}. The diagram

0 ⟶ A(ℙ_{0}^{n−1}) ⟶^{u′_{A}} A(ℙ^{n}) ⟶^{v_{A}} A_{0}(ℙ^{n}) ⟶ 0
↓^{F_{n−1}}  ↓^{F_{n}}  ↓^{g}
0 ⟶ ⨁_{i=0}^{n−1} A(pt) ⟶^{a_{n−1,n}} ⨁_{i=0}^{n} A(pt) ⟶ A(pt) ⟶ 0

shows that we will be done as soon as it is proved that g is an isomorphism. For 1 ≤ i ≤ n, consider the cohomology localization sequence of the pair (ℙ^{n}, 𝔸_{i}^{n}):

(3.3)  A_{ℙ_{i}^{n−1}}^{·}(ℙ^{n}) ⟶^{v_{i}^{A}} A^{·}(ℙ^{n}) ⟶^{u_{i}^{A}} A^{·}(𝔸_{i}^{n}) ,

where u_{i} : 𝔸_{i}^{n} → ℙ^{n} and v_{i} : (ℙ^{n}, ∅) → (ℙ^{n}, 𝔸_{i}^{n}) are the natural maps. As A^{·}(𝔸_{i}^{n}) ≅ A^{·}(pt) by (c1), this is a split short exact sequence, the splitting for u_{i}^{A} given by 1 ↦ 1. The element ξ_{n} ∈ A^{·}(ℙ^{n}) maps to zero via u_{i}^{A} as the restriction of O(−1) to 𝔸_{i}^{n} is isomorphic to the trivial line bundle. Thus ξ_{n} comes from a uniquely determined element ξ̄_{n,i} ∈ A_{ℙ_{i}^{n−1}}^{·}(ℙ^{n}). Note that

ℙ_{1}^{n−1} ∩ … ∩ ℙ_{n}^{n−1} = {0} and consider the diagram

A_{ℙ_{1}^{n−1}}^{·}(ℙ^{n}) ⊕ A_{ℙ_{2}^{n−1}}^{·}(ℙ^{n}) ⊕ … ⊕ A_{ℙ_{n}^{n−1}}^{·}(ℙ^{n}) ⟶^{∪} A_{0}^{·}(ℙ^{n})
v_{1}^{A}⊕...⊕v_{n}^{A}↓  ↓_{v^{A}}
A^{·}(ℙ^{n}) ⊕ A^{·}(ℙ^{n}) ⊕ … ⊕ A^{·}(ℙ^{n})  ⟶^{∪} A^{·}(ℙ^{n})

which commutes since ∪ is compatible with pull-backs. It follows that the element

t̄h_{n} := ξ̄_{n,1} ∪ ξ̄_{n,2} ∪ … ∪ ξ̄_{n,n} ∈ A_{0}^{·}(ℙ^{n})

satisfies v^{A}(t̄h_{n}) = ξ_{n}^{n}. Now apply (cap3), with X = Y = ℙ^{n}, U = U′ = ℙ^{n}−0, V = V′ = ∅ and f = v, to a = t̄h_{n} and any b ∈ A(ℙ^{n}) and get the commutativity of the diagram

A(ℙ^{n}) ⟶^{ξ_{n}^{n}∩−} A(ℙ^{n}) ⟶ A(pt)
v_{A}↓  ↓^{1}  ↓^{1}
A_{0}(ℙ^{n}) ⟶^{t̄h_{n}∩−} A(ℙ^{n}) ⟶ A(pt)

The composition of the top arrows is f_{n,n}. As g is the unique arrow satisfying f_{n,n} = gv_{A}, we can conclude that g equals the composition of the bottom arrows. Let j : 𝔸^{n} → ℙ^{n} denote the inclusion map, and let j_{i} : (𝔸^{n}, 𝔸^{n} − 𝔸_{i}^{n−1}) → (ℙ^{n}, ℙ^{n} − ℙ_{i}^{n−1}), with 1 ≤ i ≤ n, and j̃ : (𝔸^{n}, 𝔸^{n} − 0) → (ℙ^{n}, ℙ^{n} − 0) denote the corresponding maps of pairs. Define ξ_{n,i} ∈ A_{𝔸_{i}^{n−1}}^{·}(𝔸^{n}) to be the image of

