KOSZUL COMPLEXES AND SYMMETRIC FORMS OVER THE PUNCTURED AFFINE SPACE

PAUL BALMER AND STEFAN GILLE

Abstract. Let X be a regular separated ℤ[1/2]-scheme of finite Krull dimension and let 𝕌_{X}^{n} ⊂ 𝔸_{X}^{n} be the punctured affine n-space over X. We show that the total graded Witt ring W^{tot}(𝕌_{X}^{n}) is a free W^{tot}(X)-module with two generators 1 and ε. The second generator satisfies the equation ε^{2} = 1 when n = 1 and ε^{2} = 0 when n ≥ 2.

Introduction

Let X be scheme. We are studying the (total) graded Witt ring

W^{tot}(X) := ⨁_{i∈ℤ} W^{i}(X)

where the groups W^{i} are the derived Witt groups of Balmer [2, 3] with the multiplicative structure of Gille-Nenashev [11]. See more in Section 2.

We fix an integer n ≥ 1 for the entire article. Consider the following open subset 𝕌_{ℤ}^{n} ⊂ 𝔸_{ℤ}^{n} of the affine space 𝔸_{ℤ}^{n} = Spec(ℤ[T_{1}, …, T_{n}]) :

𝕌_{ℤ}^{n} := ⋃_{j=1}^{n} Spec (ℤ[T_{1}, …, T_{n}, T_{j}^{−1} ]) ⊂ 𝔸_{ℤ}^{n} .

For any scheme X, define by base-change the open subscheme 𝕌_{X}^{n} ⊂ 𝔸_{X}^{n}, called the punctured affine space over X, i.e. define 𝕌_{X}^{n} to be the following pull-back :

𝕌_{X}^{n} := X × 𝕌_{ℤ}^{n}

Our main results are summarized in Theorem 8.13 below, which says :

Theorem. If X is regular, contains \frac{1}{2} and has finite Krull dimension, there is a decomposition W^{tot}(𝕌_{X}^{n}) = W^{tot}(X) ⊕ W^{tot}(X) · ε for some Witt class ε = εX^{(n)} in W^{n−1}(𝕌_{X}^{n}). If n=1, we have ε^{2} = 1. If n ≥ 2, we have ε^{2} = 0 and an isomorphism

W^{tot}(𝕌_{X}^{n}) ≅ \frac{W^{tot}(X) [ε]}{ε^{2}}

of graded rings, with the generator ε in degree n − 1.

When n = 1, and at least for X affine, 𝕌_{X}^{n} consists of the “Laurent polynomials” over X, in which case the above Theorem is due to Ranicki [16], see also [15]. Note that the decomposition of W^{tot}(𝕌_{X}^{n}) for n = 1 as two copies of W^{tot}(X) remains true for n ≥ 2, but that the second copy is shifted by n − 1, which is of course not visible when n = 1. Note also that the ring structure is different from the Laurent case when n ≥ 2. The special case n = 1 also shows that this result cannot hold in general for non-regular schemes, see Ojanguren-Panin [15, § 8].

Date: September 02, 2004.

The second goal of the article is an explicit description of the Witt class εX^{(n)} ∈ W^{n−1}(𝕌_{X}^{n}). It is constructed as the Witt class of a symmetric space denoted EX^{(n)} = (EX^{(n)}, ξX^{(n)}), which exists for any scheme X, not necessarily regular. Here EX^{(n)} is a complex in the derived category of vector bundles over 𝕌_{X}^{n} and the form ξX^{(n)} is a symmetric quasi-isomorphism. Both are constructed via a suitable truncation of the Koszul complex over 𝔸_{X}^{n} for the regular sequence (T_{1}, …, T_{n}). This explicit description is necessary to prove the following fact (see Theorem 8.10) :

Theorem. Assume n ≥ 2. The symmetric space EX^{(n)} is locally metabolic on 𝕌_{X}^{n}.

This result does not hold for n = 1 and is used to establish the announced relation (εX^{(n)})^{2} = 0, see Theorem 8.12. On the other hand, Theorem 9.2 says :

Theorem. The symmetric space EX^{(n)} cannot be extended from 𝕌_{X}^{n} to 𝔸_{X}^{n}, not even up to Witt equivalence. In particular, EX^{(n)} is not metabolic on 𝕌_{X}^{n}.

