The Chow-Witt ring

Jean Fasel

Received: May 18, 2006 Communicated by Ulf Rehmann

Abstract. We define a ring structure on the total Chow-Witt group of any integral smooth scheme over a field of characteristic different from 2. 2000 Mathematics Subject Classification: 14C15, 14C17, 14C99, 14F43 Keywords and Phrases: Chow-Witt groups, Chow groups, Grothendieck-Witt groups and Witt groups

Contents 1 Introduction 276

1.1 Conventions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 2 Preliminaries 278

2.1 Witt groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278 2.2 Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279 2.3 Supports . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

3 Chow-Witt groups 283 4 The exterior product 290 5 Intersection with a smooth subscheme 296

5.1 The Gysin-Witt map . . . . . . . . . . . . . . . . . . . . . . . . 296 5.2 Functoriality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

6 The ring structure 304 7 Basic properties 307

Documenta Mathematica 12 (2007) 275–312

1 Introduction

Let A be a commutative noetherian ring of Krull dimension n and P a projective A-module of rank d. One can ask the following question: does P admit a free factor of rank one? Serre proved a long time ago that the answer is always positive when d > n. So in fact the first interesting case is when P is projective of rank equal to the dimension of A. Suppose now that X is an integral smooth scheme over a field k of characteristic not 2. To deal with the above question, Barge and Morel introduced the Chow-Witt groups C̃H^{j}(X) of X (called at that time groupes de Chow des cycles orientés, see [BM]) and associated to each vector bundle E of rank n an Euler class c̃_{n}(E) in C̃H^{n}(X). It was proved recently that if X = Spec(A) we have c̃_{n}(P) = 0 if and only if P ≃ Q ⊕ A (see [Mo] for n ≥ 4, [FS] for n = 3 and [BM] or [Fa] for the case n = 2). It is therefore important to provide more tools, such as a ring structure and a pull-back for regular embeddings, to compute the Chow-Witt groups and the Euler classes. To define C̃H^{p}(X) consider the fibre product of the complex in Milnor K-theory

\xymatrix{0 \ar[r] & K_{p}^{M}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} K_{p−1}^{M}(k(x_{1})) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} K_{p−n}^{M}(k(x_{n})) \ar[r] & 0}

and the Gersten-Witt complex restricted to the fundamental ideals

\xymatrix{0 \ar[r] & I^{p}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} I^{p−1}(O_{X,x_{1}}) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} I^{p−n}(O_{X,x_{n}}) \ar[r] & 0}

over the quotient complex

\xymatrix{0 \ar[r] & I^{p}/I^{p+1}(k(X)) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} I^{p−n}/I^{p+1−n}(O_{X,x_{n}}) \ar[r] & 0.}

The group C̃H^{p}(X) is defined as the p-th cohomology group of this fibre product. Roughly speaking, an element of C̃H^{p}(X) is the class of a sum of varieties of codimension p with a quadratic form defined on each variety. We oviously have a map C̃H^{p}(X) → CH^{p}(X). Using the functoriality of the two complexes we see that the Chow-Witt groups satisfy good functorial properties (see [Fa]). For example, we have a pull-

back morphism f^{∗} : C̃H^{j}(X) → C̃H^{j}(Y) associated to each flat morphism

f : Y → X and a push-forward morphism g_{∗} : C̃H^{j}(Y, L) → C̃H^{j+r}(X) associated to each proper morphism g : Y → X, where r = dim(X) − dim(Y) and L is a suitable line bundle over Y. Using this functorial behaviour, it is possible to produce a good intersection theory. This is what we do in this paper using the classical strategy (see for example [Fu] or [Ro]). First we define an exterior product

C̃H^{j}(X) × C̃H^{i}(Y) → C̃H^{i+j}(X × Y)

and then a Gysin-like homomorphism i^{!} : C̃H^{d}(X) → C̃H^{d}(Y) associated to a closed embedding i : Y → X of smooth schemes. The product is then defined as the composition

\xymatrix{C̃H^{j}(X) × C̃H^{i}(X) \ar[r] & C̃H^{i+j}(X × X) \ar[r]^{△^{!}} & C̃H^{i+j}(X)}

where △ : X → X × X is the diagonal embedding. To define the exterior product, we first note that Rost already defined an exterior product on the homology of the complex in Milnor K-theory ([Ro]). Thus it is enough to define an exterior product on the homology of the Gersten-Witt complex and show that both exterior products coincide over the quotient complex. We use the usual product on derived Witt groups ([GN]) and show that this product passes to homology using the Leibnitz rule proved by Balmer (see [Ba2]). The definition of the Gysin-like map is done by following the ideas of Rost ([Ro]). It uses the deformation to the normal cone to modifiy any closed embedding to a nicer closed embedding and uses also the long exact sequence associated to a triple (Z, X, U) where Z is a closed subset of X and U = X \Z. The product that we obtain has the meaning of intersecting varieties with quadratic forms defined on them. It is therefore not a surprise that the natural map C̃H^{tot}(X) → CH^{tot}(X) turns out to be a ring homomorphism. There is however a surprise: the product that we obtain is a priori neither commutative nor anticommutative. This comes from the fact that the product of triangulated Grothendieck-Witt groups GW^{i} × GW^{j} → GW^{i+j} does not satisfy any commutativity property. The organization of this paper is as follows: In section 2, we recall some basic results on triangular Witt groups. This includes the construction of the Gersten-Witt complex, and some results on products and consanguinity. In section 3, we construct the Chow-Witt groups, recall some results and prove some basic facts. The definition of the exterior product takes place in section 4 and the definition of the Gysin-Witt map in section 5. In this part, we also prove the functoriality of this map. Finally we put all the pieces together in section 6 and prove some basic results in section 7.

I would like to thank Paul Balmer, Stefan Gille and Ivo Dell’Ambrogio for several careful readings of earlier versions of this work. I also would like to thank the referee for some useful comments. This research was supported by Swiss National Science Foundation, grant PP002-112579.

1.1 Conventions

All schemes are smooth and integral over a field k of characteristic different from 2, or are localizations of such schemes. For any two schemes X and Y we will always denote by X × Y the fibre product X ×_{Spec(k)} Y.

2 Preliminaries

2.1 Witt groups

We recall here some basic facts on Witt groups of triangulated categories following the exposition of [Ba2]. We suppose that for any triangulated category C and any objects A, B of C the group Hom(A, B) is uniquely 2-divisible. We also suppose that all triangulated categories are essentially small.

Definition 2.1. Let C be a triangulated category. A duality on C is a triple (D, δ, ϖ) where δ = ±1, D : C → C is a δ-exact contravariant functor and ϖ : 1 ≃ D^{2} is an isomorphism of functors satisfying D(ϖ_{A}) ∘ ϖ_{DA} = id_{DA} and T (ϖ_{A}) = ϖ_{T A} for all A ∈ C. A triangulated category C with a duality (D, δ, ϖ) is written (C, D, δ, ϖ).

Example 2.2. Let X be a regular scheme and P(X) the category of locally free coherent O_{X}-modules. Let D^{b}(P(X)) be the triangulated category of bounded complexes of objects of P(X). Then the usual duality ^{∨} on P(X) defined by P^{∨} = Hom_{O_{X}} (P, O_{X}) induces a 1-exact duality on D^{b}(P(X)). We also denote this derived duality by ^{∨}. Moreover, the canonical isomorphism ev : P → P^{∨∨} for any locally free module P induces a canonical isomorphism ϖ : 1 →^{∨∨} in D^{b}(P(X)). More generally, if L is any invertible module over X, then the duality Hom_{O_{X}} (_, L) on P(X) also induces a duality on D^{b}(P(X)).

Definition 2.3. Let (C, D, δ, ϖ) be a triangulated category with duality. For any i ∈ ℤ, define (D^{(i)}, δ^{(i)}, ϖ^{(i)}) by D^{(i)} = T^{i} ∘ D, δ^{(i)} = (−1)^{i}δ and ϖ^{(i)} = δ^{i}(−1)^{i(i+1)/2}ϖ. It is easy to check that (D^{(i)}, δ^{(i)}, ϖ^{(i)}) is a duality on C. It is called the i^{th}-shifted duality of (D, δ, ϖ).

Definition 2.4. Let (C, D, δ, ϖ) be a triangulated category with duality, A ∈ C and i ∈ ℤ. A morphism φ : A → D^{(i)}A is i-symmetric if the following diagram commutes:

[figure: A; φ; ϖ_{A}^{(i)}; D^{(i)}A; (D^{(i)})^{2}(A); D^{(i)}φ; D^{(i)}A.]

The couple (A, φ) is called an i-symmetric pair.

Definition 2.5. We denote by Symm^{i}(C) the monoid of isometry classes of i-symmetric pairs, equipped with the orthogonal sum.

Definition 2.6. An i-symmetric form is an i-symmetric pair (A, φ) where φ is an isomorphism.

Theorem 2.7. Let (C, D, δ, ϖ) be a triangulated category with duality and let (A, ϕ) be an i-symmetric pair. Choose an exact triangle containing ϕ

\xymatrix{A \ar[r]^{ϕ} & D^{(i)}A \ar[r]^{α} & C \ar[r]^{β} & T A.}

Then there exists an (i + 1)-symmetric isomorphism ψ : C → D^{(i+1)}C such that the following diagram commutes

[figure: A; ϕ; ϖ^{(i)}; D^{(i)}A; α; C; β; ψ; T A; T ϖ^{(i)}; D^{(i)}(D^{(i)}A); D^{(i)}ϕ; D^{(i)}A; δ^{(i+1)}D^{(i+1)}β; D^{(i+1)}C; D^{(i+1)}α; T (D^{(i)}(D^{(i)}A))]

where the rows are exact triangles and the second one is the dual of the first. Moreover, the (i + 1)-symmetric form (C, ψ) is unique up to isometry. It is denoted by cone(A, ϕ).

Proof. See [Ba1], Theorem 1.6.

Example 2.8. Let A ∈ C. For any i, the morphism 0 : A → D^{(i)}A is symmetric and then cone(A, 0) is well defined.

Corollary 2.9. The above construction gives a well defined homomorphism of monoids d^{i} : Symm^{(i)}(C) → Symm^{(i+1)}(C) such that d^{i+1}d^{i} = 0.

Definition 2.10. Let (C, D, δ, ϖ) be a triangulated category with duality. The Witt group W^{i}(C) is defined as Ker(d^{i})/Im(d^{i+1}). Remark that Ker(d^{i}) is just the monoid of isometry classes of i-symmetric forms.

Definition 2.11. Let (C, D, δ, ϖ) be a triangulated category with duality. The Grothendieck-Witt group GW^{i}(C) is defined as the quotient of Ker(d^{i}) by the submonoid generated by the elements cone(A, ϕ) − cone(A, 0) where A ∈ C and ϕ is (i − 1)-symmetric (0 is also seen as an (i − 1)-symmetric morphism).

Example 2.12. Let (D^{b}(P(X)),^{∨} , 1, ϖ) be the triangulated category with duality defined in Example 2.2. Its Witt groups are the Witt groups W^{i}(X) of the scheme X as defined in [Ba1].

2.2 Products

Given a pairing ⊗ : C × D → M of triangulated categories with duality and assuming that this pairing satisfies some nice conditions, the authors of [GN] define a pairing of Witt groups. We briefly recall some definitions (see 1.2 and 1.11 in [GN]):

Definition 2.13. Let C, D and M be triangulated categories. A product between C and D with codomain M is a covariant bi-functor

⊗ : C × D → M

exact in both variables and satisfying the following condition: the functorial isomorphisms r_{A,B} : A ⊗ T B ≃ T (A ⊗ B) and l_{A,B} : T A ⊗ B ≃ T (A ⊗ B) make the diagram

\xymatrix{T A ⊗ T B \ar[r]^{l_{A,T B}} \ar[d]_{r_{T A,B}} & T (A ⊗ T B) \ar[d]^{T (r_{A,B})} \\ T (T A ⊗ B) \ar[r]_{T (l_{A,B})} & T^{2}(A ⊗ B)}

skew-commutative.

