A product in triangular Witt theory

S. Gille^{1} and A. Nenashev^{2}

May 2001

0 Introduction

Witt theory for triangulated categories with duality was introduced recently by Balmer [1]. In this theory, we can shift the duality structure on a triangulated category K and obtain an infinite series of Witt groups W^{i}(K) (i ∈ ℤ), which proves to be 4-periodic: W^{i}(K) ≃ W^{i+4}(K) naturally. The triangulated categories environment – as opposed to the Witt theory of exact categories with duality introduced earlier – enables one to prove some fundamental properties of the Witt groups of schemes as e. g. localization [1]. Further study of the triangular Witt groups requires more natural structures on them; the objective of the present paper is to introduce products

W^{i}(K) × W^{j}(K) ⟶ W^{i+j}(K)

provided K has a product ⊠ compatible with the other structures.

In fact, we introduce two products, left and right, related by the formula

u ⋆_{l} v = (−1)^{ij}u ⋆_{r} v

for u ∈ W^{i}(K) and v ∈ W^{j}(K). Let T and ♯ denote the duality and translation functors in K respectively. Both products are equally natural for the following reason. Given two forms ϕ : A ⟶^{≃} T^{i}A^{♯} and ψ : B ⟶^{≃} T^{j}B^{♯} representing elements of W^{i}(K) and W^{j}(K) respectively, we want their product to be an isomorphism

A ⊠ B ⟶^{≃} T^{i+j}(A ⊠ B)^{♯} ,

and can (attempt to) define it as the composite map

A ⊠ B ⟶^{ϕ⊠ψ} T^{i}(A^{♯}) ⊠ T^{j}(B^{♯}) ⟶ T^{i+j}(A^{♯} ⊠ B^{♯}) ⟶ T^{i+j}(A ⊠ B)^{♯} .  (0.1)

Examining how to define an isomorphism

T^{i}(A^{♯}) ⊠ T^{j}(B^{♯}) ⟶ T^{i+j}(A^{♯} ⊠ B^{♯}) ,

one can see that this map should be composed of “elementary” maps of two kinds:

𝔩_{A,B} : T (A) ⊠ B ⟶^{≃} T (A ⊠ B)  and  𝔯_{A,B} : A ⊠ T (B) ⟶^{≃} T (A ⊠ B) .

These two maps commute by −1 if we consider them for chain complexes of locally free modules over a scheme, where the product is given by the tensor product of complexes, i. e. the following diagram is skew-commutative:

(T P) ⊗ (T Q) \overline{}T (P ⊗ (T Q))

−1

[figure: T ((T P) ⊗ Q); T^{2}(P ⊗ Q) ,]

where P and Q are complexes of vector bundles over a scheme X.

1Mathematisches Institut, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany e-mail: gilles@math.uni-muenster.de

2e-mail: nenashev@mathematik.uni-bielefeld.de

As the latter provides our main example, we have to require skew-commutativity of the similar diagram as an axiom in our definition of a triangulated category with product. This skewcommutativity is also essential for our proof of the existence of a product on the triangular Witt groups (Section 3.1).

Balmer has shown that the group W^{0}(A) = W^{0}(D^{b}(VB_{X})) coincides with the classical Witt group of the scheme X as defined by Knebusch in [5]. The left and right product are the same on W^{0} and it turns out that the product on the usual Witt group of a scheme coincides with this one.

In Section 1 we supply axioms relating duality, product, and triangulated category structures, without claiming for any originality. In Section 2 we remind the definition of triangular Witt groups and duality preserving functors; the latter are just functors which induce maps of the Witt groups. In Section 3 we define the products

⋆_{l} , ⋆_{r} : W^{i}(K) × W^{j}(K) ⟶ W^{i+j}(K) .

We avoid the straightforward definition by (0.1) and show instead that taking the product with an object A endowed with a form ϕ : A ⟶^{≃} T^{i}(A^{♯}) yields duality preserving functors, which provide maps W^{j}(K) → W^{i+j}(K). Assembling such maps for all spaces [A, ϕ], we get the desired pairings (A⊠− corresponds to the left product and −⊠A to the right product); the use of duality preserving functors allows us to avoid too long routine verifications. In 3.2 we show that these pairings are compatible with localization. In 3.3 we supply conditions under which the products ⋆_{l} and ⋆_{r} become associative and skew commutative.

In section 4 we consider the bounded derived category of finite rank vector bundles over a scheme X. It turns out that the (derived) tensor product is compatible with the canonical duality on this category. Hence one can apply the results of the preceding sections. As an application of this fact we show in the last section that a Springer type theorem for the derived Witt groups is true, too (Theorem 5.1). Acknowledgments: We thank Jens Hornbostel for some useful comments on an earlier draft of this work. The second author thanks the Universities of Bielefeld and Münster for their hospitality and the Alexander von Humboldt Foundation and SFB-478 “Geometrische Strukturen in der Mathematik” for financial support.

Notation and conventions: We denote the translation functor in triangulated categories by T if not specified otherwise. When we say that a diagram in an additive category

[figure: A; α; B; C; γ; δ; D; β]

is (−1)^{m}-commutative, we mean that β · α = (−1)^{m}δ · γ.

We assume throughout that the morphism groups in all the categories under consideration are (unique) 2-divisible. In particular we assume that \frac{1}{2} is in the global section of all the schemes we consider.

1 TPD-categories

We first recall the definition of a triangulated category with product respectively with duality. After that we introduce TPD-categories which are triangulated categories endowed with both structures. We assume some familiarity of the reader with triangulated categories, say at the level of [4], chapter IV.

1.1 Triangulated categories with duality respectively with product

The following definition is due to Verdier.

Definition 1.1 Let K and K̃ be triangulated categories with translation functors T respectively T̃. A covariant functor F : K ⟶ K̃ is called δ-exact (δ = ±1), if it is an additive functor and there exists an isomorphism of functors

θ : F T ⟶ T̃F ,

such that for every exact triangle

A ⟶^{α} B ⟶^{β} C ⟶^{γ} T A

in K, the triangle

F A ⟶^{F (α)} F B ⟶^{F (β)} F C ⟶^{δ·θ_{A}·F (γ)} T̃F A

is exact in K̃.

A contravariant δ-exact functor F : K ⟶ K̃ is defined similarly, i. e. there is an isomorphism

θ : T̃^{−1}F ⟶ F T

such that for every exact triangle as above the triangle

F C ⟶^{F (β)} F B ⟶^{F (α)} F A ⟶^{δ·T̃F (γ)·T̃(θ_{A})} T̃F C

is exact in K̃.

Remark 1.2 Using θ we can define isomorphisms of functors

θ^{(i)} : F T^{i} ⟶ T̃^{i}F

for all i ∈ ℤ in the covariant case. We set θ^{(0)} := id, θ^{(1)} := θ and

θ^{(−1)} := (T̃^{−1}(θ_{T^{−1}}))^{−1} .

Obviously θ^{(−1)} is an isomorphism F T^{−1} ⟶^{≃} T̃^{−1}F.

For |i| ≥ 1 these isomorphisms of functors are defined inductively as follows:

θ^{(i)} := T̃^{sgn(i)}(θ^{(i−sgn(i))}) · θ_{T^{i−sgn(i)}}^{(sgn(i))} ,

where sgn(i) = 1 if i > 0 and = −1 if i < 0. By induction, we see that for |i| ≥ 1

θ^{(i)} = T̃^{i−sgn(i)}(θ^{(sgn(i))}) · θ_{T^{sgn(i)}}^{(i−sgn(i))} .