In other words, this symmetric space EX^{(n)} is a little miracle happening over 𝕌_{X}^{n} , which vanishes when restricted to smaller open subschemes and which cannot be extended to the bigger scheme 𝔸_{X}^{n}.

Our last goal is a description of the generator εX^{(n)} ∈ W^{n−1}(𝕌_{X}^{n}) in “classical” terms. Recall a few facts. First, the derived Witt groups are 4-periodic : W^{i} = W^{i+4}. Secondly, W^{0} and W^{2} are naturally isomorphic to the usual Witt groups W_{us}^{+} and W_{us}^{−} of symmetric and skew-symmetric vector bundles respectively, as defined by Knebusch [14]. Thirdly, W^{1} and W^{3} = W^{−1} are groups of formations, see Walter [18]. So, describing “in classical terms” our generator εX^{(n)} in W^{n−1} amounts to produce an explicit element of the above nature, i.e. a ±1-symmetric form or formation, depending on the congruence of n modulo 4. This short symmetric space is denoted FX^{(n)} and appears in Section 7.

There are two appendices. In the first one, we show that when n ≥ 4 our locally free O_{𝕌_{X}^{n}} -module EX^{(n)} cannot be extended to a locally free O_{𝔸_{X}^{n}} -module and in particular EX^{(n)} is not free. The second appendix contains the compatibility between product and 4-periodicity, a fact which we use several times in this work.

Contents

Introduction 1 1. Conventions and notations 3 2. Recalling derived Witt groups 3 3. Basic facts about Koszul complexes 7 4. Koszul cut in two 10 5. The Koszul symmetric space KX^{(n)} over 𝔸_{X}^{n} 14 6. The half-Koszul symmetric space EX^{(n)} over 𝕌_{X}^{n} 15 7. The short symmetric space FX^{(n)} over 𝕌_{X}^{n} 16 8. Witt groups of the punctured affine space 18 9. Witt non-triviality of the (half) Koszul spaces 23 Appendix A. The locally free module EX^{(n)} 24 Appendix B. Product and 4-periodicity 25 References 27 Acknowledgment. We would like to thank Manuel Ojanguren for useful references. We would further like to thank the referee for useful suggestions. Both authors are supported by the Swiss National Science Foundation, grant 620-066065.

1. Conventions and notations

We collect here the notations which are kept unchanged in all sections.

First of all, recall that we have fixed an integer n ≥ 1. We decompose it as

n = 4 q + r + 1  (1)

where q ∈ ℕ and r ∈ {−1, 0, 1, 2}. Note that n − 1 ≡ r mod 4. We also baptize

[\frac{n}{2}] =: ℓ .  (2)

Convention 1.1. Unless mentioned, a ring means a commutative ring with unit.

Convention 1.2. As always, when using a notation defined for schemes X in the affine case, X = Spec(R), we shall drop “Spec” as for instance : VB_{R}, 𝔻^{b}(VB_{R}), W^{i}(R) instead of VB_{Spec(R)}, 𝔻^{b}(VB_{Spec(R)}), W^{i}(Spec(R)), and so on. See 2.12.

Convention 1.3. We shall say that a scheme is regular if it is noetherian and separated and if all its local rings are regular.

Notation 1.4. Let f : Y → X be a morphism of schemes. We denote by 𝔸_{X}^{n} the affine n-space and by 𝕌_{X}^{n} the punctured affine n-space as in the introduction. The obvious structure morphisms and base-change morphisms will be denoted :

\xymatrix{𝕌_{Y}^{n} \ar[rr]^{σ_{Y}} \ar[r]_{ι_{Y}} \ar[d]_{υ_{f}} & 𝔸_{Y}^{n} \ar[r]_{π_{Y}} \ar[d]_{α_{f}} & Y \ar[d]^{f} \\ 𝕌_{X}^{n} \ar[r]_{ι_{X}} \ar[d]_{υ_{X}} & 𝔸_{X}^{n} \ar[r]_{π_{X}} \ar[d]_{α_{X}} & X \ar[d] \\ 𝕌_{ℤ}^{n} \ar[r]_{ι_{ℤ}} & 𝔸_{ℤ}^{n} \ar[r]_{π_{ℤ}} & Spec(ℤ) .}  (3)

2. Recalling derived Witt groups

This section is a quick course on triangular Witt groups over schemes, included only for the reader’s convenience. Here, X is a scheme with structure bundle O_{X}.