Definition 2.14. Let C, D and M be triangulated categories with dualities. Where there is no possible confusion, we drop the subscripts for D, δ and ϖ. A dualizing pairing between C and D with codomain M is a product ⊗ with isomorphisms

η_{A,B} : DA ⊗ DB ≃ D(A ⊗ B)

natural in A and B which make the following diagrams commute

1.

\xymatrix{A ⊗ B \ar[r]^{ϖ_{A}⊗ϖ_{B}} \ar[d]_{ϖ_{A⊗B}} & D^{2}A ⊗ D^{2}B \ar[d]^{η_{DA,DB}} \\ D^{2}(A ⊗ B) \ar[r]_{D(η_{A,B})} & D(DA ⊗ DB)}

2.

\xymatrix{T (DT A ⊗ DB) \ar[d]_{δ_{C}δ_{M}T (η_{T A,B})} & DA ⊗ DB \ar[l]_{l_{DT A,DB}} \ar[d]_{η_{A,B}} \ar[r]^{r_{DA,DT B}} & T (DA ⊗ DT B) \ar[d]^{δ_{L}δ_{M}T (η_{A,T B})} \\ T D(T A ⊗ B) & D(A ⊗ B) \ar[l]^{T D(l_{A,B})} \ar[r]_{T D(r_{A,B})} & T D(A ⊗ T B).}

Theorem 2.15. Let C, D and M be triangulated categories with duality. Let ⊗ : C × D → M be a dualizing pairing between C and D with codomain M. Then ⊗ induces for all i, j ∈ ℤ a pairing

⋆ : W^{i}(C) × W^{j}(D) → W^{i+j}(M).

Proof. See [GN], Theorem 2.9.

Example 2.16. Let (D^{b}(P(X)),^{∨} , 1, ϖ) be the triangulated category with duality defined in Example 2.2. The usual tensor product induces a dualizing pairing of triangulated categories and then a product W^{i}(X)×W^{j}(X) → W^{i+j}(X). Suppose that L and N are invertible modules over X. Then Hom_{O_{X}} (_, L), Hom_{O_{X}} (_, N) and Hom_{O_{X}} (_, L ⊗ N) give dualities ^{♯}, ^{♮} and ^{♭} on D^{b}(P(X)). The tensor product gives a dualizing pairing

⊗ : (D^{b}(P(X)),^{♯} , 1, ϖ) × (D^{b}(P(X)),^{♮} , 1, ϖ) → (D^{b}(P(X)),^{♭} , 1, ϖ).

2.3 Supports

We briefly recall the notion of triangulated category with supports following [Ba2].

Definition 2.17. Let X be a topological space. A triangulated category defined over X is a pair (C, Supp) where C is a triangulated category and Supp assigns to each object A ∈ C a closed subset Supp(A) of X such that the following rules are satisfied:

(S1) Supp(A) = ∅ ⟺ A ≃ 0.

(S2) Supp(A ⊕ B) = Supp(A) ∪ Supp(B).

(S3) Supp(A) = Supp(T A).

(S4) For every distinguished triangle

\xymatrix{A \ar[r] & B \ar[r] & C \ar[r] & T A}

we have Supp(C) ⊂ Supp(A) ∪ Supp(B).

When I is a saturated triangulated subcategory of C and S is the multiplicative system of morphisms whose cone is in I, then we can construct a support on the category S^{−1}C := C/I. This is done in [Ba3] when C has a tensor product. However we will only need some basic facts, so we prove the following lemma:

Lemma 2.18. let C be a triangulated category defined over a topological space X. Let I be a saturated triangulated subcategory of C and let Supp(I) = ∪_{A∈I}Supp(A). Suppose that Supp(A) ⊂ Supp(I) implies A ∈ I. Let S be the multiplicative system in C of morphisms f such that cone(f) ∈ I and let

\xymatrix{I \ar[r] & C \ar[r] & C/I}

be the exact sequence of triangulated categories obtained by inverting S. Then C/I is a triangulated category defined over X′ = X \ Supp(I) (with the induced topology).

Proof. We define Supp_{S}(A) := Supp(A) ∩ X′ for any object A ∈ C/I and show that Supp_{S} satisfies the properties of Definition 2.17. It is easy to see that the rules (S1), (S2) and (S3) are satisfied. We only have to prove (S4). First observe that if s : A → B is a morphism in S and

\xymatrix{A \ar[r]^{s} & B \ar[r] & C \ar[r] & T A}

is an exact triangle in C containing s, then Supp_{S}(A) = Supp_{S}(B) (use (S4) for the category C). This shows that Supp_{S}(A) = Supp_{S}(A′) if A ≃ A′ in C/I. By definition of the triangulation of C/I, any exact triangle

\xymatrix{A \ar[r]^{α} & B \ar[r] & C \ar[r] & T A}

in C/I is isomorphic to the localization of an exact triangle in C. This shows that Supp_{S}(C) ⊂ Supp_{S}(A) ∪ Supp_{S}(B).

Example 2.19. Let D^{b}(P(X)) be the usual triangulated category. Define the support of an object P ∈ D^{b}(P(X)) as the union of the support of all the cohomology groups of P, i.e

Supp(P) = ⋃_{i} Supp(H^{i}(P)).

Then it is easy to see that (D^{b}(P(X)), Supp) is a triangulated category with support. Denote by D^{b}(P(X))^{(k)} the full subcategory of D^{b}(P(X)) of objects whose support is of codimension ≥ k. Then D^{b}(P(X))^{(k)} is a saturated triangulated category and the following sequence

D^{b}(P(X))^{(k)} → D^{b}(P(X)) → D^{b}(P(X))/D^{b}(P(X))^{(k)}

satisfies the conditions of Lemma 2.18. So D^{b}(P(X))/D^{b}(P(X))^{(k)} is a triangulated category over X′ = {x ∈ X | codim(x) ≤ k − 1}.

The following definitions are also due to Balmer (see [Ba2]):

Definition 2.20. Let (C, Supp) be a triangulated category over X and assume that C has a structure of triangulated category with duality (C, D, δ, ϖ). Then we say that C is a triangulated category with duality defined over X if

(S5) Supp(A) = Supp(DA) for every object A.

Definition 2.21. Let (C, Supp_{C}), (D, Supp_{D}) and (M, Supp_{M}) be triangulated categories defined over X. Suppose that

⊗ : C × D → M

is a pairing of triangulated categories. The pairing ⊗ is defined over X if

(S6) Supp_{M}(A ⊗ B) = Supp_{C}(A) ∩ Supp_{D}(B).

Example 2.22. The triangulated category D^{b}(P(X)) with the support defined in Example 2.19 and the pairing of Example 2.16 satisfy the condition (S5) and (S6).

Definition 2.23. The degeneracy locus of a symmetric pair (A, α) is defined to be the support of the cone of α:

DegLoc(α) = Supp(cone(α)).

Definition 2.24. Let (C, Supp) be a triangulated category with duality defined over X. The consanguinity of two symmetric pairs α and β is defined to be the following subset of X:

Cons(α, β) = (Supp(α) ∩ DegLoc(β)) ∪ (DegLoc(α) ∩ Supp(β)).

We are now ready to state the Leibnitz formula:

Theorem 2.25 (Leibnitz formula). Assume that we have a dualizing pairing ⊗ : C × D → F of triangulated categories with dualities over X. Let α and β be two symmetric pairs such that DegLoc(α) ∩ DegLoc(β) = ∅. Then we have an isometry

δ_{F} · d(α ⋆ β) = δ_{C} · d(α) ⋆ β + δ_{D} · α ⋆ d(β)

where δ_{C}, δ_{D}, δ_{F} are the signs involved in the dualities of C, D and F.

Proof. See [Ba2], Theorem 5.2.

3 Chow-Witt groups

Let (D^{b}(P(X)),^{∨} , 1, ϖ) be the triangulated category with the usual duality of Example 2.2 and consider its full subcategory D^{b}(P(X))^{(i)} of objects with supports of codimension ≥ i (here we use the support defined in Example 2.19). Then the duality on D^{b}(P(X)) induces dualities on D^{b}(P(X))^{(i)} for any i ([Ba1]). It is also clear that D^{b}(P(X))^{(i+1)} ⊂ D^{b}(P(X))^{(i)} for any i.

Definition 3.1. For all i ∈ ℕ, denote by D_{i}^{b}(X) the triangulated category D^{b}(P(X))^{(i)}/D^{b}(P(X))^{(i+1)}.

Suppose that (A, α) is an i-symmetric form in D_{i}^{b}(X). Then there exists an i-symmetric pair (B, β) such that the localization of (B, β) is (A, α) (by localization we mean the map Symm^{i}(D^{b}(P(X))^{(i)}) → Symm^{i}(D_{i}^{b}(X)) induced by the functor D^{b}(P(X))^{(i)} → D_{i}^{b}(X)). Applying 2.7, we get an (i+1)-symmetric form (C, ψ). By construction, C ∈ D^{b}(P(X))^{(i+1)}. Localizing this form we get a form (C, ψ) in W^{i+1}(D_{i+1}^{b}(X)). At first sight, this construction depends on some choices but in fact this is not the case (see [Ba1], Corollary 4.16). Hence we get a well defined homomorphism

d^{i} : W^{i}(D_{i}^{b}(X)) → W^{i+1}(D_{i+1}^{b}(X)).

Theorem 3.2. Let X be a regular scheme of dimension n. Then we have a complex

\xymatrix{0 \ar[r] & W^{0}(D_{0}^{b}(X)) \ar[r]^{d^{0}} & W^{1}(D_{1}^{b}(X)) \ar[r]^{d^{1}} & … \ar[r]^{d^{n}} & W^{n}(D_{n}^{b}(X)) \ar[r] & 0.}

Proof. See [BW], Theorem 3.1 and Paragraph 8.

Let A be a regular local ring. We denote by W^{fl}(A) the Witt group of finite length modules over A (see [QSS] for more informations about Witt groups of finite length modules). The following proposition holds:

Proposition 3.3. We have isomorphisms

W^{i}(D_{i}^{b}(X)) ≃ ⨁_{x∈X^{(i)}} W^{fl}(O_{X,x}).

Proof. See [BW], Theorem 6.1 and Theorem 6.2.

Remark 3.4. Since we use the isomorphism of the above proposition, we briefly recall how to obtain a symmetric complex from a finite length module. For more details, see [BW] or [Fa], Chapter 3. Choose a point x ∈ X^{(i)}, a finite length O_{X,x}-module M and a symmetric isomorphism ϕ : M → Ext_{O_{X,x}}^{i}(M, O_{X,x}). Let P_{•} be a resolution of M by locally free coherent O_{X,x}-modules. Then P_{•} can be chosen of the form

\xymatrix{0 \ar[r] & P_{i} \ar[r] & … \ar[r] & P_{0} \ar[r] & M \ar[r] & 0.}

Dualizing this complex and shifting i times gives the following diagram

\xymatrix{0 \ar[r] & P_{i} \ar[r] \ar@{-->}[d]_{∃} & … \ar[r] & P_{0} \ar[r] \ar@{-->}[d]_{∃} & M \ar[r] \ar[d]_{ϕ} & 0 \\ 0 \ar[r] & P_{0}^{∨} \ar[r] & … \ar[r] & P_{i}^{∨} \ar[r] & Ext_{O_{X,x}}^{i}(M, O_{X,x}) \ar[r] & 0.}

Using ϕ we get a symmetric morphism φ : P_{•} → (P_{•})^{∨}. Thus we have constructed an i-symmetric pair in the category D^{b}(P(O_{X,x})) from the pair (M, ϕ). Since D_{i}^{b}(X) ≃ ∐_{x∈X^{(i)}} D^{b}(P(O_{X,x})) ([BW], Proposition 7.1), we can see the pair

(P_{•}, φ) as a symmetric pair in D_{i}^{b}(X).