Clearly we could also take this equation for an inductive definition. Note that

θ^{(j+k)} = T̃^{k}(θ^{(j)}) · θ_{T^{j}}^{(k)} ,

for all j, k ∈ ℤ.

We define now triangulated categories with product.

Definition 1.3 A triangulated category with product is a pair (K, ⊠), where K is a triangulated category with translation functor T and ⊠ is a bi-covariant functor

⊠ : K × K ⟶ K

which is 1-exact in both variables and satisfies 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)

associated with the 1-exact functors A ⊠ − and − ⊠ B as in Definition 1.1 make the diagram

[figure: (T A) ⊠ (T B); 𝔩_{A,T B}; T (A ⊠ (T B)); T ((T A) ⊠ B); 𝔯_{T A,B}; T (𝔩_{A,B}); T^{2}(A ⊠ B); T (𝔯_{A,B})]

skew-commutative for any A and B in K, i. e.

T (𝔯_{A,B}) · 𝔩_{A,T B} = −T (𝔩_{A,T B}) · 𝔯_{T A,B} .  (1.1)

Remark 1.4 The skew-commutativity of the diagram above is true in the main example we have in mind, namely the derived category of locally free modules over a scheme, where the product is given by the derived tensor product (cf. section 4).

Surprisingly, this skew-commutativity becomes important when we want to get a product on the derived Witt groups of a scheme which coincides with the product on the usual Witt group.

The functorial isomorphisms 𝔯 and 𝔩 involved in the definition of a triangulated category with product have also multiple or shifted versions 𝔯^{(i)} and 𝔩^{(i)}, which are defined as in Remark 1.2. They satisfy the following relation.

Lemma 1.5 For all i, j ∈ ℤ and A, B ∈ K, the following diagram is (−1)^{ij}-commutative:

[figure: (T^{i}A) ⊠ (T^{j}B); 𝔩_{A,T^{j}B}^{(i)}; T^{i}(A ⊠ (T^{j}B)); T^{j}((T^{i}A) ⊠ B); 𝔯_{T^{i}A,B}^{(j)}; T^{j}(𝔩_{A,B}^{(i)}); T^{i+j}(A ⊠ B) .; T^{i}(𝔯_{A,B}^{(j)})]

Proof: It follows from equation (1.1) and the definition of 𝔩^{(−1)} and 𝔯^{(−1)} that

T^{i}(𝔯_{A,B}^{(j)}) · 𝔩_{A,T^{j}B}^{(i)} = −T^{j}(𝔩_{A,B}^{(i)}) · 𝔯_{T^{i}A,B}^{(j)}

for i = ±1 and j = ±1. The lemma follows by induction. □

We end this subsection with the definition of a triangulated category with duality.

Definition 1.6 (Balmer [1]) Let K be a triangulated category with translation functor T, δ = ±1, ♯ : K ⟶ K a δ-exact contravariant functor satisfying the strict condition ♯T = T^{−1}♯ and ϖ : id ⟶^{≃} ♯♯ an isomorphism of functors.

The triple (K, ♯, ϖ) is called a triangulated category with δ-exact duality ♯, if T ϖ_{M} = ϖ_{T M} and

(ϖ_{M})^{♯} · ϖ_{M^{♯}} = id_{M^{♯}} .

Notation 1.7 1. To specify that the duality is δ-exact we will often use the notation (K, ♯, δ, ϖ) instead of (K, ♯, ϖ).

2. At some places in this work we write D(A) instead of A^{♯} to denote the dual of the object A ∈ K.

Remark 1.8 It is an easy exercise to show that T^{i}♯ is then a (−1)^{i}δ-exact duality on K making

K^{(i)} := (K, T^{i}♯, (−1)^{\frac{i(i+1)}{2}} δ^{i}ϖ)

a triangulated category with duality. Clearly K^{(0)} = (K, ♯, ϖ).

1.2 TPD-categories

Definition 1.9 Let (K, ♯, ϖ, ⊠) be a triangulated category with δ-exact duality ♯, product ⊠ and translation functor T. Let further 𝔯_{A,B} and 𝔩_{A,B} be as in Definition 1.3 above.

K is called a TPD-category, if there are functorial (in A and B) isomorphisms

η_{A,B} : A^{♯} ⊠ B^{♯} ⟶ (A ⊠ B)^{♯}

which make the following two diagrams commutative:

1.

[figure: A ⊠ B; ϖ_{A} ⊠ ϖ_{B}; A^{♯♯} ⊠ B^{♯♯}; (A ⊠ B)^{♯♯}; ϖ_{A⊠B}; (η_{A,B})^{♯}; (A^{♯} ⊠ B^{♯})^{♯}; η_{A^{♯},B^{♯}}]

2.

[figure: T ((T A)^{♯} ⊠ B^{♯}); 𝔩_{(T A)^{♯},B^{♯}}; A^{♯} ⊠ B^{♯}; 𝔯_{A^{♯},(T B)^{♯}}; T (A^{♯} ⊠ (T B)^{♯}); T (((T A) ⊠ B)^{♯}); T η_{T A,B}; T (𝔩_{A,B}^{♯}); (A ⊠ B)^{♯}; η_{A,B}; T (𝔯_{A,B}^{♯}); T ((A ⊠ T B)^{♯}) .; T η_{A,T B}]

We will need the following iterated version of diagram 2 above.

Lemma 1.10 For all A, B in K and all i ∈ ℤ the following diagram commutes:

[figure: T^{i}((T^{i}A)^{♯} ⊠ B^{♯}); 𝔩_{(T^{i}A)^{♯},B^{♯}}^{(i)}; A^{♯} ⊠ B^{♯}; 𝔯_{A^{♯},(T^{i}B)^{♯}}^{(i)}; T^{i}(A^{♯} ⊠ (T^{i}B)^{♯}); T^{i}(((T^{i}A) ⊠ B)^{♯}); T^{i}η_{T^{i}A,B}; T^{i}(𝔩_{A,B}^{(i)})^{♯}; (A ⊠ B)^{♯}; η_{A,B}; T^{i}(𝔯_{A,B}^{(i)})^{♯}; T^{i}((A ⊠ T^{i}B)^{♯}) .; T^{i}η_{A,T^{i}B}]

Proof: By induction (cf. Remark 1.2). □

2 Triangular Witt groups

2.1 Witt groups of triangulated categories

A symmetric space in a triangulated category with duality (K, ♯, δ, ϖ) is defined as a pair (V, φ), in which V ∈ K and φ : V ⟶^{≃} V^{♯} is a symmetric isomorphism, i. e. φ^{♯} · ϖ_{V} = φ. Morphisms between symmetric spaces are isometries, defined in the usual way. We will use the following

Notation 2.1 When we say that (A, ϕ) is a symmetric space for the i-th shifted duality T^{i}♯ or simply a T^{i}♯-symmetric space, we mean that it is a δ^{i}(−1)^{\frac{i(i+1)}{2}} ϖ-symmetric space for the shifted duality T^{i}♯. We will also say that it is a symmetric space over K^{(i)}.

To define Witt groups we have the specify which spaces are regarded as trivial. The definition is the following one.