2.1. Categories and dualities.

We denote by the symbol VB_{X} the exact category of locally free O_{X}-modules of finite rank, i.e. vector bundles. The usual duality on VB_{X} is abbreviated

(−)^{∨} := Hom_{O_{X}} (−, O_{X}) .

𝔻^{b}(VB_{X}) stands for the bounded derived category of VB_{X}. We use homological notations for complexes. The translation functor Σ : 𝔻^{b}(VB_{X}) ⟶ 𝔻^{b}(VB_{X}), also written P_{•} ⟼ P_{•}[1], is given by (P_{•}[1])_{j} = P_{j−1}; as usual, Σ changes the sign of all differentials : d_{j}^{P [1]} = −d_{j−1}^{P}.

Let P_{•} = (P_{•}, dP_{•}) be a complex in 𝔻^{b}(VB_{X}). Its dual D_{X}(P_{•}) is the complex

\xymatrix{D_{X}(P_{•}) := & … \ar[r] & P_{−j}^{∨} \ar[r]^{^{d_{−j+1}^{P}^{∨}}} & P_{−(j−1)}^{∨} \ar[r] & … \\  &  & deg j & deg (j − 1)}

and similarly for morphisms of complexes. In other words, D_{X} is the derived functor of (−)^{∨} = Hom_{O_{X}} ( − , O_{X}). This defines a 1-exact duality on 𝔻^{b}(VB_{X}) turning it into a triangulated category with duality in the sense of [2]. Recall that “1-exact”

means that the dual of an exact triangle \xymatrix{N_{•} \ar[r]^{ρ} & P_{•} \ar[r]^{ϕ} & Q_{•} \ar[r]^{φ} & N_{•}[1]} is given by

\xymatrix{D_{X}(Q_{•}) \ar[r]^{D_{X}(ϕ)} & D_{X}(P_{•}) \ar[r]^{D_{X}(ρ)} & D_{X}(N_{•}) \ar[r]^{D_{X}(φ)[1]} & D_{X}(Q_{•})[1] .}

The isomorphism between the identity and the double dual, ϖ : id_{𝔻^{b}(VB_{X})} ⟶^{≃} D_{X}D_{X}, is given in each degree j by the canonical (evaluation) isomorphism can_{P_{j}} : P_{j} ⟶ P_{j}^{∨∨}. We consider VB_{X} as a subcategory of 𝔻^{b}(VB_{X}) via the natural embedding VB_{X} ⟶ 𝔻^{b}(VB_{X}), which we denote c_{0}. The restriction of the duality D_{X} to this subcategory is the original duality of VB_{X} and the restriction of ϖ is the above can .

Definition 2.2. Let P_{•} be a complex in 𝔻^{b}(VB_{X}). Let i ∈ ℤ, and ϕ : P_{•} ⟶ D_{X}(P_{•})[i] be a morphism in 𝔻^{b}(VB_{X}). We say that ϕ is an symmetric i-form on the complex P_{•} if

D_{X}(ϕ)[i] · ϖ_{P_{•}} = (−1)^{\frac{i(i+1)}{2}} ϕ .

We then say that (P_{•}, ϕ) is a symmetric i-pair. If ϕ is moreover an isomorphism we say that (P_{•}, ϕ) is a symmetric i-space over X. Two symmetric i-pairs (P_{•}, ϕ) and (Q_{•}, ψ) are called isometric if there exists in 𝔻^{b}(VB_{X}) an isometry between them, that is, an isomorphism h : P_{•} ⟶^{≃} Q_{•} such that ϕ = D_{X}(h)[i] · ψ · h.

Remark 2.3. Note that if (P_{•}, ϕ) is a symmetric i-pair then (P_{•}[2], ϕ[2]) is a symmetric (i + 4)-pair because D_{X}(P_{•})[1] = D_{X}(P_{•}[−1] ) for all P_{•} ∈ 𝔻^{b}(VB_{X}).