Definition 3.5. The complex

\xymatrix{0 \ar[r] & W^{fl}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} W^{fl}(O_{X,x_{1}}) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} W^{fl}(O_{X,x_{n}}) \ar[r] & 0}

is called the Gersten-Witt complex of X. We denote it by C(X, W).

This complex is obtained by using the usual duality ^{∨} on the triangulated category D^{b}(P(X)) (Example 2.2). For any invertible module L over X, we have a duality derived from the functor ^{♯} = Hom_{O_{X,x}}(_, L) and we can apply the same construction to get a Gersten-Witt complex.

Definition 3.6. Let X be a regular scheme and L an invertible O_{X}-module. We denote by C(X, W, L) the Gersten-Witt complex obtained from the duality ^{♯}.

Theorem 3.7. Let A be a regular local k-algebra and X = Spec(A). Then for any i > 0 we have H^{i}(C(X, W)) = 0.

Proof. See [BGPW], Theorem 6.1.

Let A be a regular local ring of dimension n. Denote by F the residue field of A. Then any choice of a generator ξ ∈ Ext_{A}^{n}(F, A) gives an isomorphism α_{ξ} : W (F) → W^{fl}(A). Recall that I(F) is the fundamental ideal of W (F). If n ≤ 0, put I^{n}(F) = W (F).

Definition 3.8. For any n ∈ ℤ let I_{fl}^{n}(A) be the image of I^{n}(F) by α_{ξ}.

Remark 3.9. It is easily seen that I_{fl}^{n}(A) does not depend on the choice of the generator ξ ∈ Ext_{A}^{n}(F, A).

Proposition 3.10. The differential d of the Gersten-Witt complex satisfies d(I_{fl}^{m}(O_{X,x})) ⊂ I_{fl}^{m−1}(O_{X,y}) for any m ∈ ℤ, x ∈ X^{(i)} and y ∈ X^{(i−1)}.

Proof. See [Gi3], Theorem 6.4 or [Fa], Theorem 9.2.4.

Definition 3.11. Let L be an invertible O_{X}-module. We denote by C(X, I^{d}, L) the complex

\xymatrix{0 \ar[r] & I_{fl}^{d}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} I_{fl}^{d−1}(O_{X,x_{1}}) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} I_{fl}^{d−n}(O_{X,x_{n}}) \ar[r] & 0.}

Remark 3.12. In particular, we have C(X, I^{0}, L) = C(X, W, L).

Theorem 3.13. Let A be an essentially smooth local k-algebra. Then for any i > 0 we have H^{i}(C(X, I^{d})) = 0.

Proof. See [Gi3], Corollary 7.7.

Of course, there is an inclusion C(X, I^{d+1}, L) → C(X, I^{d}, L) and therefore we get a quotient complex.

Definition 3.14. Denote by C(X, \overline{I}^{d}) the complex C(X, I^{d}, L)/C(X, I^{d+1}, L).

Remark 3.15. For any invertible module L the complexes C(X, I^{d})/C(X, I^{d+1}) and C(X, I^{d}, L)/C(X, I^{d+1}, L) are canonically isomorphic (see [Fa], Corollary E.1.3), so we can drop the L in C(X, \overline{I}^{d}).

Remark 3.16. The complex C(X, \overline{I}^{d}) is of the form

\xymatrix{0 \ar[r] & I_{fl}^{d}(k(X))/I_{fl}^{d+1}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} I_{fl}^{d−1}(O_{X,x_{1}})/I_{fl}^{d}(O_{X,x_{1}}) \ar[r] & ….}

Remark 3.17. As a consequence of Theorem 3.13, we immediately see that H^{i}(C(X, \overline{I}^{d})) = 0 for i > 0 if X = Spec(A) where A is an essentially smooth local k-algebra. Let F be a field and denote by K_{i}^{M}(F) the i-th Milnor K-theory group of F. If i < 0 it is convenient to put K_{i}^{M}(F) = 0.

Definition 3.18. Let X be a scheme. Then for any d we have a complex

\xymatrix{0 \ar[r] & K_{d}^{M}(k(X)) \ar[r] & ⨁_{x_{1}∈X^{(1)}} K_{d−1}^{M}(k(x_{1})) \ar[r] & … \ar[r] & ⨁_{x_{n}∈X^{(n)}} K_{d−n}^{M}(k(x_{n})) \ar[r] & 0.}

We denote it by C(X, K_{d}^{M}).

Proof. See [Ka], Proposition 1 or [Ro], Paragraph 3.

We also have the exactness of this complex when X is the spectrum of a smooth local k-algebra:

Theorem 3.19. Let A be a smooth local k-algebra. Then for all i > 0 we have H^{i}(C(X, K_{d}^{M})) = 0.

Proof. See [Ro], Theorem 6.1.

If F is a field, recall that we have a homomorphism due to Milnor

s : K_{j}^{M}(F) → I^{j}(F)/I^{j+1}(F)

given by s({a_{1}, …, a_{j}}) =< 1, −a_{1} > ⊗ … ⊗ < 1, −a_{j} >. The following is true:

Lemma 3.20. The homomorphisms s induce a morphism of complexes

s : C(X, K_{d}^{M}) → C(X, \overline{I}^{d}).

Proof. See [Fa], Proposition 10.2.5.

Definition 3.21. Let C(X, G^{d}, L) be the fibre product of C(X, K_{d}^{M}) and C(X, I^{d}, L) over C(X, \overline{I}^{d}):

\xymatrix{C(X, G^{d}, L) \ar[r] \ar[d] & C(X, I^{d}, L) \ar[d]^{π} \\ C(X, K_{d}^{M}) \ar[r]_{s} & C(X, \overline{I}^{d}).}

Definition 3.22. Let X be a smooth scheme and L an invertible O_{X}- module. The j-th Chow-Witt group C̃H^{j}(X, L) of X twisted by L is the group H^{j}(C(X, G^{j}, L)).

Remark 3.23. Denote by GW^{j}(D_{j}^{b}(X), L) the j-th Grothendieck-Witt group of the category D_{j}^{b}(X) with the duality derived from Hom_{O_{X}} (_, L) (see Definition 2.11). It is not hard to see that C(X, G^{j}, L) is isomorphic to GW^{j}(D_{j}^{b}(X), L) and therefore the complex C(X, G^{j}, L) is

\xymatrix{… \ar[r] & C(X, G^{j}, L)_{j−1} \ar[r] & GW^{j}(D_{j}^{b}(X), L) \ar[r]^{d^{j}} & W^{j+1}(D_{j+1}^{b}(X), L) \ar[r] & …}

Hence C̃H^{j}(X, L) is a quotient of Ker(d^{j}) and a subquotient of GW^{j}(D_{j}^{b}(X), L).

We also have the exactness of the complex C(X, G^{d}, L) in the local case:

Theorem 3.24. Let A be a smooth local k-algebra and X = Spec(A). Then H^{i}(C(X, G^{j})) = 0 for all j and all i > 0.

Proof. As C(X, G^{j}) is the fibre product of the complexes C(X, K_{j}^{M}) and

C(X, I^{j}) over C(X, \overline{I}^{j}), we have an exact sequence of complexes

\xymatrix{0 \ar[r] & C(X, G^{j}) \ar[r] & C(X, I^{j}) ⊕ C(X, K_{j}^{M}) \ar[r] & C(X, \overline{I}^{j}) \ar[r] & 0}

inducing a long exact sequence in cohomology. It follows then from Theorem 3.13 and Theorem 3.19 that H^{i}(C(X, G^{j})) = 0 if i > 1. For i = 1, we have an exact sequence

\xymatrix{H^{0}(C(X, I^{j})) ⊕ H^{0}(C(X, K_{j}^{M})) \ar[r] & H^{0}(C(X, \overline{I}^{j})) \ar[r] & H^{1}(C(X, G^{j})) \ar[r] & 0.}

The exact sequence of complexes

\xymatrix{0 \ar[r] & C(X, I^{j+1}) \ar[r] & C(X, I^{j}) \ar[r] & C(X, \overline{I}^{j}) \ar[r] & 0}

shows that H^{0}(C(X, I^{j})) maps onto H^{0}(C(X, \overline{I}^{j})).

Definition 3.25. Let X be a smooth scheme and L an invertible O_{X}-module. We define the sheaf G_{L}^{j} on X by G_{L}^{j}(U) = H^{0}(C(U, G^{j}, L)).

We have:

Theorem 3.26. Let X be a smooth scheme of dimension n. Then for any i we have

H_{Zar}^{i}(X, G_{L}^{j}) ≃ H^{i}(C(X, G^{j}, L)).

Proof. Define sheaves C_{l} by C_{l}(U) = C(U, G^{j}, L)_{l} for any l ≥ 0. It is clear that the C_{l} are flasque sheaves. We have a complex of sheaves over X

\xymatrix{0 \ar[r] & G_{L}^{j} \ar[r] & C_{0} \ar[r] & C_{1} \ar[r] & … \ar[r] & C_{n} \ar[r] & 0.}

Theorem 3.24 shows that this complex is a flasque resolution of G_{L}^{j}. Thus the theorem is proved.

Suppose that f : X → Y is a flat morphism. Since it preserves codimensions, it induces a morphism of complexes

f^{∗} : C(Y, G^{j}, L) → C(X, G^{j}, f^{∗}L)

for any j ∈ ℕ and any line bundle L over Y ([Fa], Corollary 10.4.2). Hence we have:

Theorem 3.27. Let f : X → Y be a flat morphism and L a line bundle over Y. Then, for any i, j we have homomorphisms

f^{∗} : H^{i}(C(Y, G^{j}, L)) → H^{i}(C(X, G^{j}, f^{∗}L)).

In particular, if E is a vector bundle over Y and π : E → Y is the projection, we have isomorphisms

π^{∗} : H^{i}(C(Y, G^{j}, L)) → H^{i}(C(E, G^{j}, π^{∗}L)).

Proof. We have a morphism of complexes f^{∗} : C(Y, G^{j}, L) → C(X, G^{j}, f^{∗}L) which gives the induced homomorphisms in cohomology. For the proof of homotopy invariance, see Corollary 11.3.2 in [Fa].

Proposition 3.28. Let f : X → Y and g : Y → Z be flat morphisms. Then (gf)^{∗} = f^{∗}g^{∗}.

Proof. See [Fa], Proposition 3.4.9.

Suppose that f : X → Y is a finite morphism with dim(Y) − dim(X) = r. Consider the morphism of locally ringed spaces \overline{f} : (X, O_{X}) → (Y, f_{∗}O_{X}) induced by f. If X is smooth, then L = \overline{f}^{∗}Ext_{O_{Y}}^{r} (f_{∗}O_{X}, O_{Y}) is an invertible module over Y ([Gi2], Corollary 6.6) and we get a morphism of complexes (of degree r)

f_{∗} : C(X, G^{j−r}, L ⊗ f^{∗}N) → C(Y, G^{j}, N)

for any invertible module N over Y ([Fa], Corollary 5.3.7).