Definition 2.2 (Balmer) A symmetric space (V, φ) is called neutral, if there exists an exact triangle

T^{−1}L^{♯} ⟶^{ν} L ⟶^{ι} V ⟶^{ι^{♯}φ} L^{♯}

in K with T^{−1}(ν^{♯}) = δ(ϖ_{L} · ν). Then L ⟶^{ι} V is called a Lagrangian of the space (V, φ).

The isometry classes of symmetric spaces in K with the orthogonal sum as operation form a monoid MW (K, ♯, ϖ). It is easy to check that the sum of two neutral spaces is again neutral, hence the isometry classes NW (K, ♯, ϖ) of neutral spaces form a submonoid.

Definition 2.3 (Balmer) We set

W (K, ♯, ϖ) := \frac{MW (K, ♯, ϖ)}{NW (K, ♯, ϖ)} .

and define for i ∈ ℤ the i-th triangular Witt group of (K, ♯, ϖ) as:

W^{i}(K) := W (K^{(i)})

(cf. Remark 1.8 for the definition of K^{(i)}). Clearly W^{0}(K) = W (K, ♯, ϖ).

The monoid W^{i}(K) is in fact a group, since the orthogonal sum (V, φ) ⊥ (V, −φ) is neutral. The class of a T^{i}♯-symmetric space (V, φ) in W^{i}(K) will be denoted by [V, φ].

We end this subsection with the following remarkable result, whose analog is wrong for the usual Witt groups of exact categories.

Theorem 2.4 ([1], Theorem 3.5) [(V, φ)] = 0 in W^{i}(K) if and only if the space (V, φ) is neutral.

More information on this subject can be found in the papers [1] and [2].

2.2 Duality preserving functors

Definition 2.5 A duality preserving functor between triangulated categories with duality

(A, D_{A}, δ_{A}, ϖ_{A})  and  (B, D_{B}, δ_{B}, ϖ_{B})

is a pair (F, ρ), where F is a 1-exact covariant functor A ⟶ B and

ρ : F D_{A} ⟶ D_{B}F

is an isomorphism of functors, such that the diagrams

1.

[figure: F; F (ϖ_{A}); F D_{A}D_{A}; D_{B}D_{B}F; ϖ_{B}; D_{B}ρ; D_{B}F D_{A}; ρ_{D_{A}}]

and 2.

[figure: F T_{A}D_{A}; ρ_{T_{A}^{−1}}; D_{B}F T_{A}^{−1}; T_{B}F D_{A}; (δ_{A}δ_{B}) · θ_{D_{A}}; T_{B}ρ; T_{B}D_{B}F; D_{B}T_{B}^{−1}θ_{T_{A}^{−1}}]

commute, where θ : F T_{A} ⟶^{≃} T_{B}F as in Definition 1.1.

The isomorphism of functors ρ is called a duality transformation for F.

For i ∈ ℤ, we define the i-th shifted duality transformation of F:

ρ^{(i)} := (δ_{A}δ_{B})^{i} · T_{B}^{i}(ρ) · θ_{D_{A}}^{(i)} .

We leave it to the reader to check that then (F, ρ^{(i)}) is a duality preserving functor for the shifted duality:

(F, ρ^{(i)}) : A^{(i)} ⟶ B^{(i)} .

Theorem 2.6 Let F : A ⟶ B be a duality preserving functor with a duality transformation ρ. Then F induces a homomorphism

F_{∗} : W^{i}(A) ⟶ W^{i}(B) [A, φ] ⟼ [F (A), ρ_{A}^{(i)} · F (φ)]

for all i ∈ ℤ, where (A, φ) is a symmetric space over K^{(i)}. If F is an equivalence of categories, then F_{∗} is an isomorphism for all i ∈ ℤ.

Proof: By the remark before the theorem it is enough to prove this for i = 0. We leave the straightforward verifications to the reader with the hint to use Theorem 2.4 to show that the homomorphism F_{∗} is injective if F is an equivalence. □

Remark 2.7 It is important to note that the homomorphism induced by a duality preserving functor on the Witt groups depends on the duality transformation ρ. E. g. assume we have a finite dimensional vector space V over a field endowed with two bilinear forms which are not Witt equivalent. Then the exact functor V ⊗ − can be made in two different ways duality preserving such that the induced maps on the Witt groups are not the same (cf. the next section).

Example 2.8 Let (K, ♯, δ, ϖ) be a triangulated category with duality ♯ and translation functor T. Then

(T^{2}, id) : (K, ♯, δ, ϖ) ⟶ (K, T^{4}♯, δ, ϖ)

is a duality preserving functor. It induces isomorphisms W^{n}(K) ⟶^{≃} W^{n+4}(K) for all n ∈ ℤ, which proves the 4-periodicity of the Witt groups.

3 Two product structures

We fix in this section a TPD-category (K, ♯, δ, ϖ, ⊠). We use 𝔩 and 𝔯 to denote the functorial isomorphisms (T A) ⊠ B ⟶^{≃} T (A ⊠ B) respectively A ⊠ (T B) ⟶^{≃} T (A ⊠ B) and η to denote the isomorphism A^{♯} ⊠ B^{♯} ⟶^{≃} (A ⊠ B)^{♯}.

3.1 − ⊠ A is a duality preserving functor

Let A ∈ K, i ∈ ℤ, and ϕ : A ⟶^{≃} T^{i}(A^{♯}) be a T^{i}♯-symmetric form in K. We define a duality transformation ℜ(A, ϕ) for the exact functor − ⊠ A by setting

ℜ(A, ϕ)_{B} : B^{♯} ⊠ A ⟶^{id_{B^{♯}} ⊠ϕ} B^{♯} ⊠ (T^{i}A^{♯}) ⟶^{𝔯_{B^{♯},A^{♯}}^{(i)}} T^{i}(B^{♯} ⊠ A^{♯}) ⟶^{T^{i}(η_{B,A})} T^{i}(B ⊠ A)^{♯} .

for any B ∈ K.

With this notation we have the following

Theorem 3.1 The pair (− ⊠ A, ℜ(A, ϕ)) is a duality preserving functor

K = (K, ♯, δ, ϖ) ⟶ (K, T^{i}♯, (−1)^{i}δ, (−1)^{\frac{i(i+1)}{2}} δ^{i}ϖ) = K^{(i)} .