Let f : Y ⟶ X be a morphism of schemes. There is a natural isomorphism of functors η_{f} : f^{∗}D_{X} ⟶^{≃} D_{Y} f^{∗} which is induced by the natural isomorphism of locally free O_{Y} -modules f^{∗}Hom_{O_{X}} (P, O_{X}) ⟶^{≃} Hom_{O_{Y}} (f^{∗}P, O_{Y}). If now (P_{•}, ϕ) is a symmetric i-space over X then the isomorphism

f^{∗}(P_{•}) ⟶^{f^{∗}ϕ} f^{∗}(D_{X}(P_{•})[i]) = f^{∗}(D_{X}(P_{•}))[i] ⟶^{η_{f,P} [i]} D_{Y} (f^{∗}P_{•})[i]

is a symmetric i-form and so f^{∗}(P_{•}, ϕ) := (f^{∗}(P_{•}) , η_{f,P} [i] · f^{∗}(ϕ)) a symmetric i-space over Y.

2.4. “Short” i-forms : Forms and formations.

We present examples of symmetric i-pairs (P_{•}, ϕ) in four cases i = −1, 0, 1, 2.

\xymatrix{i = 0 & deg 0 &  & i = 1 & deg 1 & deg 0 \\ ⋯ 0 \ar[r] & P_{0} \ar[r] \ar[d]^{ϕ_{0}=ϕ_{0}^{∨}} & 0 ⋯ & ⋯ 0 \ar[r] & P_{1} \ar[r]^{d} \ar[d]_{ϕ_{1}} & P_{0} \ar[r] \ar[d]^{−ϕ_{1}^{∨}} & 0 ⋯ \\ ⋯ 0 \ar[r] & _{P_{0}_{∨}} \ar[r] & 0 ⋯ & ⋯ 0 \ar[r] & _{P_{0}_{∨}} \ar[r]_{−d^{∨}} & _{P_{1}_{∨}} \ar[r] & 0 ⋯}

\xymatrix{i = 2 & deg 1 &  & i = −1 & deg 0 & deg -1 \\ ⋯ 0 \ar[r] & P_{1} \ar[r] \ar[d]^{ϕ_{1}=−ϕ_{1}^{∨}} & 0 ⋯ & ⋯ 0 \ar[r] & P_{0} \ar[r]^{d} \ar[d]_{ϕ_{0}} & P_{−1} \ar[r] \ar[d]^{ϕ_{0}^{∨}} & 0 ⋯ \\ ⋯ 0 \ar[r] & _{P_{1}_{∨}} \ar[r] & 0 ⋯ & ⋯ 0 \ar[r] & _{P_{−1}_{∨}} \ar[r]_{−d^{∨}} & _{P_{0}_{∨}} \ar[r] & 0 ⋯}

In each case, the complexes P_{•} and P_{•}^{∨} are depicted horizontally and the symmetric i-form ϕ : P_{•} ⟶ D_{X}(P_{•})[i] vertically. The symmetric pairs of the left-hand column are classical symmetric and skew-symmetric forms embedded in 𝔻^{b}(VB_{X}) via the functor c_{0} (slightly pushed to the left for i = 2). These symmetric i-pairs are i-spaces exactly when ϕ_{0} and ϕ_{1} is an isomorphism. The symmetric i-pairs of the right-hand column are i-spaces when ϕ is a quasi-isomorphism, i.e. when its cone is an exact complex; these are formations; we call them symmetric if i = −1 and skew-symmetric if i = 1. The four types of i-form presented above will be called short, for the obvious reasons.

2.5. Product of symmetric spaces.

The precise definition of this product is given in [11], where the reader will also find an explanation for the existence of two different products – the left and the right one – which differ by signs. To fix the ideas, we will use here the left product. Let (P_{•}, ϕ) be a symmetric i-form and (Q_{•}, ψ) a symmetric j-form. The product

(P_{•}, ϕ) ⋆ (Q_{•}, ψ)

is then a symmetric (i + j)-form on the tensor product (of complexes) P_{•} ⊗_{O_{X}} Q_{•} and we denote it by (P_{•} ⊗_{O_{X}} Q_{•} , ϕ ⋆ ψ). Up to signs and identifications like for instance P_{•} ⊗_{O_{X}} (Q∨_{•}[j]) ≃ (P_{•} ⊗_{O_{X}} Q∨_{•})[j], the morphism of complexes ϕ ⋆ ψ is equal to the tensor product ϕ ⊗ ψ. Via c_{0}, this product coincides on short 0-spaces with the usual tensor product of symmetric spaces as defined in Knebusch [14].

2.6. Symmetric cones.

We now recall the important cone construction. Let ϕ : P_{•} ⟶ D_{X}(P_{•})[i] be a symmetric i-form (maybe not an isomorphism). Let Q_{•} be the mapping cone of ϕ.