Proposition 3.29. Let f : X → Y be a finite morphism between smooth schemes. Let dim(Y) − dim(X) = r and N be an invertible module over Y. Then the morphism of complexes f_{∗} induces a homomorphism

f_{∗} : H^{i−r}(C(X, G^{j−r}, L ⊗ f^{∗}N)) → H^{i}(C(Y, G^{j}, N)).

In particular, we have ([Fa], Remark 9.3.5):

Proposition 3.30. Let f : X → Y be a closed immersion of codimension r between smooth schemes. Then f induces an isomorphism

f_{∗} : H^{i−r}(C(X, G^{j−r}, L ⊗ f^{∗}N)) → H_{X}^{i}(C(Y, G^{j}, N))

for any i, j and any invertible module N over Y.

Important remark 3.31. If f : X → Y is a closed immersion, then f_{∗} will always be the map with support:

f_{∗} : H^{i−r}(C(X, G^{j−r}, L ⊗ f^{∗}N)) → H_{X}^{i}(C(Y, G^{j}, N))

The transfer for finite morphisms is functorial ([Fa], proposition 5.3.8):

Proposition 3.32. Let f : X → Y and g : Y → Z be finite morphisms. Then g_{∗}f_{∗} = (gf)_{∗}.

Remark 3.33. Let X be a smooth scheme and D be a smooth effective Cartier divisor on X. Let i : D → X be the inclusion and L(D) be the line bundle over X associated to D. Then there is a canonical section s ∈ L(D) (see [Fu], Appendix B.4.5) and an exact sequence

\xymatrix{0 \ar[r] & O_{X} \ar[r]^{s} & L(D) \ar[r] & i_{∗}O_{D} \ar[r] & 0.}

Applying Hom_{O_{X}} (_, L(D)) and shifting, we obtain the following diagram

\xymatrix{0 \ar[r] & O_{X} \ar[r]^{s} \ar[d]_{≃} & L(D) \ar[r] \ar[d]_{≃} & i_{∗}O_{D} \ar[r] \ar@{-->}[d] & 0 \\ 0 \ar[r] & Hom_{O_{X}} (L(D), L(D)) \ar[r]_{s} & Hom_{O_{X}} (O_{X}, L(D)) \ar[r] & Ext_{O_{X}}^{1} (i_{∗}O_{D}, L(D)) \ar[r] & 0}

which shows that Ext_{O_{X}}^{1} (i_{∗}O_{D}, O_{X}) ⊗ L(D) ≃ i_{∗}O_{D}. Proposition 3.30 shows that we then have an isomorphism

i_{∗} : H^{i−1}(C(D, G^{j−1}, i^{∗}L(D))) → H_{D}^{i}(C(X, G^{j})).

Lemma 3.34. Let g : X → Y be a flat morphism and f : Z → Y a finite morphism. Consider the following fibre product

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

Then (f′)_{∗}(g′)^{∗} = g^{∗}f_{∗}.

Proof. See [Fa], Corollary 12.2.8.

Remark 3.35. Of course, in the above fibre product we suppose that V is also smooth and integral. Such a strong assumption is not necessary in general, but this case is sufficient for our purposes.

Remark 3.36. It is possible to define a map f_{∗} when the morphism f is proper (see [Fa]) but we don’t use this fact here.

4 The exterior product

Let X and Y be two schemes. The fibre product X × Y comes equipped with two projections p_{1} : X × Y → X and p_{2} : X × Y → Y.

Lemma 4.1. For any i, j ∈ ℕ, the pairing

⊠ : D_{i}^{b}(X) × D_{j}^{b}(Y) → D_{i+j}^{b}(X × Y)

given by P ⊠ Q = p_{1}^{∗}P ⊗ p_{2}^{∗}Q is a dualizing pairing of triangulated categories with duality.

Proof. Straight verification.

Corollary 4.2. For any i, j ∈ ℕ, the pairing

⊠ : D_{i}^{b}(X) × D_{j}^{b}(Y) → D_{i+j}^{b}(X × Y)

induces a pairing

⋆ : W^{i}(D_{i}^{b}(X)) × W^{j}(D_{j}^{b}(Y)) → W^{i+j}(D_{i+j}^{b}(X × Y)).

Proof. Clear by Theorem 2.15.

Corollary 4.3. Let ψ ∈ W^{j}(D_{j}^{b}(Y)). Then we have a homomorphism

μ_{ψ} : W^{i}(D_{i}^{b}(X)) → W^{i+j}(D_{i+j}^{b}(X × Y))

given by μ_{ψ}(φ) = φ ⋆ ψ.

Recall that we have isomorphisms W^{i}(D_{i}^{b}(X)) ≃ ⨁_{x∈X^{(i)}} W^{fl}(O_{X,x}) (Proposi-

tion 3.3).

Definition 4.4. For any s ∈ ℤ, denote by I^{s}(D_{i}^{b}(X)) the preimage of ⨁_{x∈X^{(i)}} I_{fl}^{s}(O_{X,x}) under the above isomorphism.

Proposition 4.5. For any m, p ∈ ℕ the product

⋆ : W^{i}(D_{i}^{b}(X)) × W^{j}(D_{j}^{b}(Y)) → W^{i+j}(D_{i+j}^{b}(X × Y))

induces a product

⋆ : I^{m}(D_{i}^{b}(X)) × I^{n}(D_{j}^{b}(Y)) → I^{m+n}(D_{i+j}^{b}(X × Y)).

Proof. Let x ∈ X^{(i)} and y ∈ Y^{(j)}. It is clear that the product can be computed locally (use [GN], Theorem 3.2). So we can suppose that X = Spec(A) and Y = Spec(B) where A and B are local in x and y respectively. Recall that we have the following diagram

\xymatrix{X × Y \ar[r]^{p_{2}} \ar[d]_{p_{1}} & Y \ar[d] \\ X \ar[r] & Spec(k).}

Let P be an A-projective resolution of k(x) and Q be a B-projective resolution of k(y). Consider a symmetric form ρ : k(x) → Ext_{A}^{i}(k(x), A) and a symmetric form μ : k(y) → Ext_{B}^{j}(k(y), B). Then p_{1}^{∗}(ρ) is a symmetric isomorphism supported by the complex P ⊗_{k} B and p_{2}^{∗}(μ) is a symmetric isomorphism supported by the complex A ⊗_{k} Q. The complex (P ⊗_{k} B) ⊗_{A⊗_{k}B} (A ⊗_{k} Q) (which is isomorphic to P ⊗_{k} Q) has its homology concentrated in degree 0, and this homology is isomorphic to k(x)⊗_{k} k(y). Let u be a point of Spec(k(x)⊗_{k} k(y)). Then the restriction of p_{1}^{∗}ρ⊗p_{2}^{∗}μ to u is a finite length module M whose support is on u with a symmetric form

M → Ext_{(A⊗B)_{u}}^{i+j}(M, (A ⊗ B)_{u}).

Taking its class in the Witt group, we obtain a k(u)-vector space V with a symmetric form ψ : V → Ext_{(A⊗B)_{u}}^{i+j}(V, (A ⊗ B)_{u}). Now choose a unit a ∈ k(x)^{×}. Consider the image a_{u} of a under the homomorphism k(x) → k(u). The class of p_{1}^{∗}(aρ) ⊗ p_{2}^{∗}(μ) is the symmetric form

a_{u}ψ : V → Ext_{(A⊗B)_{u}}^{i+j}(V, (A ⊗ B)_{u}).

As the same property holds for any unit b ∈ k(y)^{×}, we conclude that

p_{1}^{∗}(< 1, −a_{1} > ⊗ … ⊗ < 1, −a_{n} > ρ) ⊗ p_{2}^{∗}(< 1, −b_{1} > ⊗ … ⊗ < 1, −b_{m} > μ)

is equal to < 1, −(a_{1})_{u} > ⊗ … ⊗ < 1, −(b_{m})_{u} > ψ.

Recall that for any scheme X we have a Gersten-Witt complex (Definition 3.5)

\xymatrix{C(X, W) : & … \ar[r] & W^{r}(D_{r}^{b}(X)) \ar[r]^{d_{X}^{r}} & W^{r+1}(D_{r+1}^{b}(X)) \ar[r] & …}

and a complex C(X, I^{d}):

\xymatrix{… \ar[r] & ⨁_{x_{r}∈X^{(r)}} I_{fl}^{d−r}(O_{X,x_{r}}) \ar[r] & ⨁_{x_{r+1}∈X^{(r+1)}} I_{fl}^{d−r−1}(O_{X,x_{r+1}}) \ar[r] & ….}

The above proposition gives:

Corollary 4.6. The product

⋆ : C(X, W) × C(Y, W) → C(X × Y, W)

induces for any r, s ∈ N a product

⋆ : C(X, I^{r}) × C(Y, I^{s}) → C(X × Y, I^{r+s}).

Now we investigate the relations between ⋆ and the differentials of the complexes.

Proposition 4.7. Let ψ ∈ W^{j}(D_{j}^{b}(Y)) be such that d_{Y}^{j} (ψ) = 0. Then the following diagram commutes

\xymatrix{W^{i}(D_{i}^{b}(X)) \ar[r]^{d_{X}^{i}} \ar[d]_{(−1)^{j}μ_{ψ}} & W^{i+1}(D_{i+1}^{b}(X)) \ar[d]^{μ_{ψ}} \\ W^{i+j}(D_{i+j}^{b}(X × Y)) \ar[r]_{d_{X×Y}^{i+j}} & W^{i+j+1}(D_{i+j+1}^{b}(X × Y)).}

Proof. Let φ ∈ W^{i}(D_{i}^{b}(X)). Let X^{(≥i+1)} be the set of points of X of codimension ≥ i+1, Y^{(≥j+1)} the points of Y of codimension ≥ j+1 and (X×Y)^{(≥i+j+1)} the set of points of X × Y of codimension ≥ i + j + 1. By Lemma 2.18, the triangulated categories D_{i}^{b}(X), D_{j}^{b}(Y) and D_{i+j}^{b}(X × Y) are defined over the topological spaces X \X^{(≥i+1)}, Y \Y^{(≥j+1)} and (X ×Y)\(X ×Y)^{(≥i+j+1)}. Let α ∈ Symm^{i}(D^{b}(P(X))^{(i)}) and β ∈ Symm^{j}(D^{b}(P(Y))^{(j)}) be symmetric pairs representing φ and ψ. By definition, DegLoc(α) is of codimension ≥ i + 1, DegLoc(β) is of codimension ≥ j + 1 and dβ is neutral. It is easily seen that Supp(dp_{1}^{∗}α)∩Supp(dp_{2}^{∗}β) = ∅ in the topological space (X×Y)\(X×Y)^{(≥i+j+1)}. Theorem 2.25 implies that

(−1)^{i+j}d(p_{1}^{∗}α ⋆ p_{2}^{∗}β) = (−1)^{i}dp_{1}^{∗}α ⋆ p_{2}^{∗}β + (−1)^{j}p_{1}^{∗}α ⋆ dp_{2}^{∗}β.

Using Theorem 2.15, we see that we have in W^{i+j}(D_{i+j}^{b}(X × Y)) the equality

(−1)^{j}d_{X×Y}^{i+j} (p_{1}^{∗}φ ⋆ p_{2}^{∗}ψ) = p_{1}^{∗}d_{X}^{i}(φ) ⋆ p_{2}^{∗}ψ.

The following corollary is obvious.