ξ̄_{n,i} under the map j_{i}^{A} : A_{ℙ_{i}^{n−1}}^{·}(ℙ^{n}) → A_{𝔸_{i}^{n−1}}^{·}(𝔸^{n}). The diagram

A_{ℙ_{1}^{n−1}}^{·}(ℙ^{n}) ⊕ … ⊕ A_{ℙ_{n}^{n−1}}^{·}(ℙ^{n}) ⟶^{∪} A_{0}^{·}(ℙ^{n})
j_{1}^{A}⊕...⊕j_{n}^{A}↓  ↓j̃^{A}
A_{𝔸_{1}^{n−1}}^{·}(𝔸^{n}) ⊕ … ⊕ A_{𝔸_{n}^{n−1}}^{·}(𝔸^{n}) ⟶^{∪} A_{0}^{·}(𝔸^{n})

shows that the element

th_{n} := ξ_{n,1} ∪ … ∪ ξ_{n,n} ∈ A_{0}^{·}(𝔸^{n})

satisfies j̃^{A}(t̄h_{n}) = th_{n}. Consider the diagram

A_{0}(𝔸^{n}) ⟶^{th_{n}∩−} A(𝔸^{n}) ⟶^{∼} A(pt)
j̃_{A}↓  ↓^{j_{A}}  ↓^{1}
A_{0}(ℙ^{n}) ⟶^{t̄h_{n}∩−} A(ℙ^{n}) ⟶ A(pt)

that commutes by (cap3). Recall that our current goal is to prove that g, which equals the composition of the bottom arrows in the diagram, is an isomorphism. As j̃_{A} is an isomorphism by excision and A(𝔸^{n}) → A(pt) is an isomorphism by homotopy invariance, it now suffices to prove that th_{n} ∩ − : A_{0}(𝔸^{n}) → A(𝔸^{n}) is an isomorphism. This will be done in the next section.

4. Proof: Part II

First we will obtain another description for the elements ξ_{n,i} and th_{n}. Consider the short exact sequnce

0 ⟶ A_{0}^{·}(ℙ^{1}) ⟶ A^{·}(ℙ^{1}) ⟶ A^{·}(ℙ^{1} − 0) ⟶ 0 ,

which is the one-dimensional version of (3.3). The element ξ_{1} = c(O(−1)) ∈ A^{·}(ℙ^{1}) maps to zero and comes therefore from a uniquely determined element t̄ ∈ A_{0}^{·}(ℙ^{1}). Let t ∈ A_{0}^{·}(𝔸^{1}) denote its image under the restriction map A_{0}^{·}(ℙ^{1}) → A_{0}^{·}(𝔸^{1}). (As the one-dimensional case plays a distinguished role, we change the notation and denote these elements by t̄ and t.) If we think of ℙ^{1} and 𝔸^{1} as coordinate axes ℙ_{i}^{1} and 𝔸_{i}^{1} in ℙ^{n} and 𝔸^{n} accordingly, 1 ≤ i ≤ n, then we will denote the corresponding elements by t̄_{i} ∈ A_{0}^{·}(ℙ_{i}^{1}) and t_{i} ∈ A_{0}^{·}(𝔸_{i}^{1}). Denote by pr_{i} : 𝔸^{n} → 𝔸_{i}^{1} the projection to the i-th coordinate and consider the map pr_{i}^{A} : A_{0}^{·}(𝔸_{i}^{1}) → A_{𝔸_{i}^{n−1}}^{·}(𝔸^{n}). It is proved in [PS] that pr_{i}^{A}(t_{i}) = ξ_{n,i}, and we can therefore rewrite th_{n} in the form

(4.1)  th_{n} = pr_{1}^{A}(t_{1}) ∪ pr_{2}^{A}(t_{2}) ∪ … ∪ pr_{n}^{A}(t_{n}).