Proof: We have to verify the axioms in Definition 2.5. The first diagram in the definition of duality preserving functors

[figure: B ⊠ A; ϖ_{B} ⊠ id_{A}; B^{♯♯} ⊠ A; (B ⊠ A)^{♯♯}; δ^{i}(−1)^{\frac{i(i+1)}{2}} ϖ_{B⊠A}; T^{i}(ℜ(A, ϕ)_{B}^{♯}); T^{i}((B^{♯} ⊠ A)^{♯}) .; ℜ(A, ϕ)_{B^{♯}}]

commutes for the following reasons

T^{i}(ℜ(A, ϕ)_{B}^{♯}) · δ^{i}(−1)^{\frac{i(i+1)}{2} +i}ϖ_{B⊠A}

= δ^{i}(−1)^{\frac{i(i+1)}{2}} T^{i}((id_{B^{♯}} ⊠ϕ)^{♯}) · T^{i}((𝔯_{B^{♯},A^{♯}}^{(i)})^{♯}) · η_{B,A}^{♯} · ϖ_{B⊠A}

= δ^{i}(−1)^{\frac{i(i+1)}{2}} T^{i}((id_{B^{♯}} ⊠ϕ)^{♯}) · T^{i}((𝔯_{B^{♯},A^{♯}}^{(i)})^{♯}) · η_{B^{♯},A^{♯}} · ϖ_{B} ⊠ ϖ_{A}  by Definition 1.9
= δ^{i}(−1)^{\frac{i(i+1)}{2}} T^{i}((𝔯_{B^{♯},T^{−i}A}^{(i)})^{♯}) · (id_{B^{♯}} ⊠T^{−i}(ϕ))^{♯} · η_{B^{♯},A^{♯}} · ϖ_{B} ⊠ ϖ_{A} 𝔯_{B^{♯},}^{(i)} is natural
= δ^{i}(−1)^{\frac{i(i+1)}{2}} T^{i}((𝔯_{B^{♯},T^{−i}A}^{(i)})^{♯}) · η_{B^{♯},A} · id_{B^{♯♯}} ⊠T^{i}(ϕ^{♯}) · ϖ_{B} ⊠ ϖ_{A}  η is natural
= T^{i}((𝔯_{B^{♯},T^{−i}A}^{(i)})^{♯}) · η_{B^{♯},A} · ϖ_{B} ⊠ ϕ  ϕ is T^{i}♯-symmetric
= T^{i}(η_{B^{♯},A}) · 𝔯_{B^{♯♯},A^{♯}}^{(i)} · ϖ_{B} ⊠ ϕ  by Lemma 1.10

= ℜ(A, ϕ)_{B^{♯}} · ϖ_{B} ⊠ id_{A} .

Consider the second diagram in the definition:

[figure: T (B^{♯}) ⊠ A; ℜ(A, ϕ)_{T^{−1}B}; T^{i}(((T^{−1}B) ⊠ A)^{♯}); T (B^{♯} ⊠ A); (−1)^{i}𝔩_{B^{♯},A}; T (ℜ(A, ϕ)_{B}); T^{i+1}((B ⊠ A)^{♯}) .; T^{i+1}(𝔩_{T^{−1}B,A}^{♯})]

We have: T^{i+1}(𝔩(A)_{T^{−1}B}^{♯}) · ℜ(A, ϕ)_{T^{−1}B}

= T^{i+1}(𝔩_{T^{−1}B,A}^{♯}) · T^{i}(η_{T^{−1}B,A}) · 𝔯_{(T^{−1}B)^{♯},A^{♯}}^{(i)} · ϕ ⊠ id_{(T^{−1}B)^{♯}}
= T^{i+1}(η_{B,A}) · T^{i}(𝔩_{B^{♯},A^{♯}}) · 𝔯_{T (B^{♯}),A^{♯}}^{(i)} · ϕ ⊠ id_{T (B^{♯})}  by Definition 1.9
= (−1)^{i} T^{i+1}(η_{B,A}) · T (𝔯_{B^{♯},A^{♯}}^{(i)}) · 𝔩_{B^{♯},T^{i}(A^{♯})} · ϕ ⊠ id_{T (B^{♯})}  by Lemma 1.5
= (−1)^{i} T^{i+1}(η_{B,A}) · T (𝔯_{B^{♯},A^{♯}}^{(i)}) · T (ϕ ⊠ id_{B^{♯}}) · 𝔩_{B^{♯},A}  𝔩 is natural.
= (−1)^{i} T (ℜ(A, ϕ)_{B}) · 𝔩_{B^{♯},A}

So this diagram commutes too and we are done. □

Let now (B, ψ) be a T^{j}♯-symmetric space. We define then

𝔏(B, ψ)_{A} : B ⊠ A^{♯} ⟶^{ψ⊠id_{A^{♯}}} T^{j}B^{♯} ⊠ A^{♯} ⟶^{𝔩_{B^{♯},A^{♯}}^{(j)}} T^{j}(B^{♯} ⊠ A^{♯}) ⟶^{T^{j}(η_{B,A})} T^{j}((B ⊠ A)^{♯}) .

We get with the same reasoning as above:

Theorem 3.2 The pair (B ⊠ −, 𝔏(B, ψ)) is a duality preserving functor

K = (K, ♯, δ, ϖ) ⟶ (K, T^{j}♯, (−1)^{j}δ, (−1)^{\frac{j(j+1)}{2}} δ^{j}ϖ) = K^{(j)} .

We denote the induced homomorphisms on the Witt groups by ℜ(A, ϕ)_{∗} and 𝔏(B, ψ)_{∗} respectively.

𝔏(B, ψ)_{∗} sends the Witt class of the space (A, ϕ) to

[B ⊠ A , T^{i+j}(η_{B,A}) · T^{i}(𝔩_{B^{♯},A^{♯}}^{(j)}) · 𝔯_{T^{j}(B^{♯}),A^{♯}}^{(i)} · ψ ⊠ ϕ]

and ℜ(A, ϕ)_{∗} sends the Witt class of (B, ψ) to

[B ⊠ A , T^{i+j}(η_{B,A}) · T^{j}(𝔯_{B^{♯},A^{♯}}^{(i)}) · 𝔩_{B^{♯},T^{i}(A^{♯})}^{(j)} · ψ ⊠ ϕ] .

Theorem 3.3

ℜ(A, ϕ)_{∗}([B, ψ]) = (−1)^{ij}𝔏(B, ψ)_{∗}([A, ϕ]) .

Proof: Follows from the remarks above and Lemma 1.5. □

Corollary 3.4 ℜ(A, ϕ)_{∗} and 𝔏(B, ψ)_{∗} depend only on the Witt class of (A, ϕ) respectively (B, ψ).

Proof: We show this only for ℜ. Clearly it is enough to show

(A, ϕ) is neutral ⟹ ℜ(A, ϕ)_{∗} = 0 .

By the theorem ℜ(A, ϕ)_{∗}([B, ψ]) = (−1)^{ij}𝔏(B, ψ)_{∗}([A, ϕ]) and the right hand side of this equation is zero for all [B, ψ] if [A, ϕ] = 0. □

Hence we can give the following

Definition 3.5 Let x ∈ W^{i}(K) and y ∈ W^{j}(K). Then the left product is defined by

x ⋆_{l} y := 𝔏(x)_{∗}(y)

and the right product by

x ⋆_{r} y := ℜ(y)_{∗}(x) .

This extends linearly to a left respectively right product on

W^{∗}(K) := ⨁_{ℓ∈ℤ}W^{ℓ}(K) .

Note that x ⋆_{l} y = (−1)^{ij}x ⋆_{r} y by Theorem 3.3. Both the right and the left product make W^{∗}(K) a graded algebra.

3.2 Functorial properties of the product

We consider only the left product. Clearly we have the same results for the right product.

We start by recalling the localization sequence. For this we assume that our triangulated category K satisfies the enriched octahedron axiom (e. g. [1], Section 1). This axiom is true in all the triangulated categories of our interest such as the derived categories of exact categories or their full subcategories and localizations.

Let S ⊆ Mor K be a localizing class of morphisms compatible with the triangulated structure of K (cf. [4], p 251). Then S^{−1}K is also a triangulated category and we have an exact sequence of triangulated categories

I_{S} ⟶^{ι} K ⟶^{q} S^{−1}K ,

where I_{S} is the kernel of the quotient functor q : K ⟶ S^{−1}K, i. e. the full subcategory of K of objects M with qM ≃ 0.