Corollary 4.8. Let ψ ∈ I^{m}(D_{j}^{b}(Y)) be such that d_{j}^{Y} (ψ) = 0. Then the following diagram commutes

\xymatrix{I^{p}(D_{i}^{b}(X)) \ar[r]^{d_{X}^{i}} \ar[d]_{(−1)^{j}μ_{ψ}} & I^{p−1}(D_{i+1}^{b}(X)) \ar[d]^{μ_{ψ}} \\ I^{p+m}(D_{i+j}^{b}(X × Y)) \ar[r]_{d_{X×Y}^{i+j}} & I^{p+m−1}(D_{i+j+1}^{b}(X × Y)).}

We now have to deal with the complex in Milnor K-theory. Let C(X, K_{r}^{M}), C(Y, K_{s}^{M}) and C(X ×Y, K_{r+s}^{M}) be the complexes in Milnor K-theory associated to X, Y and X × Y. In [Ro], Rost defines a product

⊙ : C(X, K_{r}^{M})^{i} × C(Y, K_{s}^{M})^{j} → C(X × Y, K_{r+s}^{M})^{i+j}

as follows: Let u ∈ (X ×Y)^{(i+j)}, x ∈ X^{(i)}, y ∈ Y^{(j)} be such that x and y are the projections of u. Let ρ = {a_{1}, …, a_{r−i}} ∈ K_{r−i}^{M}(k(x)) and μ = {b_{1}, …, b_{s−j}} ∈ K_{s−j}^{M}(k(y)). Then

(ρ ⊙ μ)_{u} = l((k(x) ⊗_{k} k(y))_{u}){(a_{1})_{u}, …, (a_{r−i})_{u}, (b_{1})_{u}, …, (b_{s−j})_{u}}

where the (a_{l})_{u} and (b_{t})_{u} are the images of the a_{l} and b_{t} under the inclusions k(x) → k(u) and k(y) → k(u), and l((k(x) ⊗_{k} k(y))_{u}) is the length of the module k(x) ⊗_{k} k(y) localized in u.

Lemma 4.9. For any ρ ∈ C(X, K_{r}^{M})^{i} and μ ∈ C(Y, K_{s}^{M})^{j} we have

d(ρ ⊙ μ) = d(ρ) ⊙ μ + (−1)^{j}ρ ⊙ d(μ).

Proof. See [Ro], Paragraph 14.4.

Corollary 4.10. Let μ ∈ C(Y, K_{s}^{M})^{j} be such that dμ = 0. Then the following diagram commutes:

\xymatrix{C(X, K_{r}^{M})^{i} \ar[r]^{d_{X}^{i}} \ar[d]_{⊙μ} & C(X, K_{r}^{M})^{i+1} \ar[d]^{⊙μ} \\ C(X × Y, K_{r+s}^{M})^{i+j} \ar[r]_{d_{X×Y}^{i+j}} & C(X × Y, K_{r+s}^{M})^{i+j+1}.}

Documenta Mathematica 12 (2007) 275–312

Proof. Obvious.

Now we compare the products ⋆ and ⊙.

Proposition 4.11. The following diagram commutes:

\xymatrix{C(X, K_{r}^{M})^{i} × C(Y, K_{s}^{M})^{j} \ar[r]^{⊙} \ar[d]_{s_{(r−i)}×s_{(s−j)}} & C(X × Y, K_{r+s}^{M})^{i+j} \ar[d]^{s_{(r+s−i−j)}} \\ C(X, \overline{I}^{r})^{i} × C(Y, \overline{I}^{s})^{j} \ar[r]_{⋆} & C(X × Y, \overline{I}^{r+s})^{i+j}.}

Proof. Let {a_{1}, …, a_{r−i}} ∈ K_{r−i}^{M}(k(x)) and {b_{1}, …, b_{s−j}} ∈ K_{s−j}^{M}(k(y)). Let ρ′ be a symmetric isomorphism

ρ′ : k(x) → Ext_{O_{X,x}}^{i}(k(x), O_{X,x})

and μ′ a symmetric isomorphism

μ′ : k(y) → Ext_{O_{Y,y}}^{j}(k(y), O_{Y,y}).

We then have ρ := s_{(r−i)}({a_{1}, …, a_{r−i}}) =< 1, −a_{1} > ⊗ … ⊗ < 1, −a_{r−i} > ρ′ and μ := s_{(s−j)}({b_{1}, …, b_{s−j}}) =< 1, −b_{1} > ⊗ … ⊗ < 1, −b_{s−j} > μ′. Choose a point u in (X × Y)^{(i+j)} lying over x and y. The proof of Proposition 4.5 shows that

(ρ ⋆ μ)_{u} = s_{(r+s−i−j)}({(a_{1})_{u}, …, (a_{r−i})_{u}, (b_{1})_{u}, …, (b_{s−j})_{u}})φ

where φ : M → Ext_{O_{X×Y,u}}^{i+j}(M, O_{X×Y,u}) is a symmetric isomorphism and M is a k(u)-vector space. But dim_{k(u)}M ≡ l((k(x) ⊗ k(y))_{u}) (mod 2) where l denotes the length. So we have in C(X × Y, \overline{I}^{r+s})^{i+j} the equality

(ρ ⋆ μ)_{u} = s_{(r+s−i−j)}({(a_{1})_{u}, …, (a_{r−i})_{u}, (b_{1})_{u}, …, (b_{s−j})_{u}})l((k(x) ⊗ k(y))_{u}).

The right hand term is equal to s_{(r+s−i−j)}({a_{1}, …, a_{r−i}} ⊙ {b_{1}, …, b_{s−j}}) by definition.

Corollary 4.12. The products

⋆ : C(X, I^{r}) × C(Y, I^{s}) → C(X × Y, I^{r+s})

and

⊙ : C(X, K_{r}^{M}) × C(Y, K_{s}^{M}) → C(X × Y, K_{r+s}^{M})

give a product

⋄ : C(X, G^{r}) × C(Y, G^{s}) → C(X × Y, G^{r+s}).

Corollary 4.13. Let μ ∈ C(Y, G^{s})^{j} such that d_{Y}^{j} μ = 0. Then μ induces a product

_⋄ μ : H^{i}(C(X, G^{r})) → H^{i+j}(C(X × Y, G^{r+s})).

Proof. This a direct consequence of Proposition 4.11, Corollary 4.8 and Corollary 4.10.

Next we have to check that _⋄ μ is well defined on the cohomolgy class of μ.

Lemma 4.14. Let γ ∈ C(Y, G^{s})^{j−1} and μ = d_{Y}^{j−1}γ. Then _⋄ μ = 0.

Proof. Suppose that α is such that d_{X}^{i}α = 0. By Corollary 4.8 and Corollary 4.10 we have up to signs d_{X×Y}^{i+j−1}(α ⋄ γ) = α ⋄ d^{j−1}γ = α ⋄ μ. So α ⋄ μ is trivial in H^{i+j}(C(X × Y, G^{r+s})).

Finally:

Theorem 4.15. Let X and Y be smooth schemes. Then for any i, j, r, s ∈ ℕ the product

⋄ : C(X, G^{r}) × C(Y, G^{s}) → C(X × Y, G^{r+s})

induces an exterior product

× : H^{i}(C(X, G^{r})) × H^{j}(C(Y, G^{s})) → H^{i+j}(C(X × Y, G^{r+s})).

This exterior product can also be defined with complexes twisted by invertible modules.

Theorem 4.16. Let X and Y be smooth schemes. Let L and N be invertible modules over X and Y respectively. For any i, j, r, s ∈ ℕ, the pairing

⋄ : C(X, G^{r}, L) × C(Y, G^{s}, N) → C(X × Y, G^{r+s}, p_{1}^{∗}L ⊗ p_{2}^{∗}N)

induces an exterior product

× : H^{i}(C(X, G^{r}, L)) × H^{j}(C(Y, G^{s}, N)) → H^{i+j}(C(X × Y, G^{r+s}, p_{1}^{∗}L ⊗ p_{2}^{∗}N)).

Proof. Left to the reader.

If i = r and j = s, we obtain the following corollary:

Corollary 4.17. Let X and Y be smooth schemes. Then for any i, j ∈ ℕ the product

⋄ : C(X, G^{i}) × C(Y, G^{j}) → C(X × Y, G^{i+j})

gives an exterior product

× : C̃H^{i}(X) × C̃H^{j}(Y) → C̃H^{i+j}(X × Y).

Next we prove some properties of this exterior product:

Proposition 4.18. The exterior product × is associative.

Proof. It clearly suffices to prove that the exterior products ⋆ and ⊙ are associative. For ⋆ this is clear because of the associativity of the tensor product (up to isomorphism). For the second, see (14.2) in [Ro].

Now we deal with the commutativity. Let X and Y be smooth schemes and let τ : X × Y → Y × X be the flip. We have:

Lemma 4.19. Let μ ∈ H^{i}(C(X, K_{r}^{M})) and η ∈ H^{j}(C(Y, K_{s}^{M})). Then we have τ^{∗}(η ⊙ μ) = (−1)^{(r−i)(s−j)}(μ ⊙ η).

Proof. This is clear from the definition.

Lemma 4.20. Let μ ∈ H^{i}(C(X, I^{r})) and η ∈ H^{j}(C(Y, I^{s})). Then we have τ^{∗}(η ⋆ μ) = (−1)^{ij}(μ ⋆ η).

Proof. It is clear by the skew-commutativity of the product of Witt groups ([GN], Theorem 3.1).

Remark 4.21. Of course, the associativity and the anticommutativity of the exterior product are also true for the twisted product of Theorem 4.16.

5 Intersection with a smooth subscheme

5.1 The Gysin-Witt map

The goal of this section is to define for any closed embedding i : Y → X of smooth schemes a Gysin-Witt map i^{!} : H^{r}(C(X, G^{j})) → H^{r}(C(Y, G^{j})). In order to define such a map, we adapt the ideas of Rost ([Ro], Paragraph 11). First we briefly recall the properties of the deformation to the normal cone. For more details, see [Fu] (Chapter 5) or [Ro] (Chapter 10). Let Y be a closed subscheme of a smooth scheme X. Then there is a smooth scheme D(X, Y), a closed imbedding j : Y ×𝔸^{1} ↪ D(X, Y) and a flat morphism ρ : D(X, Y) → 𝔸^{1} such that the following diagram commutes

\xymatrix{Y × 𝔸^{1} \ar[r]^{j} \ar[dr]_{pr} & D(X, Y) \ar[d]^{ρ} \\  & 𝔸^{1}}

and

(1) ρ^{−1}(𝔸^{1} −0) = X ×(𝔸^{1} −0) and the restriction of j is the closed imbedding i × Id : Y × (𝔸^{1} − 0) ↪ X × (𝔸^{1} − 0).

(2) ρ^{−1}(0) = N_{Y} X, where N_{Y} X is the normal cone to Y in X and the restriction of j is the embedding as the zero section s_{0} : Y → N_{Y} X.

The scheme D(X, Y) can be obtained as follows: Consider the blow-up M of X × 𝔸^{1} along Y × 0 and the blow-up X̃ of X × 0 along Y × 0. Then define D(X, Y) to be M \ X̃. If Y is smooth in a smooth scheme X, then it is locally of complete intersection and N_{Y} X is a vector bundle over Y of rank dim(X) − dim(Y). Moreover, N_{Y} X is Cartier divisor on D(X, Y). If 𝔸^{1} = Spec(k[t]), then the projection ρ : D(X, Y) → 𝔸^{1} gives a homomorphism k[t] → O_{D(X,Y)}(D(X, Y)). We still denote by t the image of t under this homomorphism. We have an exact sequence

\xymatrix{0 \ar[r] & O_{D(X,Y)} \ar[r]^{t} & O_{D(X,Y)} \ar[r] & κ_{∗}O_{N_{Y} X} \ar[r] & 0}

where κ : N_{Y} X → D(X, Y) is the inclusion. Remark 3.33 shows that Ext_{O_{D(X,Y )}}^{1}(κ_{∗}O_{N_{Y} X}, O_{D(X,Y)}) ≃ κ_{∗}O_{N_{Y} X} with generator the Koszul complex associated to the global section t. Let U = 𝔸^{1} − 0 and consider the form

< 1, −t >: O_{U}^{2} → O_{U}^{2}

in W^{0}(D^{b}(U)). Now let X be a smooth scheme and consider the projection η : X × U → U. Then η^{∗}(< 1, −t >) ∈ W^{0}(D^{b}(X × U)) and we also denote it by < 1, −t >. Since the support of this form is X × U, the tensor product gives a functor

< 1, −t > ⊗_: D_{i}^{b}(X × U) → D_{i}^{b}(X × U).