(NB: In [PS] th_{n} is defined by the above formula and then it is proved that pr_{i}^{A}(t_{i}) can be replaced by ξ_{n,i}, with a different notation.) To proceed further we first need to prove a technical lemma which is the homology counterpart of [PS, Lemma 3.3.2]. Let Y ∈ Sm and Z ⊂ Y be a closed subset. Let p : Y × 𝔸^{1} → Y and pr : Y × 𝔸^{1} → 𝔸^{1} denote the projections. Consider the map pr^{A} : A_{0}^{·}(𝔸^{1}) → A_{Y ×0}^{·}(Y × 𝔸^{1}) and the image pr^{A}(t) of t ∈ A_{0}^{·}(𝔸^{1}) under this map. The cap-product

∩ : A_{Y ×0}^{·}(Y × 𝔸^{1}) × A_{Z×0}(Y × 𝔸^{1}) → A_{Z×𝔸^{1}}(Y × 𝔸^{1})

induces the map pr^{A}(t) ∩ − : A_{Z×0}(Y × 𝔸^{1}) → A_{Z×𝔸^{1}}(Y × 𝔸^{1}).

Lemma. The map pr^{A}(t) ∩ − : A_{Z×0}(Y × 𝔸^{1}) → A_{Z×𝔸^{1}}(Y × 𝔸^{1}) is an isomorphism.

Proof. As p_{A} : A_{Z×𝔸^{1}}(Y × 𝔸^{1}) → A_{Z}(Y) is an isomorphism by (h1), the assertion of the lemma is equivalent to the claim that the composed map

T := p_{A} ∘ (pr^{A}(t) ∩ −) : A_{Z×0}(Y × 𝔸^{1}) → A_{Z}(Y)

is an isomorphism. It is this claim that we will actually prove. We will make use of the localization sequence of the triple (Y × ℙ^{1}, Y × ℙ^{1} − Z × 0, (Y − Z) × ℙ^{1}): (4.2)

… ⟶ A_{Z×𝔸_{∞}^{1}}(Y × ℙ^{1} − Z × 0) ⟶^{α_{A}} A_{Z×ℙ^{1}}(Y × ℙ^{1}) ⟶^{β_{A}} A_{Z×0}(Y × ℙ^{1}) ⟶ …,

where 𝔸_{∞}^{1} := ℙ^{1} −0 and α and β are the corresponding inclusion maps of pairs.

Consider the inclusion i : (Y × 𝔸_{∞}^{1}, (Y − Z) × 𝔸_{∞}^{1}) → (Y × ℙ^{1} − Z × 0, (Y − Z) × ℙ^{1}). One checks that i satisfies the excision conditions (Zarisky version), whence i_{A} : A_{Z×𝔸_{∞}^{1}}(Y × 𝔸_{∞}^{1}) → A_{Z×𝔸_{∞}^{1}}(Y × ℙ^{1} − Z × 0) is an isomorphism. Let p̃ : (Y × ℙ^{1} − Z × 0, (Y − Z) × ℙ^{1}) → (Y, Y − Z) and p′ = p̃i : (Y × 𝔸_{∞}^{1}, (Y − Z) × 𝔸_{∞}^{1}) → (Y, Y − Z) be the projections. As p′_{A} is an isomorphism by (h1), p′_{A} = p̃_{A}i_{A} shows that p̃_{A} : A_{Z×𝔸_{∞}^{1}}(Y × ℙ^{1} − Z × 0) → A_{Z}(Y) is an isomorphism too. Let p̄: (Y ×ℙ^{1}, (Y −Z)×ℙ^{1}) → (Y, Y −Z) denote the projection. Then p̃ = p̄α and p̃_{A} = p̄_{A}α_{A}, which implies that α_{A} is a split monomorphism. It follows that β_{A} is surjective and (4.2) is a short exact sequence. Let p̄r : Y × ℙ^{1} → ℙ^{1} denote the projection. Consider the diagram

(4.3)  \begin{matrix}A_{0}^{·}(ℙ^{1}) & ⟶ & A^{·}(ℙ^{1}) \\ p̄r^{A}↓ & & ↓p̄r^{A} \\ A_{Y ×0}^{·}(Y × ℙ^{1}) & ⟶ & A^{·}(Y × ℙ^{1}) \end{matrix}

and the cap-products

∩ : A_{Y ×0}^{·}(Y × ℙ^{1}) × A_{Z×0}(Y × ℙ^{1}) → A_{Z×ℙ^{1}}(Y × ℙ^{1})
∩ : A^{·}(Y × ℙ^{1}) × A_{Z×ℙ^{1}}(Y × ℙ^{1}) → A_{Z×ℙ^{1}}(Y × ℙ^{1}) .