If S^{♯} = S the categories I_{S} and S^{−1}K are also triangulated categories with duality. Their dualities are induced by ♯ and we denote them (following the convention in [1]) by ♯, too. We have then a long exact sequence

… ⟶ W^{m}(I_{S}) ⟶^{ι_{∗}} W^{m}(K) ⟶^{q_{∗}} W^{m}(S^{−1}K) ⟶^{∂} W^{m+1}(I_{S}) ⟶ ….

This sequence is natural with respect to duality preserving functors F : K ⟶ K which satisfy F (S) ⊆ S. We get the following

Theorem 3.6 Let (B, ψ) be a T^{j}♯-symmetric space and assume id_{B} ⊠ s ∈ S for all s ∈ S. Then we have a commutative diagram with exact rows:

[figure: …; W^{m}(I_{S}); W^{m}(K); W^{m}(S^{−1}K); ∂; W^{m+1}(I_{S}); …; …; W^{m+j}(I_{S}); 𝔏(B, ψ)_{∗}; W^{m+j}(K); 𝔏(B, ψ)_{∗}; W^{m+j}(S^{−1}K); 𝔏(B, ψ)_{∗}; ∂; W^{m+j+1}(I_{S}); 𝔏(B, ψ)_{∗}; …]

(we denote the induced duality preserving functors on I_{S} and S^{−1}K also by 𝔏(B, ψ)).

We make now the following assumption: For all Y ∈ K and t ∈ S the morphisms id_{Y} ⊠t and t ⊠ id_{Y} are again in S.

It follows from Corollary 3.4 that the homomorphism

𝔏(B, ψ)_{∗} : W^{m}(S^{−1}K) ⟶ W^{m+j}(S^{−1}K)

depends on the Witt class of the space (B, ψ) only. We want to show that the same is true for the restricted homomorphism W^{m}(I_{S}) ⟶ W^{m+j}(I_{S}). Note that we use here the assumption above to guarantee that the image of this restriction is in W^{m+j}(I_{S}). For this it is enough to show that if (B, ψ) is neutral, i. e. [B, ψ] = 0 in W^{j}(K), the induced morphism 𝔏(B, ψ)_{∗} : W^{m}(I_{S}) → W^{m+j}(I_{S}) is the zero map.

For let L ⟶^{ι} B be a Lagrangian of the neutral space (B, ψ) and (X, φ) a T^{m}♯-symmetric space in I_{S}. Then (X, φ) is a symmetric space in K and (− ⊠ X, ℜ(X, φ)) a duality preserving functor K ⟶ K (Theorem 3.2). We leave it to the reader to check that L ⊠ X ⟶^{ι⊠id_{X}} B ⊠ X is a Lagrangian for

ℜ(X, φ)_{∗}(B, ψ) = (−1)^{ij}𝔏(B, ψ)_{∗}(X, φ)

in K. But by our assumption L ⊠ X is in I_{S} and so this is also a Lagrangian in I_{S}. Our claim follows.

Summarizing we have shown that W^{∗}(I_{S}) and W^{∗}(S^{−1}K) are graded left modules over the graded algebra W^{∗}(K). We denote that (scalar-) product also by ⋆_{l}. With this notation we formulate Theorem 3.7 We have for all i, j ∈ ℤ a commutative diagram

[figure: W^{i}(K) × W^{j}(S^{−1}K); ⋆_{l}; W^{i+j}(S^{−1}K); W^{i}(K) × W^{j+1}(I_{S}); id ×∂; ⋆_{l}; W^{i+j+1}(I_{S}); ∂]

Proof: Follows from Theorem 3.6. □

Remark 3.8 The functorial properties of the localization sequence are not in [1]. But the result we use here can easily be deduced from the definition of the connecting homomorphism in the localization sequence and the cone construction (Theorem 2.6 in [1]). We recommend the reader to have a look at this paper.

3.3 Commutativity and associativity of the product

We continue with the notation of the preceding subsections.

3.3.1 Commutativity

Suppose we are given functorial isomorphisms

π_{A,B} : A ⊠ B ⟶^{≃} B ⊠ A

with the following properties:

(c-1) π_{B,A} is inverse to π_{A,B}.

(c-2) The diagram

[figure: A^{♯} ⊠ B^{♯}; π_{A^{♯},B^{♯}}; B^{♯} ⊠ A^{♯}; (A ⊠ B)^{♯}; η_{A,B}; π_{A,B}^{♯}; (B ⊠ A)^{♯}; η_{B,A}]

commutes for any A and B in K.

(c-3) The diagrams

[figure: T A ⊠ B; π_{T A,B}; B ⊠ T A; A ⊠ T B; π_{A,T B}; T B ⊠ A; T (A ⊠ B); 𝔩_{A,B}; T (π_{A,B}); T (B ⊠ A); 𝔯_{B,A}; T (A ⊠ B); 𝔯_{A,B}; T (π_{A,B}); T (B ⊠ A); 𝔩_{B,A}]

commute for any A and B in K.

Note that commutativity of one of the two diagrams implies commutativity of the other one by virtue of (c-1).

Remark 3.9 As usually we have iterated versions of the diagrams in (c-3). We leave it to the reader to write down these diagrams and verify their commutativity by induction.

Then the product is symmetric in the following sense.

Theorem 3.10 Let x ∈ W^{i}(K) and y ∈ W^{j}(K). Then x ⋆_{r} y = y ⋆_{l} x.

Proof: Let x = [A, ϕ] and y = [B, ψ]. We show that the spaces which represent the product are isometric by means of the isomorphism π_{A,B}. For consider the diagram

A ⊠ B  \overline{π_{A,B}} B ⊠ A

[figure: T^{i}A^{♯} ⊠ T^{j}B^{♯}; ϕ ⊠ ψ; π_{T^{i}A^{♯},T^{j}B^{♯}}; T^{j}B^{♯} ⊠ T^{i}A^{♯}; ψ ⊠ ϕ]

[figure: T^{i}(A^{♯} ⊠ T^{j}B^{♯}); 𝔩_{A^{♯},T^{j}B^{♯}}^{(i)}; T^{i}(π_{A^{♯},T^{j}B^{♯}}); T^{i}(T^{j}B^{♯} ⊠ A^{♯}); 𝔯_{T^{j}B^{♯},A^{♯}}^{(i)}]

[figure: T^{i+j}(A^{♯} ⊠ B^{♯}); T^{i}(𝔯_{A^{♯},B^{♯}}^{(j)}); T^{i+j}(π_{A^{♯},B^{♯}}); T^{i+j}(B^{♯} ⊠ A^{♯}); T^{i}(𝔩_{B^{♯},A^{♯}}^{(j)})]

[figure: T^{i+j}(A ⊠ B)^{♯}; T^{i+j}(η_{A,B}); T^{i+j}(π_{A,B}^{♯}); T^{i+j}(B ⊠ A)^{♯} .; T^{i+j}(η_{B,A})]

By definition the left column represents x ⋆_{r} y and the right column represents y ⋆_{l} x. Now the first square commutes by naturality of π. The second square is the i-iterated version of the left diagram in (c-3) and the third square is the j-iterated version of the right diagram in (c-3) shifted by T^{i}. Hence these squares commute, too. The last square commutes by (c-2). □

We have the following

Corollary 3.11 Both products ⋆_{l} and ⋆_{r} are skew-commutative, i.e.,

u ⋆_{l} v = (−1)^{ij}v ⋆_{l} u  and  u ⋆_{r} v = (−1)^{ij}v ⋆_{r} u

for any u ∈ W^{i}(K), v ∈ W^{j}(K).