Using the fact that < 1, −t > is symmetric, we see that this functor is duality preserving (see [GN], Definition 1.8 and Lemma 1.14) and therefore induces for any i a homomorphism

< 1, −t > ⊗_: W^{i}(D_{i}^{b}(X × U)) → W^{i}(D_{i}^{b}(X × U)).

For some sign reasons that will be made clearer in Lemma 5.10, we will in fact consider for any i the homomorphism

m_{t} : W^{i}(D_{i}^{b}(X × U)) → W^{i}(D_{i}^{b}(X × U))

defined by m_{t}(α) = (−1)^{i+1} < 1, −t > ⊗α.

Lemma 5.1. For any i, j ∈ ℕ the homomorphism m_{t} induces a homomorphism

I^{j}(D_{i}^{b}(X × U)) → I^{j+1}(D_{i}^{b}(X × U))

and the following diagram commutes

\xymatrix{I^{j}(D_{i}^{b}(X × U)) \ar[r]^{d^{i}} \ar[d]_{−m_{t}} & I^{j−1}(D_{i+1}^{b}(X × U)) \ar[d]^{m_{t}} \\ I^{j+1}(D_{i}^{b}(X × U)) \ar[r]_{d^{i}} & I^{j}(D_{i+1}^{b}(X × U)).}

Proof. The first assertion is clear. Now < 1, −t > is a global isomorphism and we can use Theorem 2.10 in [GN] (or Theorem 2.25 in the present paper) to see that

d^{i}(< 1, −t > ⊗α) =< 1, −t > ⊗d^{i}α

for any α ∈ I^{j}(D_{i}^{b}(X ×U)). The first term is (−1)^{i+1}d^{i}(m_{t}(α)) and the second one is (−1)^{i+2}m_{t}(d^{i}α).

Now consider t ∈ O_{X×U}^{∗}. For any i and any x ∈ X×U, we have a multiplication by t:

n_{t} : K_{i}^{M}(k(x)) → K_{i+1}^{M}(k(x))

defined by n_{t}({a_{1}, …, a_{i}}) = {t, a_{1}, …, a_{i}}.

Lemma 5.2. For any i, j ∈ N the following diagram commutes

\xymatrix{C(X × U, K_{j}^{M})^{i} \ar[r]^{d^{i}} \ar[d]_{−n_{t}} & C(X × U, K_{j}^{M})^{i+1} \ar[d]^{n_{t}} \\ C(X × U, K_{j+1}^{M})^{i} \ar[r]_{d^{i}} & C(X × U, K_{j+1}^{M})^{i+1}.}

Proof. See [Ro], Proposition 4.6.

Corollary-Definition 5.3. The homomorphisms m_{t} and n_{t} induce for any i, j ∈ N a homomorphism

{t} : H^{i}(C(X × U, G^{j})) → H^{i}(C(X × U, G^{j+1})).

We call this homomorphism multiplication by t.

Proof. It suffices to show that m_{t} and n_{t} give the same operation on the complex C(X × U, \overline{I}^{j}). It is straightforward.

We will need the following lemma:

Lemma 5.4. Let f : X → Y be a flat morphism of smooth schemes. Then for any i, j the following diagram commutes

\xymatrix{H^{i}(C(Y × U, G^{j})) \ar[r]^{\{t\}} \ar[d]_{(f×Id)^{∗}} & H^{i}(C(Y × U, G^{j+1})) \ar[d]^{(f×Id)^{∗}} \\ H^{i}(C(X × U, G^{j})) \ar[r]_{\{t\}} & H^{i}(C(X × U, G^{j+1})).}

Proof. First observe that (f × Id)^{∗}(< 1, −t >) =< 1, −t > by definition. Then for any α ∈ I^{r}(D_{i}^{b}(X ×U)) we have (f ×Id)^{∗}(m_{t}α) = m_{t}((f ×Id)^{∗}α) (use [GN], Theorem 3.4). On the other hand, we have (f × Id)^{∗}(n_{t}(α)) = n_{t}((f × Id)^{∗}α) for any α ∈ K_{r}^{M}(k(y)) ([Ro], Lemma 4.3). Putting this together, we get the conclusion.

Let Y → X be a closed embedding of smooth schemes and consider the deformation to the normal cone space D(X, Y). Then N_{Y} X is a Cartier divisor and its complement in D(X, Y) is X ×U. We have a long exact sequence associated to this triple ([Fa], Corollary 10.4.9):

\xymatrix{H^{i}(C(D(X, Y), G^{j+1})) \ar[r] & H^{i}(C(X × U, G^{j+1})) \ar[r]^{∂} & H_{N_{Y} X}^{i+1}(C(D(X, Y), G^{j+1}))}

Combining the isomorphism of Proposition 3.30 and the isomorphism

O_{N_{Y} X} → \overline{κ}^{∗}Ext_{O_{D(X,Y )}}^{1}(κ_{∗}O_{N_{Y} X}, O_{D(X,Y)})

mapping 1 to the Koszul complex associated to the global section t of O_{D(X,Y)}, we finally get an isomorphism

κ_{∗} : H^{i}(C(N_{Y} X, G^{j})) → H_{N_{Y} X}^{i+1}(C(D(X, Y), G^{j+1})).

Let q : N_{Y} X → Y and π : X × U → X be the projections and consider the following composition:

\xymatrix{H^{i}(C(X, G^{j})) \ar[d]_{π^{∗}} \ar@{-->}[rr] &  & H^{i}(C(Y, G^{j})) \\ H^{i}(C(X × U, G^{j})) \ar[r]_{\{t\}} & H^{i}(C(X × U, G^{j+1})) \ar[r]_{(κ_{∗})^{−1}∂} & H^{i}(C(N_{Y} X, G^{j})). \ar[u]_{(q^{∗})^{−1}}}

Definition 5.5. Let Y be a smooth subscheme of a smooth scheme X with inclusion i : Y → X. We denote by i^{!} : H^{r}(C(X, G^{j})) → H^{r}(C(Y, G^{j})) and call Gysin-Witt map the composition (q^{∗})^{−1}(κ_{∗})^{−1}∂{t}π^{∗}.

Remark 5.6. Let i : Y → X be a closed immersion of smooth schemes and let L be an invertible O_{X}-module. Then we have a twisted version of the Gysin-Witt map:

i^{!} : H^{r}(C(X, G^{j}, L)) → H^{r}(C(Y, G^{j}, i^{∗}L)).

5.2 Functoriality

The goal of this section is to prove that for any inclusions of smooth schemes

\xymatrix{Z \ar[r]^{i} & Y \ar[r]^{j} & X} we have (ji)^{!} = i^{!}j^{!}. The strategy is not new. We follow the exposition of the sections 11, 12 and 13 in [Ro]. First we prove some lemmas:

Lemma 5.7. Let i : Y → X be a closed immersion and g : V → X be a flat morphism. Consider the following fibre product

\xymatrix{W \ar[r]^{i′} \ar[d]_{g′} & V \ar[d]^{g} \\ Y \ar[r]_{i} & X.}

Then we have (g′)^{∗}i^{!} = (i′)^{!}g^{∗}.

Proof. Let D(X, Y) be the deformation to the normal cone associated to the inclusion i : Y ↪ X and D(V, W) be the deformation associated to i′ : W ↪ V. Let U = 𝔸^{1} − 0. Because of the universal properties of blow-ups, we see that g and g′ give a morphism D(g) : D(V, W) → D(X, Y) such that the following diagram commutes:

\xymatrix{D(V, W) \ar[d]_{D(g)} & V × U \ar[d]^{g×1} \ar[l]_{ι′} \\ D(X, Y) & X × U \ar[l]_{ι}}

where ι and ι′ are the inclusions of the respective open subsets. We also get a morphism N(g) : N_{W} V → N_{Y} X such that these diagrams commute:

\xymatrix{N_{W} V \ar[r]^{q′} \ar[d]_{N(g)} & W \ar[d]^{g′} & N_{W} V \ar[r]^{κ′} \ar[d]_{N(g)} & D(V, W) \ar[d]^{D(g)} \\ N_{Y} X \ar[r]_{q} & Y & N_{Y} X \ar[r]_{κ} & D(X, Y).}

Now use Propositions 3.28 and 3.34, Lemma 5.4, the naturality of the connecting homomorphism ∂ and the diagram

\xymatrix{W \ar[d]_{g′} & N_{W} V \ar[r]^{κ′} \ar[d]^{N(g)} \ar[l]_{q′} & D(V, W) \ar[d]^{D(g)} & V × U \ar[d]^{g×1} \ar[l]_{ι′} \ar[r]^{π′} & V \ar[d]^{g} \\ Y & N_{Y} X \ar[r]_{κ} \ar[l]^{q} & D(X, Y) & X × U \ar[l]^{ι} \ar[r]_{π} & X}

to conclude (observe that D(g) and N(g) are flat because of [Ro], Remark 10.1).

Lemma 5.8. Let \xymatrix{Z \ar[r]^{i} & Y \ar[r]^{j} & X} be inclusions of smooth schemes. Then we have inclusions a : N_{Z}Y → N_{Z}X, c : i^{∗}N_{Y} X → N_{Y} X and isomorphisms N_{(i^{∗}N_{Y} X)}(N_{Y} X) ≃ N_{Z}Y ⊕ i^{∗}N_{Y} X ≃ N_{(N_{Z}Y)}(N_{Z}X).

Proof. The first two assertions are straight computations (see also [Ne]). The relation (2.1) in [Ne] shows that we have canonical isomorphisms

N_{(i^{∗}N_{Y} X)}(N_{Y} X) ≃ N_{Z}Y ⊕ i^{∗}N_{Y} X ≃ N_{(N_{Z}Y)}(N_{Z}X).