The element p̄r^{A}(t̄) maps to p̄r^{A}(ξ_{1}) via the bottom arrow in (4.3). By (cap3), it follows that the diagram

(4.4)  \begin{matrix}A_{Z×ℙ^{1}}(Y × ℙ^{1}) & ⟶^{β_{A}} & A_{Z×0}(Y × ℙ^{1}) \\ ^{p̄r^{A}(ξ_{1})∩−}↓ & & ↓^{p̄r^{A}(t̄)∩−} \\ A_{Z×ℙ^{1}}(Y × ℙ^{1}) & ⟶^{1} & A_{Z×ℙ^{1}}(Y × ℙ^{1}) \end{matrix}

commutes. Denote T̄ = p̄_{A}∘(p̄r^{A}(t̄)∩−), f_{0}^{Z} = p̄_{A}, and f_{1}^{Z} = p̄_{A}∘(p̄r^{A}(ξ_{1})∩−). We are now prepared to consider the diagram

A_{Z×𝔸_{∞}^{1}}(Y × ℙ^{1} − Z × 0) ⟶^{α_{A}} A_{Z×ℙ^{1}}(Y × ℙ^{1}) ⟶^{β_{A}} A_{Z×0}(Y × ℙ^{1})
p̃_{A}↓  ↓( \begin{matrix}f_{0}^{Z} \\ f_{1}^{Z} \end{matrix} )  ↓T̄
A_{Z}(Y)  \underrightarrow{( \begin{matrix}1 \\ 0 \end{matrix} )} A_{Z}(Y) ⊕ A_{Z}(Y) ⟶^{(0,1)}  A_{Z}(Y)

where the rows are short exact sequences, with undisplayed zeros on both sides. The right square commutes since (4.4) commutes. To prove the commutativity on the left we must check that f_{1}^{Z}α_{A} = 0 (recall that p̄_{A}α_{A} = p̃_{A}). Consider the diagram

A_{Z×𝔸_{∞}^{1}}(Y × 𝔸_{∞}^{1}) ⟶^{α_{A}i_{A}} A_{Z×ℙ^{1}}(Y × ℙ^{1})
i^{A}(p̄r^{A}(ξ_{1}))∩−↓  ↓^{p̄r^{A}(ξ_{1})∩−}
A_{Z×𝔸_{∞}^{1}}(Y × 𝔸_{∞}^{1}) ⟶^{α_{A}i_{A}} A_{Z×ℙ^{1}}(Y × ℙ^{1})

which commutes by (cap3). But i^{A}(p̄r^{A}(ξ_{1})) ∈ A^{·}(Y × 𝔸_{∞}^{1}) vanishes as O(−1) restricted to 𝔸^{1} is trivial. Thus (p̄r^{A}(ξ_{1}) ∩ −)α_{A}i_{A} = 0. As i_{A} is an isomorphism, (p̄r^{A}(ξ_{1})∩−)α_{A} = 0, whence f_{1}^{Z}α_{A} = 0 and the big diagram commutes. Now we claim that the arrow ( \begin{matrix}f_{0}^{Z} \\ f_{1}^{Z} \end{matrix} ) is an isomorphism. The absolute (without supports) version of this is postulated in (ch4). The ‘with supports’ version can be deduced from (ch4) by applying the five-lemma to obvious localization sequences. Recall that p̃_{A} is an isomorphism and conclude that T̄ is an isomorphism. To complete the proof, consider the diagram

A_{Z×0}(Y × 𝔸^{1}) ⟶^{pr^{A}(t)∩−} A_{Z×𝔸^{1}}(Y × 𝔸^{1}) ⟶ A_{Z}(Y)
↓  ↓  ↓^{1}
A_{Z×0}(Y × ℙ^{1}) ⟶^{p̄r^{A}(t̄)∩−} A_{Z×ℙ^{1}}(Y × ℙ^{1}) ⟶ A_{Z}(Y)

where 𝔸^{1} now denotes ℙ^{1} − ∞ (as opposed to 𝔸_{∞}^{1}). It commutes by (cap3) as p̄r^{A}(t̄) maps to pr^{A}(t) via the map A_{Y ×0}^{·}(Y × ℙ^{1}) → A_{Y ×0}^{·}(Y × 𝔸^{1}). The top composition is T, the bottom one is T̄, and the left vertical arrow is an isomorphism by excision. We conclude that T is an isomorphism. The lemma is proved.