Proof: Follows from the theorem and Corollary 3.4. □

3.3.2 Associativity

Suppose we are given functorial isomorphisms

α_{A,B,C} : (A ⊠ B) ⊠ C ⟶^{≃} A ⊠ (B ⊠ C)

which make the following diagrams commutative:

A product in triangular Witt theory 13 (a-1l)

(T A ⊠ B) ⊠ C α_{T A,B,C} T A ⊠ (B ⊠ C)
[figure: T (A ⊠ B) ⊠ C; 𝔩_{A,B} ⊠ id_{C}; T (A ⊠ (B ⊠ C)); 𝔩_{A,B⊠C}]
[figure: T ((A ⊠ B) ⊠ C); 𝔩_{A⊠B,C}; T (α_{A,B,C}); T (A(⊠B ⊠ C)); =]

(a-1m)

(A ⊠ T B) ⊠ C \overline{α_{A,T B,C}} A ⊠ (T B ⊠ C)
[figure: T (A ⊠ B) ⊠ C; 𝔯_{A,B} ⊠ id_{C}; A ⊠ T (B ⊠ C); id_{A} ⊠𝔩_{B,C}]
[figure: T ((A ⊠ B) ⊠ C); 𝔩_{A⊠B,C}; T (α_{A,B,C}); T (A ⊠ (B ⊠ C)); 𝔯_{A,B⊠C}]

(a-1r)

(A ⊠ B) ⊠ T C \overline{α_{A,B,T C}} A ⊠ (B ⊠ T C)
[figure: T ((A ⊠ B) ⊠ C); 𝔯_{A⊠B,C}; A ⊠ T (B ⊠ C); id_{A} ⊠𝔯_{B,C}]
[figure: T ((A ⊠ B) ⊠ C); =; T (α_{A,B,C}); T (A ⊠ (B ⊠ C)); 𝔯_{A,B⊠C}]

(a-2)

(A^{♯} ⊠ B^{♯}) ⊠ C^{♯} \overline{\begin{matrix}α_{A^{♯},B^{♯},C^{♯}} \\  \end{matrix}}A^{♯} ⊠ (B^{♯} ⊠ C^{♯})
[figure: (A ⊠ B)^{♯} ⊠ C^{♯}; η_{A,B} ⊠ id_{C^{♯}}; A^{♯} ⊠ (B ⊠ C)^{♯}; id_{A^{♯}} ⊠η_{B,C}]
[figure: ((A ⊠ B) ⊠ C)^{♯}; η_{A⊠B,C}; (α_{A,B,C})^{♯}; (A ⊠ (B ⊠ C))^{♯} .; η_{A,B⊠C}]

Again we have iterated versions of the diagrams (a-1l), (a-1m), (a-1r) and (a-2), i. e. similar diagrams for 𝔯^{(i)} and 𝔩^{(i)}. We leave this to the reader.

This makes both the left and the right product associative.

Theorem 3.12 Let a ∈ W^{i}(K), b ∈ W^{j}(K) and c ∈ W^{k}(K). Then

(a ⋆_{l} b) ⋆_{l} c = a ⋆_{l} (b ⋆_{l} c)

and

(a ⋆_{r} b) ⋆_{r} c = a ⋆_{r} (b ⋆_{r} c) .

Proof: The proof is straightforward, but cumbersome. For the sake of completeness we give the details for the left product. Set a = [A, ϕ], b = [B, ψ] and c = [C, χ].

Then (a ⋆_{l} b) ⋆_{l} c is represented by [(A ⊠ B) ⊠ C, (ϕ ⋆_{l} ψ) ⋆_{l} χ], where (ϕ ⋆_{l} ψ) ⋆_{l} χ =

T^{i+j+k}(η_{A⊠B,C}) · T^{k}(𝔩_{(A⊠B)^{♯},C^{♯}}^{(i+j)}) · 𝔯_{T^{i+j}((A⊠B)^{♯}),C^{♯}}^{(k)} · T^{i+j}(η_{A,B}) ⊠ id_{T^{k}(C^{♯})} ·T^{j}(𝔩_{A^{♯},B^{♯}}^{(i)}) ⊠ id_{T^{k}(C^{♯})} ·

· 𝔯_{T^{i}(A^{♯}),B^{♯}}^{(j)} ⊠ id_{T^{k}(C^{♯})} ·(ϕ ⊠ ψ) ⊠ χ ,

and a ⋆_{l} (b ⋆_{l} c) is represented by [A ⊠ (B ⊠ C), ϕ ⋆_{l} (ψ ⋆_{l} χ)], where ϕ ⋆_{l} (ψ ⋆_{l} χ) =

T^{i+j+k}(η_{A⊠B,C}) · T^{j+k}(𝔩_{A^{♯},(B⊠C)^{♯}}^{(i)} · 𝔯_{T^{i}(A^{♯}),(B⊠C)^{♯}}^{(j+k)} · id_{T^{i}(A^{♯})} ⊠T^{j+k}(η_{B,C}) · id_{T^{i}(A^{♯})} ⊠T^{k}(𝔩_{B^{♯},C^{♯}}^{(j)})

· id_{T^{i}(A^{♯})} ⊠𝔯_{T^{j}(B^{♯}),C^{♯}}^{(k)} · ϕ ⊠ (ψ ⊠ χ) .

We show now that α_{A,B,C} is an isometry between these two spaces, i. e.

T^{i+j+k}(α_{A,B.C}^{♯}) · ϕ ⋆_{l} (ψ ⋆_{l} χ) · α_{A,B,C} = (ϕ ⋆_{l} ψ) ⋆_{l} χ .

Using the naturality of 𝔩^{(m)}, 𝔯^{(m)} and α we see that the left hand side of this is equal to

D · T^{j+k}(𝔩_{A^{♯},(B⊠C)^{♯}}^{(i)} · 𝔯_{T^{i}(A^{♯}),B^{♯}⊠C^{♯}}^{(j+k)} · id_{T^{i}(A^{♯})} ⊠T^{k}(𝔩_{B^{♯},C^{♯}}^{(j)}) · id_{T^{i}(A^{♯})} ⊠𝔯_{T^{j}(B^{♯}),C^{♯}}^{(k)}·
· α_{T^{i}(A^{♯}),T^{j}(B^{♯}),T^{k}(C^{♯})} · (ϕ ⊠ ψ) ⊠ χ ,

where D = T^{i+j+k}(α_{A,B,C}^{♯} · η_{A,B⊠C} · id_{A^{♯}} ⊠η_{B,C}). Applying axioms (a-1m) and (a-1r) (iterated versions), the naturality of 𝔩^{(m)} and 𝔯^{(m)} and then axiom (a-2) we see that this is equal to (taking the last formula of Remark 1.2 into account)

D · T^{j+k}(𝔩_{A^{♯},B^{♯}⊠C^{♯}}^{(i)} · α_{T^{i}(A^{♯}),B^{♯},C^{♯}}) · T^{k}(𝔩_{T^{i}(A^{♯})⊠B^{♯},C^{♯}}^{(j)}) · 𝔯_{T^{i}(A^{♯}),B^{♯}}^{(j)}) · 𝔯_{T^{i}(A^{♯})⊠T^{j}(B^{♯}),C^{♯}}^{(k)} · (ϕ ⊠ ψ) ⊠ χ .