Then, there exists an isomorphism ψ such that the following diagram commutes :

[figure: P_{•}; ϕ; (−1)^{\frac{i(i+1)}{2}} ·ϖ_{P}; ≅; D_{X}(P_{•})[i]; u; =; Q_{•}; v; ≃; ψ; P_{•}[1]; (−1)^{\frac{i(i+1)}{2}} ·ϖ_{P} [1]; ≅; D_{X}D_{X}(P_{•}); D_{X}(ϕ)[i]; D_{X}(P_{•})[i]; −D_{X}(v)[i+1]; D_{X}(Q_{•})[i + 1]; (−1)^{i}D_{X}(u)[i+1]; D_{X}D_{X}(P_{•})[1]]

If the isomorphism ψ is moreover a symmetric (i + 1)-form, we call such a diagram a cone diagram (over ϕ) and we say that (Q_{•}, ψ) is a symmetric cone of the pair (P_{•}, ϕ).

Note that both rows of the diagram are exact triangles in 𝔻^{b}(VB_{X}) : The upper one by definition and the lower one is the dual of the upper row, shifted i times. Assume for a moment that 2 is invertible over our scheme X. Then we can always choose the isomorphism ψ to be a symmetric (i + 1)-form, see [2]. Moreover, if (Q′_{•}, ψ′) is another symmetric cone of ϕ, then there exists an isometry (Q_{•}, ψ) ≃ (Q′_{•}, ψ′). We say then that (Q_{•}, ψ) is the symmetric cone of ϕ, in symbols :

(Q_{•}, ψ) = cone ϕ = cone(P_{•}, ϕ) .

2.7. Witt groups.

The usual Witt group of symmetric (respectively skew-symmetric) spaces W_{us}(X) (respectively W_{us}^{−}(X)) classifies these spaces up to isometry and modulo metabolic ones. More information about these Witt groups can be found in the fundamental paper of Knebusch [14]. The i-th derived Witt group W^{i}(X) classifies symmetric i-spaces up to isometry and modulo neutral spaces, i.e. spaces with Lagrangian (cf. [2], Sect. 2). In fact, a symmetric i-space is neutral exactly if it is a symmetric cone of some symmetric (i − 1)-form, as described above. Observe that this definition does not require 2 to be invertible in X. We denote by [P_{•}, ϕ] the Witt class of (P_{•}, ϕ).

The Witt groups are contravariant functors. If f : Y ⟶ X is a morphism of schemes then the assignment [P_{•}, ϕ] ⟼ [f^{∗}(P_{•}, ϕ)] defines a homomorphism f^{∗} : W^{i}(X) ⟶ W^{i}(Y) for all i ∈ ℤ.

2.8. Periodicity.

The derived Witt groups are 4-periodic. The shift by two: P_{•} ⟼ P_{•}[2] induces an isomorphism τ : W^{i}(X) ⟶^{≃} W^{i+4}(X) for all i ∈ ℤ and all schemes X. The same periodicity applies to the Witt groups with support defined below.

2.9. Agreement.

We assume now that “X contains \frac{1}{2}”, i.e. that X is a ℤ[1/2]-scheme, i.e. 2 is invertible in the ring Γ(X, O_{X}). The main result of [3] is that the functor c_{0} : VB_{X} ⟶ 𝔻^{b}(VB_{X}) induces isomorphisms :

W(X) = W_{us}(X) ⟶^{≃} W^{0}(X) [P, ϕ] ⟼ [c_{0}(P), c_{0}(ϕ)]

and

W^{−}(X) = W_{us}^{−}(X) ⟶^{≃} W^{2}(X) [Q, ψ] ⟼ [c_{0}(Q)[1] , c_{0}(ψ)[1] ] .