Lemma 5.9. Let \xymatrix{Z \ar[r]^{i} & Y \ar[r]^{j} & X} be inclusions of smooth schemes. Let a : N_{Z}Y → N_{Z}X, c : i^{∗}N_{Y} X → N_{Y} X be the inclusions and q : N_{Y} X → Y, r : N_{Z}X → Z, s_{1} : N_{(i^{∗}N_{Y} X)}(N_{Y} X) → i^{∗}N_{Y} X, s_{2} : N_{(N_{Z}Y)}(N_{Z}X) → N_{Z}Y the projections. Then we have (s_{1})^{∗}c^{!}q^{∗}j^{!} = (s_{2})^{∗}a^{!}r^{∗}(ji)^{!}

Proof. Consider the deformation to the normal cone spaces D(Y, Z) and D(X, Z). Using the universal property of blow-ups, we get a map D(Y, Z) → D(X, Z) such that the following diagram commutes

\xymatrix{N_{Z}Y \ar[r]^{a} \ar[d] & N_{Z}X \ar[d] \\ D(Y, Z) \ar[r] & D(X, Z) \\ Y × U \ar[r]_{j×1} \ar[u] & X × U \ar[u]}

where the top vertical maps are inclusions of the exceptional fiber in the deformation to the normal space and the bottom vertical maps are inclusions of open subsets. It is easy to check that the map D(Y, Z) → D(X, Z) is a closed immersion. Let D(X, Y, Z) be the deformation to the normal cone space associated to this closed immersion. Using again the universal property of blow-ups, we see that the above diagram gives a sequence

\xymatrix{D(N_{Z}X, N_{Z}Y) \ar[r] & D(X, Y, Z) & D(X, Y) × U \ar[l]}

where the first map is a closed immersion and the second one is an open immersion. Consider now the space D(X, Y, Z). We have an open immersion D(X, Z) × U → D(X, Y, Z) and a closed immersion (as the special fiber) N_{D(Y,Z)}D(X, Z) → D(X, Y, Z). In fact, this exceptional fiber is isomorphic to D(N_{Y} X, i^{∗}N_{Y} X) (see [Ne], paragraph 3.2). So we get a diagram

\xymatrix{N_{Y} X \ar[r]^{κ} & D(X, Y) & X × U \ar[l]_{ι} \ar[r]^{π} & X \\ N_{Y} X × U \ar[r]^{κ} \ar[u]^{π} \ar[d]_{ι} & D(X, Y) × U \ar[u]_{π} \ar[d]^{ι} & X × U × U \ar[r]^{π} \ar[l]_{ι} \ar[u]_{π} \ar[d]^{ι} & X × U \ar[u]_{π} \ar[d]^{ι} \\ D(N_{Y} X, i^{∗}N_{Y} X) \ar[r]^{κ} & D(X, Y, Z) & D(X, Z) × U \ar[l]_{ι} \ar[r]^{π} & D(X, Z) \\ N_{(i^{∗}N_{Y} X)}N_{Y} X \ar[u]^{κ} \ar[r]_{κ} & D(N_{Z}X, N_{Z}Y) \ar[u]_{κ} & N_{Z}X × U \ar[u]_{κ} \ar[r]_{π} \ar[l]^{ι} & N_{Z}X \ar[u]_{κ}}

where all the lines are deformations to the normal cone, the first and fourth columns are also deformations to the normal cone. This diagram is commutative (see [Ne], paragraph 3.2). The maps κ denote inclusions of special fibers, ι denote the inclusions of the complement of these special fibers and π denote the relevant projections. The map q^{∗}j^{!} is obtained by composing the operations (in cohomology) of the top row and s_{1}^{∗}b^{!} is obtained by working with the left column. Similarly, r^{∗}(ji)^{!} and s_{2}^{∗}a^{!} are deduced from the right column and the bottom row. Now all the squares appearing in this diagram are commutative and give commutative diagrams in cohomology (Proposition 3.28, Proposition 3.32, Lemma 3.34 and the naturality of the residual homomorphism ∂). Using this and Lemma 5.4, we get the result.

Lemma 5.10. Let V, X and W be smooth schemes. Consider the following commutative diagram

\xymatrix{W \ar[r]^{i} \ar[dr]_{p′} & V \ar[d]^{p} \\  & X}

where p, p′ are flat and i is a closed immersion. Suppose that the composition N_{W} V → W → X is of the same relative dimension as p. Then i^{!}p^{∗} = (p′)^{∗}.

Proof. Let D(V, W) be the deformation to the normal cone associated to i and b : D(V, W) → V ×𝔸^{1} be the blow-down map. We have a commutative diagram

[figure: W; N_{V} W; κ; q; D(V, W); b; V × 𝔸^{1}; p×Id; X × 𝔸^{1}; V × U; ι; V × U; p×Id; ι; π; X × U; π′; ι; V; p; X.]

By definition, i^{!}p^{∗} = (q^{∗})^{−1}(κ_{∗})^{−1}∂{t}π^{∗}p^{∗}. Using Proposition 3.28, we get i^{!}p^{∗} = (q^{∗})^{−1}(κ_{∗})^{−1}∂{t}(p × Id)^{∗}(π′)^{∗}. By Lemma 5.4, this gives

(q^{∗})^{−1}(κ_{∗})^{−1}∂{t}(p × Id)^{∗}(π′)^{∗} = (q^{∗})^{−1}(κ_{∗})^{−1}∂(p × Id)^{∗}{t}(π′)^{∗}.

Using Remark 10.1 in [Ro], we see that f := (p × Id)b is flat because the composition N_{W} V → W → X is of the same relative dimension as p. We have a commutative diagram

[figure: H^{i}(C(X × 𝔸^{1}, G^{j})); f^{∗}; H^{i}(C(X × U, G^{j})); ∂′; (p×Id)^{∗}; H_{X}^{i+1}(C(X × 𝔸^{1}, G^{j})); f^{∗}; H^{i}(C(D(V, W), G^{j})); H^{i}(C(V × U, G^{j})); ∂; H_{N_{V} W}^{i+1} (C(D(V, W), G^{j}))]

where the first line is the localization long exact sequence associated to the triple (X × U, X × 𝔸^{1}, X × 0) and the second line is the one associated to the triple (V × U, D(V, W), N_{V} W). Then

(q^{∗})^{−1}(κ_{∗})^{−1}∂(p × Id)^{∗}{t}(π′)^{∗} = (q^{∗})^{−1}(κ_{∗})^{−1}f^{∗}∂′{t}(π′)^{∗}.

Consider next the fibre product

\xymatrix{N_{V} W \ar[r]^{κ} \ar[d]_{p′q} & D(V, W) \ar[d]^{f} \\ X \ar[r]_{i_{0}} & X × 𝔸^{1}}

where i_{0} : X → X × 𝔸^{1} is the inclusion in 0. Using Lemma 3.34, we finally find i^{!}p^{∗} = (p′)^{∗}(i_{0})_{∗}^{−1}∂′{t}(π′)^{∗}. It remains to show that (i_{0})_{∗}^{−1}∂′{t}(π′)^{∗} = Id to finish the proof. At the level of Milnor K-theory, this is Lemma 4.5 in [Ro]. Thus we only have to prove this result at the level of Witt groups. Let α ∈ W^{i}(D_{i}^{b}(X)) be such that dα = 0 ∈ W^{i+1}(D_{i+1}^{b}(X)). Now DegLoc((π′)^{∗}α) ∩ DegLoc(< 1, −t >) is a closed subset of X × 𝔸^{1} of codimension ≥ i + 2. Therefore we can use 2.25 to compute

(−1)^{i}d(< 1, −t > ⊗α) = d(< 1, −t >) ⊗ α + (−1)^{i} < 1, −t > ⊗dα.

By assumption we have dα = 0 in W^{i+1}(D_{i+1}^{b}(X)) and then

(−1)^{i}d(< 1, −t > ⊗α) = d(< 1, −t >) ⊗ α = −dt ⊗ α

in W^{i+1}(D_{i+1}^{b}(X)). By definition of m_{t}, we find d(m_{t}(α)) = dt ⊗ α. The latter is precisely (i_{0})_{∗}α (see [GH], Lemma 2.8).

Now we have all the tools to prove the following theorem:

Theorem 5.11. Let \xymatrix{Z \ar[r]^{i} & Y \ar[r]^{j} & X} be inclusions of smooth schemes. Then (ji)^{!} = i^{!}j^{!}.

Proof. Let q : N_{Y} X → Y, p : N_{Z}Y → Z and r : N_{Z}X → Z be the projections. Consider also the projections s_{1} : N_{(i^{∗}N_{Y} X)}(N_{Y} X) → i^{∗}N_{Y} X and s_{2} : N_{(N_{Z}Y)}(N_{Z}X) → N_{Z}Y. Denote by a : N_{Z}Y → N_{Z}X and c : i^{∗}N_{Y} X → N_{Y} X the inclusions. We also have a fibre product

\xymatrix{i^{∗}N_{Y} X \ar[r]^{c} \ar[d]_{q′} & N_{Y} X \ar[d]^{q} \\ Z \ar[r]_{i} & Y.}

Then

(s_{1})^{∗}(q′)^{∗}i^{!}j^{!} = (s_{1})^{∗}c^{!}q^{∗}j^{!} = (s_{2})^{∗}a^{!}r^{∗}(ji)^{!} = (s_{2})^{∗}p^{∗}(ji)^{!}

where the first equality is due to Lemma 5.7, the second is due to Lemma 5.9 and the third to Lemma 5.10. As (s_{2})^{∗}p^{∗} induces an isomorphism in cohomology and q′s_{1} = ps_{2}, we get the result.

6 The ring structure

Let X be a smooth scheme and let △ : X → X × X be the diagonal inclusion. For any i, j, r, s we have an exterior product (Theorem 4.15)

× : H^{i}(C(X, G^{r})) × H^{j}(C(X, G^{s})) → H^{i+j}(C(X × X, G^{r+s}))

and a Gysin-Witt map (Definition 5.5)

△^{!} : H^{i+j}(C(X × X, G^{r+s})) → H^{i+j}(C(X, G^{r+s})).

Definition 6.1. We denote by · the composition △^{!} ∘ ×.

Remark 6.2. If X is a smooth scheme and L, N are invertible O_{X}-modules, then using Theorem 4.16 and Remark 5.6 we see that there is a product

· : H^{i}(C(X, G^{i}, L)) × H^{j}(C(X, G^{j}, N)) → H^{i+j}(C(X, G^{i+j}, L ⊗_{O_{X}} N)).

Remark 6.3. In particular, we have for any i, j ∈ ℕ a product

· : H^{i}(C(X, G^{i})) × H^{j}(C(X, G^{j})) → H^{i+j}(C(X, G^{i+j}))

which by definition is a product C̃H^{i}(X) × C̃H^{j}(X) → C̃H^{i+j}(X).

Remark 6.4. It is clear from our construction that we also can define a product

· : H^{i}(C(X, K_{r}^{M})) × H^{j}(C(X, K_{s}^{M})) → H^{i+j}(C(X, K_{r+s}^{M})).

This product coincide with the one defined by Rost ([Ro], Chapter 14) and the natural projections π : C(X, G^{p}) → C(X, K_{p}^{M}) give a commutative diagram

\xymatrix{H^{i}(C(X, G^{r})) × H^{j}(C(X, G^{s})) \ar[r]^{·} \ar[d]_{π×π} & H^{i+j}(C(X, G^{r+s})) \ar[d]^{π} \\ H^{i}(C(X, K_{r}^{M})) × H^{j}(C(X, K_{s}^{M})) \ar[r]_{·} & H^{i+j}(C(X, K_{r+s}^{M})).}

Remark 6.5. Our technique provides also a product on the cohomology of the Gersten-Witt complex of a scheme. That is, we have a product

· : H^{i}(C(X, W)) × H^{j}(C(X, W)) → H^{i+j}(C(X, W)).

Now we prove the associativity of the product we have defined.

Proposition 6.6. The product · is associative.

Proof. First note that the exterior product is associative (Proposition 4.18). We consider the following fibre product diagram

\xymatrix{X \ar[r]^{△} \ar[d]_{△} & X × X \ar[d]^{Id×△} \\ X × X \ar[r]_{△×Id} & X × X × X.}

We see that ((Id × △)△)^{!} = ((△ × Id)△)^{!}. Theorem 5.11 shows that we have in fact △^{!}(Id × △)^{!} = △^{!}(△ × Id)^{!}. Since (Id × △)^{!} is clearly Id × △^{!} and (△ × Id)^{!} = △^{!} × Id, the associativity is proved.

Remark 6.7. In general, the product does not satisfy any commutativity property. This is due to the fact that × and ⋆ do not commute with the flip τ : X × X → X × X (see 4.19 and 4.20). Moreover, the product is not anticommutative because the signs in 4.19 and 4.20 are not compatible. However, let α ∈ C̃H^{i}(X) and β ∈ C̃H^{j}(X). Then α · β is an element of C̃H^{i+j}(X) and is therefore represented by a sum ∑(P_{s}, ψ_{s}) ∈ Ker(d^{i+j}) where

d^{i+j} : GW^{i+j}(D_{i+j}^{b}(X)) → W^{i+j+1}(D_{i+j+1}^{b}(X))

(see Remark 3.23). Using 4.19 and 4.20, we see that β · α = ∑(P_{s}, (−1)^{ij}ψ_{s}). For a more precise statement, the reader is referred to Theorem 7.6.