Applying now axiom (a-1l) (iterated version) and the definition of the term D, we are in position to simplify the above product by means of axiom (a-2) and get using the naturality of 𝔩^{(m)} and 𝔯^{(m)}:

T^{i+j+k}(η_{(A⊠B)⊠C}) · T^{j+k}(𝔩_{(A⊠B)^{♯},C^{♯}}^{(i)} · T^{i}(η_{A,B}) ⊠ id_{C^{♯}}) · T^{k}(𝔩_{T^{i}(A^{♯}⊠B^{♯}),C^{♯}}^{(j)}) · 𝔯_{T^{i+j}(A^{♯}⊠B^{♯}),C^{♯}}^{(k)}·

· T^{j}(𝔩_{A^{♯},B^{♯}}^{(i)}) ⊠ id_{T^{k}(C^{♯})} ·𝔯_{T^{i}(A^{♯}),B^{♯}}^{(j)} ⊠ id_{T^{k}(C^{♯})} ·(ϕ ⊠ ψ) ⊠ χ .

We use again the naturality of 𝔩^{(m)} and 𝔯^{(m)} and see that this is equal to (ϕ⋆_{l} ψ)⋆_{l} χ. The theorem is proved. □

4 Derived Witt groups of a scheme

4.1 The product and duality structure

In order to apply our definitions to the derived Witt groups of schemes, we have to consider the bounded derived category D^{b}(VB_{X}) of the category VB_{X} of locally free sheaves of finite rank over a scheme X and make it a TPD-category. More generally, one can work with the bounded derived category D^{b}(A) of an exact category A and make it a TPD-category provided A is an exact category with duality ♯ and product ⊗. We leave it to the reader to give an appropriate definition of an exact category with duality and product. Note that the arguments of this section will not work unless the product is bi-exact in A.

In this section we will work with the category D^{b}(VB_{X}) only. The product in it is given by the (derived) tensor product and the duality D_{X} is the derived functor of

F ⟼ Hom_{O_{X}} (F, O_{X}) .

The natural isomorphism ϖ to the bidual is induced by the evaluation map. It is easy to check that (D^{b}(X), D_{X}, ϖ) is a triangulated category with 1-exact duality.

Definition 4.1 (Balmer [2])

W^{i}(X) := W^{i}(D^{b}(VB_{X}), D_{X}, ϖ) .

We have then the following important result.

Theorem 4.2 (Balmer [2]) The natural functor VB_{X} ⟶ D^{b}(VB_{X}) induces isomorphisms

W (X) ⟶^{≃} W^{0}(X)  and  W^{−}(X) ⟶ W^{2}(X) ,

where W (X) and W^{−}(X) are the Witt groups of symmetric respectively skew-symmetric forms on the scheme X defined by Knebusch.

Let now (P_{•}, d_{•}^{P}) and (Q_{•}, d_{•}^{Q}) be two chain complexes of locally free O_{X}-modules. Since the tensor product is a bi-exact functor on VB_{X}, we see that in degree n the tensor product P_{•} ⊗ Q_{•} is given by

(P_{•} ⊗ Q_{•})_{n} = ⨁_{i+j=n} P_{i} ⊗ Q_{j} .

For the differential we adopt the following sign convention:

d^{P ⊗Q} = d_{i}^{P} ⊗ id_{Q_{j}} +(−1)^{i} id_{P_{i}} ⊗d_{j}^{Q}

on the term P_{i} ⊗ Q_{j}.

The structural isomorphisms of Section 1 and Section 3.3 can be defined as follows:

1l) 𝔩_{P,Q} : T P ⊗ Q ⟶ T (P ⊗ Q) is given by +1 everywhere;

1r) 𝔯_{P,Q} : P ⊗ T Q ⟶ T (P ⊗ Q) is given by (−1)^{i} on P_{i} ⊗ Q_{j};

2) η_{P,Q} : D_{X}(P) ⊗ D_{X}(Q) ⟶ D_{X}(P ⊗ Q), is essentially +1 everywhere;

3) π_{P,Q} : P ⊗ Q ⟶ Q ⊗ P is given by assembling all P_{i} ⊗ Q_{j} ⟶^{(−1)^{ij}} Q_{j} ⊗ P_{i}.

4) α_{P,Q,R} : (P ⊗ Q) ⊗ R ⟶ P ⊗ (Q ⊗ R) is given by +1 everywhere.

We leave it to the reader to check that these are morphisms of complexes. We have then the following theorem, whose easy proof is left to the reader, too.

Theorem 4.3 (D^{b}(X), D_{X}, ϖ, ⊗) is a TPD-category.

As the natural isomorphisms π and α (defined above) satisfy the associativity and commutativity axioms in Section 3.3, we get

Theorem 4.4 Let X be a scheme. The left and right product make

W^{∗}(X) := ⨁_{i∈ℤ}W^{i}(X)

a graded associative ring with skew-commutative product, i. e.

x ⋆_{l} y = (−1)^{ij}y ⋆_{l} x  and  x ⋆_{r} y = (−1)^{ij}y ⋆_{r} x ,

where x ∈ W^{i}(X) and y ∈ W^{j}(X). The 1-element is the class of the D_{X}-symmetric space

O_{X} ⟶^{id} O_{X} ≃ Hom_{O_{X}} (O_{X}, O_{X}) .

Remark 4.5 On W^{0}(X) the left and right products coincide. Identifying W (X) with W^{0}(X) (Theorem 4.2) this is the same product as the one defined by Knebusch ([5]).

4.2 Remarks on sign conventions

If we had chosen the sign rule for d^{P ⊗Q} to be

d_{i}^{P} ⊗ id_{Q_{j}} +(−1)^{i+j} id_{P_{i}} ⊗ d_{j}^{Q}

on the term P_{i} ⊗Q_{j}, then the structural isomorphisms would be given by different formulas. With this tensor product D^{b}(VB_{X}) will be again a TPD-category and the product on the Witt groups will be associative and skew-commutative. Moreover

P_{i} ⊗ Q_{j} ⟶^{(−1)^{\frac{j(j+1)}{2}}} P_{i} ⊗ Q_{j}  (4.1)

defines an isomorphism between these two tensor products which gives an isomorphism of the two TPD-structures on D^{b}(VB_{X}). It follows that the products on the Witt groups defined by means of these different tensor products coincide.

In fact, we claim that the products ∗_{l} and ∗_{r} on the Witt groups of D^{b}(X) can be defined (and will be the same) by means of any definition of P ⊗ Q corresponding to any, even absolutely chaotic and non-periodic choice of signs which makes a commutative bicomplex anti-commutative.