2.10. Localization (with \frac{1}{2}).

Other Witt groups appearing in this work are the Witt groups with support. For a complex P_{•} ∈ 𝔻^{b}(VB_{X}) let

supp P_{•} := {x ∈ X | H_{j}(P_{•})_{x} ≠ 0 for at least one j } ,

be its (homological) support. Let Z be a closed subscheme of X with open complement U. The full triangulated subcategory of 𝔻^{b}(VB_{X}) which consists of complexes with support contained in Z is denoted 𝔻_{Z}^{b}(VB_{X}). The restriction of the duality D_{X} to 𝔻_{Z}^{b}(VB_{X}) is again a duality, turning 𝔻_{Z}^{b}(VB_{X}) into a triangulated category with duality. The corresponding triangular Witt groups W_{Z}^{i}(X) (i ∈ ℤ) are called the derived Witt groups of X with support in Z. They appear in the localization sequence of Balmer [2]. If X is a regular scheme then there is an exact sequence

\xymatrix{⋯ \ar[r] & W^{i}(X) \ar[r] & W^{i}(U) \ar[r]^{∂} & W_{Z}^{i+1}(X) \ar[r] & W^{i+1}(X) \ar[r] & · · · .}

The connecting morphism ∂ comes from the cone construction 2.6 as follows. Let w ∈ W^{i}(U). Then ∂(w) = [cone(P_{•}, ϕ)], where (P_{•}, ϕ) is any symmetric i-pair over X with [(P_{•}, ϕ)|_{U}] = w (the existence of (P_{•}, ϕ) is guaranteed by regularity of X). The Witt groups with support are natural and so is the localization sequence.

2.11. The graded Witt ring.

The (left) product of symmetric spaces of 2.5 yields a product structure

⋆ : W^{i}(X) × W_{Z}^{j}(X) ⟶ W_{Z}^{i+j}(X) ( [P_{•}, ϕ] , [Q_{•}, ψ] ) ⟼ [(P_{•}, ϕ) ⋆ (Q_{•}, ψ)]

for any i, j ∈ ℤ, any scheme X and closed subset Z ⊆ X. Via this pairing, W^{tot}(X) := ⨁_{i∈ℤ} W^{i}(X) is a graded skew-commutative associative W^{0}(X)-algebra,

the graded Witt ring of X and W_{Z}^{tot}(X) := ⨁_{i∈ℤ} W_{Z}^{i}(X) is a graded W^{tot}(X)-module.

Remark 2.12. Of course, Convention 1.2 applies here as well. For instance, if X = Spec(R) is affine and Z ⊂ X is defined by the ideal I we might say that a complex “has support in the ideal I” and we shall write W_{I}^{i}(R) instead of W_{Z}^{i}(X).

3. Basic facts about Koszul complexes

In this section, A is a ring, \underline{T} = (T_{1}, …, T_{n}) is any sequence in A, and I :=

∑_{i=1}^{n} AT_{i} is the ideal generated by \underline{T}. As before, we write the dual as M^{∨} :=

Hom_{A}(M, A), for any A-module M.

We first recall the definition of the Koszul complex

K_{•}(A, \underline{T}) =: (K_{•} , d_{•}) .

Let e_{1}, e_{2}, …, e_{n} be a basis of the free A-module A^{n} = ⨁_{i=1}^{n} A · e_{i}. The A-module

K_{i} = K_{i}(A, \underline{T}) := ⋀^{i} A^{n} is by definition the i-th exterior power of A^{n}. As is wellknown, the module K_{i} is free with basis {e_{j_{1}} ∧ ⋯ ∧ e_{j_{i}} | 1 ≤ j_{1} < … < j_{i} ≤ n}.

The differential d_{i} = d_{i}(A, \underline{T}) : K_{i} ⟶ K_{i−1} is given by

e_{j_{1}} ∧ … ∧ e_{j_{i}} ⟼ ∑_{k=1}^{i}(−1)^{k−1} T_{j_{k}} · e_{j_{1}} ∧ … ê_{j_{k}} … ∧ e_{j_{i}} ,

where the symbol ê_{j_{k}} indicates that e_{j_{k}} has been omitted. We consider this (homological) Koszul complex K_{•}(A, \underline{T}) :

\xymatrix{⋯ 0 \ar[r] & K_{n}(A, \underline{T}) \ar[r]^{d_{n}(A,\underline{T})} & K_{n−1}(A, \underline{T}) \ar[r] & … \ar[r]^{d_{1}(A,\underline{T})} & K_{0}(A, \underline{T}) \ar[r] & 0 ⋯}

as an element of 𝔻^{b}(VB_{A}) with K_{j}(A, \underline{T}) in degree j.

There is a structure of symmetric n-space on K_{•}(A, \underline{T}), that we now give in an economic way; see more details in Remark 3.2. For each i = 1, …, n, let K_{•}(A, T_{i}) ∈ 𝔻^{b}(VB_{A}) be the short Koszul complex for the one-element sequence (T_{i}), i.e.