Now remark that there is a canonical class 1_{X} in C̃H^{0}(X) given by the symmetric form < 1 > in GW (k(X)).

Proposition 6.8. The class 1_{X} is a left and right unit for the product ·.

Proof. Let p_{2} : X ×X → X be the second projection and consider the following commutative diagram

\xymatrix{X \ar[r]^{△} \ar[dr]_{Id} & X × X \ar[d]^{p_{2}} \\  & X.}

By Lemma 5.10, we see that △^{!}(p_{2})^{∗} = (Id)^{∗} = Id. Consider now μ ∈ H^{i}(C(X, G^{j})). It is clear that 1_{X} × μ = (p_{2})^{∗}(μ) and then 1_{X} · μ = μ. Replacing p_{2} by p_{1} shows that 1_{X} is also a right unit.

Hence we have:

Theorem 6.9. Let X be a smooth scheme and let C̃H^{∗}(X) be the total Chow-Witt group of X. Then the product · turns C̃H^{∗}(X) into a graded associative ring with unit.

Taking the twists into account, we get the following theorem:

Theorem 6.10. Let X be a smooth scheme and let ⨁_{L∈Pic(X)/2} C̃H^{∗}(X, L) be

the total twisted Chow-Witt group of X. Then the product · turns this group into a graded associative ring with unit.

Definition 6.11. Let X be a smooth scheme. We call Chow-Witt ring the ring C̃H^{∗}(X) and twisted Chow-Witt ring the ring ⨁_{L∈Pic(X)/2} C̃H^{∗}(X, L).

The following proposition is obvious:

Proposition 6.12. Let X be a smooth scheme. Then the natural homomorphism C̃H^{∗}(X) → CH^{∗}(X) is a ring homomorphism.

Remark 6.13. The same methods show that the product of Remark 6.5 gives a graded associative anticommutative ring structure on the total cohomology group H^{∗}(C(X, W)) of the Gersten-Witt complex associated to X.

7 Basic properties

We first show that the Chow-Witt ring is a functorial construction.

Definition 7.1. Let X and Y be smooth schemes and f : X → Y a morphism. Consider the graph morphism γ_{f} : X → X × Y. We define

f^{!} : C̃H^{∗}(Y) → C̃H^{∗}(X)

by f^{!}(y) = γ_{f}^{!}(1_{X} × y) for any y ∈ C̃H^{∗}(Y).

Proposition 7.2. The map f^{!} : C̃H^{∗}(Y) → C̃H^{∗}(X) is a ring homomorphism.

Proof. We only have to check that f^{!}(y·z) = f^{!}(y)·f^{!}(z) for any y, z ∈ C̃H^{∗}(Y). Consider the following commutative diagram:

\xymatrix{X \ar[r]^{γ_{f}} \ar[d]_{△_{X}} & X × Y \ar[d]^{△_{X×Y}} \\ X × X \ar[r]_{γ_{f} ×γ_{f}} & (X × Y) × (X × Y).}

Theorem 5.11 shows that γ_{f}^{!}△_{X×Y}^{!} = △_{X}^{!}(γ_{f} ×γ_{f})^{!}. Applying this to the cycle 1_{X} × y × 1_{X} × z, we obtain the result.

Remark 7.3. The proposition shows that C̃H^{∗}(_) is a functor from the category of smooth schemes to the category of rings. It is clear that the homomorphisms C̃H^{∗}(X) → CH^{∗}(X) give a natural transformation C̃H^{∗}(_) → CH^{∗}(_).

In the case where f : X → Y is a flat morphism, we can identify f^{!} more precisely.

Proposition 7.4. Let f : X → Y be a flat morphism. Then f^{!} = f^{∗}.

Proof. Consider the following commutative diagram:

\xymatrix{X \ar[r]^{γ_{f}} \ar[dr]_{f} & X × Y \ar[d]^{p} \\  & Y}

where p : X × Y → Y is the projection. Since N_{X}(X × Y) is of rank equal to the dimension of Y, we see that the relative dimension of the composition N_{X}(X ×Y) → X → Y is the same as the relative dimension of p : X ×Y → Y. Therefore we can use Lemma 5.10 to get γ_{f}^{!}p^{∗} = f^{∗}. Since p^{∗}β = 1_{X} × β for any cycle on Y, the result is proved.

Let Z ⊂ X be a closed subset of pure codimension i. As D_{Z}^{b}(X) ⊂ D^{b}(X)^{(i)}, we have a homomorphism GW_{Z}^{i}(X) → GW^{i}(D^{b}(X)^{(i)}). Composing with the localization, we obtain a homomorphism GW_{Z}^{i}(X) → GW^{i}(D_{i}^{b}(X)). As the composition GW^{i}(D^{b}(X)^{(i)}) → GW^{i}(D_{i}^{b}(X)) → W^{i+1}(D^{b}(X)^{(i+1)}) is zero (see [Ba1]), we finally obtain a homomorphism (Remark 3.23):

α_{Z} : GW_{Z}^{i}(X) → C̃H^{i}(X).

Remark 7.5. Let f : X → Y be a flat morphism and Z ⊂ Y be a closed subset of pure codimension i. The definitions of f^{∗} for the Grothendieck-Witt groups and the definition of f^{∗} for the Chow-Witt groups show that the following diagram commutes ([Fa], Theorem 3.2.2 and Corollary 10.4.2):

\xymatrix{GW_{Z}^{i}(Y) \ar[r]^{α_{Z}} \ar[d]_{f^{∗}} & C̃H^{i}(Y) \ar[d]^{f^{∗}} \\ GW_{f^{−1}Z}^{i}(X) \ar[r]_{α_{(f^{−1}Z)}} & C̃H^{i}(X).}

The next theorem shows that our intersection product is the expected one:

Theorem 7.6. Let Z, T ⊂ X be closed subschemes of respective pure codimension i and j. Suppose that Z ∩ T is of pure codimension i + j. Then the following diagram commutes

\xymatrix{GW_{Z}^{i}(X) × GW_{T}^{j} (X) \ar[d]_{α_{Z}×α_{T}} \ar[r]^{⋆} & GW_{Z∩T}^{i+j} (X) \ar[d]^{α_{Z∩T}} \\ C̃H^{i}(X) × C̃H^{j}(X) \ar[r]_{·} & C̃H^{i+j}(X).}

Proof. Let γ ∈ GW_{Z}^{i}(X) and δ ∈ GW_{T}^{j} (X). Consider the deformation to the normal cone space D(X × X, X) and the blow down map b : D(X × X, X) → X × X × 𝔸^{1}. We have the following commutative diagram

[figure: X; X; △; N_{X}(X × X); b′; q; κ; X × X; i_{0}; D(X × X, X); b; X × X × 𝔸^{1}; π′; X × X; (1); X × X × U; ι; X × X × U; ι; π]

where i_{0} is the inclusion in 0, q is the projection and the two bottom squares are fibre products. By definition, we have

α_{Z}(γ) · α_{T} (δ) = (q^{∗})^{−1}(κ_{∗})^{−1}∂{t}π^{∗}(α_{Z}(γ) × α_{T} (δ)).

Let F = b^{−1}(π′)^{−1}(p_{1}^{−1}Z ∩ p_{2}^{−1}T) in D(X × X, X) (where p_{1} and p_{2} are the projections of X × X onto X). Observe that ι^{−1}F = F ∩ (X × X × U) is non empty and of pure codimension i + j in X × X × U. Diagram (1) gives

F ∩ N_{X}(X × X) = κ^{−1}F = κ^{−1}b^{−1}(π′)^{−1}(p_{1}^{−1}Z ∩ p_{2}^{−1}T) = q^{−1}(Z ∩ T).

As Z ∩ T is of codimension i + j in X and q is flat, q^{−1}(Z ∩ T) is also of codimension i + j in N_{X}(X × X) and hence is of codimension i + j + 1 in D(X × X, X). Therefore F itself is of pure codimension i + j in D(X × X, X). By commutativity of the above diagram and Remark 7.5, we have (note that b^{∗} is defined at the level of the Grothendieck-Witt groups, but not at the level of the Chow-Witt groups):

π^{∗}(α_{Z}(γ) × α_{T} (δ)) = α_{ι^{−1}F} (ι^{∗}b^{∗}(π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ)) = ι^{∗}α_{F} (b^{∗}(π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ)).

We have to compute (κ_{∗})^{−1}∂{t}π^{∗}(α_{Z}(γ)×α_{T} (δ)). By definition of ∂, we have to consider any element ν ∈ C(D(X × X, X), G^{i+j+1})_{i+j} having the property that ι^{∗}ν = {t}π^{∗}(α_{Z}(γ) × α_{T} (δ)) and then compute d_{G}(ν) where

d_{G} : C(D(X × X, X), G^{i+j+1})_{i+j} → C(D(X × X, X), G^{i+j+1})_{i+j+1}

is the differential of the complex C(D(X × X, X), G^{i+j+1}). Consider the commutative diagram

[figure: D(X × X, X); b; X × X × 𝔸^{1}; pr; 𝔸^{1}; X × X × U; ι; X × X × U; ι; η]

and recall that N_{X}(X × X) is the principal Cartier divisor in D(X × X, X) defined by f := b^{∗}pr^{∗}(t). Consider the form b^{∗}(π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ). Its support is F. Localizing at the generic points of F (which are on X × X × U), we obtain a form ν_{0} in W^{i+j}(D_{i+j}^{b}(D(X × X, X))). We also obtain an element ν_{1} in ⨁_{x∈F^{(0)}} K_{0}(k(x)).

The above computation shows that f is a unit in k(x) for any generic point x of F. We get an element

ν := ((−1)^{i+j+1} < 1, −f > ⊗ν_{0}, {f} · ν_{1}) ∈ C(D(X × X, X), G^{i+j+1})_{i+j}

which satisfy ι^{∗}ν = {t}π^{∗}(α_{Z}(γ)×α_{T} (δ)). A straightforward computation (use Theorem 2.25 again) shows that d_{G}(ν) = df ⊗ b^{∗}(π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ) in the group GW^{i+j+1}(D_{i+j+1}^{b}(D(X × X, X))). But df = b^{∗}dt and

b^{∗}dt ⊗ b^{∗}(π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ) = b^{∗}(dt ⊗ (π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ))

([GN], Theorem 3.2). Since dt ⊗ (π′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ) = (i_{0})_{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ) ([GH], Lemma 2.8), we finally obtain

(κ_{∗})^{−1}∂{t}π^{∗}(α_{Z}(γ) × α_{T} (δ)) = α_{F ∩N_{X}(X×X)}((b′)^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ)).

We have a commutative diagram

\xymatrix{N_{X}(X × X) \ar[r]^{b′} \ar[d]_{q} & X × X \\ X. \ar[ur]_{△}}

Now △^{−1}(p_{1}^{−1}Z ∩ p_{2}^{−1}T) = Z ∩ T and using the diagram, we see that

α_{Z}(γ) · α_{T} (β) = α_{Z∩T} (△^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ)).

Hence it only remains to show that △^{∗}(p_{1}^{∗}γ ⊗ p_{2}^{∗}δ) = γ ⋆ δ to finish the proof. This is clear by [GN], Theorem 3.2.

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Jean Fasel D-Math ETH Zentrum 8092 Zürich Switzerland jean.fasel@math.ethz.ch