Let ε = {ε_{i,j}^{h}, ε_{i,j}^{v} = ±1; i, j ∈ ℤ} be a collection of signs satisfying

ε_{i,j}^{h}ε_{i−1,j}^{v} = −ε_{i,j}^{v}ε_{i,j−1}^{h}  (4.2)

for any i, j in ℤ. Imagine that we are given a commutative complex and put the signs ε_{i,j}^{h} and ε_{i,j}^{v} on the horizontal (respectively vertical) arrow which begins in the degree (i, j) term. Then (4.2) implies that we will get an anti-commutative complex and can therefore apply Tot in the canonical sense to get a (single) complex. In particular, we can define the tensor product P ⊗_{ε} Q with the differential given by

ε_{i,j}^{h}d_{i}^{P} ⊗ id_{Q_{j}} +ε_{i,j}^{v} id_{P_{i}} ⊗d_{j}^{Q}

on P_{i} ⊗ Q_{j}. Given two such collections ε and ε′, there is a unique collection of δ_{i,j} = ±1; i, j ∈ ℤ with δ_{0,0} = +1 which provides an isomorphism between ε and ε′ in the obvious sense. This yields a functorial isomorphism

δ(ε, ε′) : P ⊗_{ε} Q ⟶ P ⊗_{ε′} Q ,  (4.3)

a particular case of which is the map (4.1). We claim that there exists a unique choice of maps 1l)-4) for an arbitrary ε such that the axioms of a TPD- category and (a-1l) - (a-2), (c-1) - (c-3) are true and that the maps (4.3) make the corresponding TPD-structures on D^{b}(X) isomorphic.

Verification of the axioms is the most delicate part of this elementary calculus of signs. It amounts to some relations between certain operators on the category of all ε, such as horizontal and vertical shifts, change of all vertical respectively horizontal signs, and some symmetries. For instance, the map η corresponds to the isomorphism between ε and its image under the central symmetry based in the origin; π is given by comparing ε to its image under the symmetry with respect to the line i = j, etc.

5 An application

Theorem 5.1 Let X be a scheme over a field K and L a finite extension of K of odd degree. Then the natural homomorphism

W^{i}(X) ⟶ W^{i}(X ×_{K} L)

is injective for all i ∈ ℤ.

Bayer-Fluckiger and Lenstra [3] have given an elegant proof of the same assertion for the classical Witt groups (cf. also [6], chapter I, §10). Our proof here is just an adaption of their proof to the situation of derived Witt groups of schemes, made possible by the existence of a product structure.

5.1 Transfers

We review this only briefly in the case of derived Witt groups and refer for more information to [6], chapter I, §7.

Let R be a commutative noetherian ring (with 1) and S a R-algebra which is a finite projective R-module. We assume that there exists τ ∈ Hom_{R}(S, R) such that

S ≃ Hom_{S}(S, S) ⟶ Hom_{R}(S, R) g ⟼ τ · g

is an isomorphism of R-modules. Then for all Q ∈ VB_{S} (i. e. Q is a finitely generated projective S-module) the natural morphism

Hom_{S}(Q, S) ⟶ Hom_{R}(Q, R) g ⟼ τ · g

is an isomorphism of projective R-modules. Clearly this functorial bijection induces a functorial isomorphism of complexes

ρ_{Q_{•}} : Hom_{S}(Q_{•}, S) ⟶^{≃} Hom_{R}(Q_{•}, R) ,

where Q_{•} ∈ D^{b}(S) = D^{b}(VB_{S}). Since S is a projective and finite R-module, we have a well defined push forward functor

Tr_{S/R} : D^{b}(S) ⟶ D^{b}(R) .

The pair (Tr_{S/R}, ρ) is then a duality preserving functor which induces homomorphisms

Tr_{S/R ∗} : W^{i}(S) ⟶ W^{i}(R)

for all i ∈ ℤ, by Theorem 2.6.

Remark 5.2 Note that these homomorphisms depend on the R-linear map τ.

Let now X be a noetherian scheme over R, X_{S} = X ×_{R} S and π : X_{S} → X the projection. π induces natural homomorphisms π^{∗} : W^{i}(X) ⟶ W^{i}(X_{S}) for all i ∈ ℤ.

The R-linear map τ induces a morphism of O_{X}-modules

O_{X_{S}} ≃ Hom_{O_{X_{S}}} (O_{X_{S}}, O_{X_{S}}) ⟶ Hom_{O_{X}} (π_{∗}O_{X_{S}}, O_{X}) .

Since S is flat over R this is an isomorphism. The arguments above give then for all i ∈ ℤ transfer homomorphisms

Tr_{X_{S}/X ∗} : W^{i}(X_{S}) ⟶ W^{i}(X) .

5.2 A projection formula

We use the notation of the last section and denote the canonical duality on D^{b}(X) by D_{X}. The natural morphisms

α : X_{S} ⟶ S  and  β : X ⟶ R

induce homomorphisms α^{∗} : W^{i}(S) → W^{i}(X_{S}) and β^{∗} : W^{i}(R) → W^{i}(X). For x ∈ W^{∗}(X), r ∈ W^{∗}(R), s ∈ W^{∗}(S) and y ∈ W^{∗}(X_{S}) we define

x ⋆_{l} r := x ⋆_{l} β^{∗}(r)  and  y ⋆_{l} s := y ⋆_{l} α^{∗}(s) .

This makes W^{∗}(X_{S}) and W^{∗}(X) a graded W^{∗}(S)- respectively W^{∗}(R)-module. We have then the following

Theorem 5.3 (Projection formula) Let x ∈ W^{i}(X) and s ∈ W^{j}(S). Then

Tr_{X_{S}/X ∗}(π^{∗}(x) ⋆_{l} s) = x ⋆_{l} Tr_{S/R ∗}(s) .

Proof: It is easy to check that any element x can be represented by a (T^{i}D_{X})-symmetric form ψ on a complex Q_{•}, such that Q_{•} ⟶^{ψ} T^{i}(D_{X}Q_{•}) is an isomorphism of complexes, i. e. not a fraction. Let further s = [P_{•}, φ].

Then the left side of the equation can be represented on the (r, s)-component (r + s = n) by:

Q_{r} ⊗_{R} P_{s} ⟶ Hom_{O_{X_{S}}} (Q_{−r+i} ⊗_{R} P_{−s+j}, O_{X})
q ⊗ p ⟼ { c ⊗ d ⟼ ψ(q)(c) · τ(φ(p)(d)) } .

The reader checks at once that this form represents also the right side of the equation. □

We can now give

Proof of Theorem 5.1: It is enough to show this for a simple extension. Since L is of odd degree over K there exists a K-linear map L ⟶^{τ} K, so that the corresponding transfer morphism Tr_{L/K ∗} satisfies

Tr_{L/K ∗}([L, id]) = [K, id]

(e. g. [6], proof of Proposition 10.3.1). Our result follows from the projection formula and Theorem 4.4. □

References

[1] Balmer, P.: Triangular Witt groups I, K-theory 19 (2000), pp 311-363

[2] Balmer, P.: Triangular Witt groups II, Math. Z. 236 (2001), pp 351-382

[3] Bayer-Fluckiger, E.,Lenstra, H.W.: Forms in odd degree extensions and self dual bases, Am. J. Math 112 (1990), pp 359-373

[4] Gelfand, S.I., Manin, Y.I.: Methods of homological algebra, Springer 1996

[5] Knebusch, M.: Symmetric bilinear forms over algebraic varieties, Proc. Conf. Quadratic forms, Kingston 1976, Queen’s Pap. pure and appl. Math. 46 (1977), pp 103-283

[6] Knus, M.A.: Quadratic and hermitian forms over rings, Springer 1991