\xymatrix{K_{•}(A, T_{i}) = & … \ar[r] & 0 \ar[r] & A \ar[r]^{·T_{i}} & A \ar[r] & 0 \ar[r] & … \\  &  &  & deg 1 & deg 0 .}

This complex can be equipped with the following symmetric 1-form (see 2.2) :

\xymatrix{K_{•}(A, T_{i}) = \ar[d]^{:=}_{Θ(A,T_{i})} & ⋯ 0 \ar[r] & A \ar[r]^{·T_{i}} \ar[d]_{− id} & A \ar[r] \ar[d]^{id} & 0 ⋯ \\ D_{A}(K_{•}(A, T_{i}))[1] = & ⋯ 0 \ar[r] & A \ar[r]^{·(−T_{i})} & A \ar[r] & 0 ⋯ \\  &  & deg 1 & deg 0 ,}

where we identify A = Hom_{A}(A, A) as usual. This is the cone of the symmetric

form A ⟶^{·(−T_{i})} A = Hom_{A}(A, A), and so in particular a symmetric 1-space. It is easily checked that the tensor product of complexes K_{•}(A, T_{1})⊗_{A} … ⊗_{A} K_{•}(A, T_{n}) is equal to the Koszul complex K_{•}(A, \underline{T}) of the sequence \underline{T} = (T_{1}, …, T_{n}) and therefore we can give the following :

Definition 3.1. With the above notations, we define a symmetric n-form

Θ(A, \underline{T}) : K_{•}(A, \underline{T}) ⟶ D_{A}(K_{•}(A, \underline{T}))[n]

as the product (see 2.5)

(K_{•}(A, \underline{T}) , Θ(A, \underline{T})) := (K_{•}(A, T_{1}), Θ(A, T_{1})) ⋆ … ⋆ (K_{•}(A, T_{n}), Θ(A, T_{n})) .

This defines a symmetric n-space (K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ) which we call the canonical space on the Koszul complex K_{•}(A, \underline{T}).

Remark 3.2. To define this canonical space on K_{•}(A, \underline{T}), it is not necessary to use the product structure of the derived Witt groups. The advantage of this approach is that we see at once that the canonical n-space is a symmetric n-space, but for calculations in the sequel it might be useful to have a good description of the symmetric n-form Θ(A, \underline{T}). We define an isomorphism

ρ : K_{•}(A, \underline{T}) ⟶^{≃} D_{A}(K_{•}(A, \underline{T}))[n]

following [7], Sect. 1.6. We fix for this an isomorphism ω : ⋀^{n}(A^{n}) ⟶^{≃} A, and define an A-bilinear pairing

b_{i} : K_{i}(A, \underline{T}) × K_{n−i}(A, \underline{T}) ⟶ A

by (x, y) ⟼ ω(x ∧ y) for all 0 ≤ i ≤ n. This b_{i} induces a homomorphism ϱ_{i} : K_{i}(A, \underline{T}) ⟶ Hom_{A}(K_{n−i}(A, \underline{T}), A) = K_{n−i}(A, \underline{T})^{∨} which is an isomorphism for all 0 ≤ i ≤ n. It is straightforward to check (see [7, Prop. 1.6.10] if necessary) that

d_{n−(i−1)}(A, \underline{T})^{∨} · ϱ_{i} = (−1)^{i−1}ϱ_{i−1} · d_{i}(A, \underline{T}) .

Consider the family of morphisms (ρ_{i})_{i∈ℤ} defined by ρ_{i} := (−1)^{\frac{i(i+1)}{2} + \frac{n(n−1)}{2}} · ϱ_{i} for 0 ≤ i ≤ n and by ρ_{i} = 0 otherwise. This defines an isomorphism of complexes ρ = ρ_{•} : K_{•}(A, \underline{T}) ⟶ D_{A}(K_{•}(A, \underline{T}))[n], which coincides with the morphism of complexes Θ(A, \underline{T}) as a thrilling calculation using (in particular the sign conventions of) [11, Ex. 1.4, Rem. 1.9] shows.

By the following lemma this is easier to see if \underline{T} is a regular sequence, which is the only interesting case for us here.