Define 𝔸^{(k)} to be the k-dimensional affine subspace of 𝔸^{n} given by x_{1} = … = x_{n−k} = 0, for 0 ≤ k ≤ n. Then 𝔸^{(k+1)} ∩ 𝔸_{n−k}^{n−1} = 𝔸^{(k)}. By (4.1) and (cap1), the map th_{n} ∩ − : A_{0}(𝔸^{n}) → A(𝔸^{n}) can be decomposed as

A_{0}(𝔸^{n}) ⟶^{pr_{n}^{A}(t_{n})∩−} A_{𝔸^{(1)}}(𝔸^{n}) ⟶^{pr_{n−1}^{A}(t_{n−1})∩−} A_{𝔸^{(2)}}(𝔸^{n}) ⟶^{pr_{n−2}^{A}(t_{n−2})∩−}

… ⟶^{pr_{1}^{A}(t_{1})∩−} A(𝔸^{n}) .

A generic step of this decomposition is a map

(4.5)  pr_{n−k}^{A}(t_{n−k}) : A_{𝔸^{(k)}}(𝔸^{n}) → A_{𝔸^{(k+1)}}(𝔸^{n}) .

In the notation of the lemma, put Y = 𝔸_{n−k}^{n−1}, Z = 𝔸^{(k)}, and think of 𝔸^{1} as 𝔸_{n−k}^{1}. Then Y × 𝔸^{1} can be identified with 𝔸^{n}, Z × 𝔸^{1} with 𝔸^{(k+1)}, pr^{A}(t) becomes pr_{n−k}^{A}(t_{n−k}), and we get that (4.5) is an isomorphism. The theorem is proved. Applying (h2) we obtain Corollary (PBTH with supports). If Z is a closed subvariety in a smooth X, E is a vector bundle over X of rank n + 1, and E_{Z} is its restriction to Z, then the map

F_{n} = (f_{0}, …, f_{n}) : A_{ℙ(E_{Z})}(ℙ(E)) → A_{Z}(X) ⊕ … ⊕ A_{Z}(X)

defined the same way as F_{n} in PBTH is an isomorphism.

References

[PS] I. Panin (after I. Panin and A. Smirnov), Oriented cohomology theories of algebraic varieties, K-Theory 30 (2003), 263-312.

[PS1] I. Panin (after I. Panin and A. Smirnov), Push-forwards in oriented cohomology theories of algebraic varieties: II, preprint at http://www.math.uiuc.edu/K-theory/0619/ (2003).

[Pa] I. Panin (after I. Panin and A. Smirnov), Riemann-Roch theorems for oriented cohomology, in Proceedings of the NATO ASI “Axiomatic, enriched and motivic homotopy theory” (Ed. J. P. C. Greenlees), NATO Sci. Ser. II: Math. Phys. and Chemistry, Vol. 131 (2004), 261-333.

[Pi] K. Pimenov, Traces in oriented homology theories of algebraic varieties, preprint at http://www.math.uiuc.edu/K-theory/0667/ (2003).

[PY] I. Panin and S. Yagunov, Poincaré duality for projective varieties, preprint at http://www.math.uiuc.edu/K-theory/0576/ (2002).

[SV] A. Suslin and V. Voevodsky, Bloch-Kato conjecture and motivic cohomology with finite coefficients, The Arithmetic and Geometry of Algebraic Cycles, 117-189; NATO Sci. Ser. C: Math. Phys. Sci., Vol. 548, Kluwer Acad. Publ., Dortrecht (2002).

[V] V. Voevodsky, 𝔸^{1}-homotopy theory, Doc. Math., Extra Vol. ICM 1998 (I), 417-442.

Alexander Nenashev

York University - Glendon College Department of Mathematics 2275 Bayview Av., Toronto ON Canada M4N 3M6 nenashev@glendon.yorku.ca 498

Documenta Mathematica 9 (2004)