Lemma 3.3. Assume that \underline{T} is a regular sequence. Identify A ≅ Hom_{A}(A, A) as usual. Then for any morphism in 𝔻^{b}(VB_{A}) between the Koszul complex and its n-dual

ς : K_{•}(A, \underline{T}) ⟶ D_{A}(K_{•}(A, \underline{T}))[n] ,

there exists an s ∈ A such that ς = s · Θ(A, \underline{T}) in 𝔻^{b}(VB_{A}). If s′ ∈ A is another element with this property then s − s′ ∈ I. The morphism ς is an isomorphism in 𝔻^{b}(VB_{A}) if and only if s + I is a unit in the quotient ring A/I.

Proof. By assumption \underline{T} = (T_{1}, …, T_{n}) is a regular sequence and so the complex K_{•}(A, \underline{T}) and its dual D_{A}(K_{•}(A, \underline{T}))[n] are A-free resolutions of A/I by [7, Prop. 1.6.10 and Cor. 1.6.14]. The lemma follows because Θ(A, \underline{T}) is an isomorphism of complexes. □

Remark 3.4. It is clear that the restriction of K_{•}(A, \underline{T}) becomes zero in the derived category 𝔻^{b}(VB_{A[T_{j}^{−1}]}) for all 1 ≤ j ≤ n. Hence the complex K_{•}(A, \underline{T}) has support in the closed subset of Spec(A) defined by the ideal I = ∑_{j=1}^{n} A T_{j}. Therefore the symmetric n-space (K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ) defines an element in

[K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] ∈ W_{I}^{n}(A) .

Proposition 3.5. Let 1 ≤ i ≤ n. Define the ideal I_{i} := ∑_{k≠i} A T_{k} of A. Then

[K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] = 0 in W_{I_{i}}^{n}(A).

Proof. The group W_{I_{i}}^{n−1}(A) obviously contains the element

y := [K_{•}(A, T_{1}), Θ(A, T_{1})] ⋆ … ⋆ [K_{•}(A, T_{i−1}), Θ(A, T_{i−1})]

⋆ [K_{•}(A, T_{i+1}), Θ(A, T_{i+1})] ⋆ … ⋆ [K_{•}(A, T_{n}), Θ(A, T_{n})] .

Since the product is skew-commutative, we have

[K_{•}(A, T_{i}), Θ(A, T_{i})] ⋆ y = (−1)^{i−1}[K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] ,

where we consider [K_{•}(A, T_{i}), Θ(A, T_{i})] as an element of W^{1}(A). Therefore the result follows from the observation that this element is indeed zero in W^{1}(A).

In fact, the complex c_{0}(A) ∈ 𝔻^{b}(VB_{A}) is a Lagrangian (cf. [2], Sect. 2) of the symmetric 1-space (K_{•}(A, T_{i}), Θ(A, T_{i})) :

\xymatrix{c_{0}(A) = \ar[d] & … \ar[r] & 0 \ar[r] \ar[d] & 0 \ar[r] \ar[d] & A \ar[r] \ar[d]^{id} & 0 \ar[r] \ar[d] & … \\ K_{•}(A, T_{i}) = & … \ar[r] & 0 \ar[r] & A \ar[r]^{·T_{i}} & A \ar[r] & 0 \ar[r] & …}

and so [K_{•}(A, T_{i}), Θ(A, T_{i})] = 0 in W^{1}(A). □

In the above proof, note that the class y ∈ W_{I_{i}}^{n−1}(A) does not belong to W_{I}^{n−1}(A) and thus the argument cannot be used to deduce that [K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] = 0 in W_{I}^{n}(A). On the contrary, see Theorem 8.2.

Corollary 3.6. [K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] = 0 in W^{n}(A).

Proof. Clear since W_{I}^{n}(A) ⟶ W^{n}(A) factors via W_{I_{1}}^{n}(A) for instance. □

Remark 3.7. Our “canonical” Koszul symmetric space [K_{•}(A, \underline{T}) , Θ(A, \underline{T}) ] is only canonical up to sign. In fact, its definition obviously depends on sign conventions, as usual when working in derived categories, as well as some personal sign choices like e.g. in the definition of the symmetric 1-form on the Koszul complex of length one, at the beginning of this section. The reader should not consider this sign question as crucial and can as well use his or her own set of conventions. Applying Lemma 3.3 to A = ℤ[T_{1}, …, T_{n}] and to \underline{T} = (T_{1}, …, T_{n}), as we shall do below, we get A/I = ℤ and no other unit than ±1 can really enter the game.

Indeed, our main result Theorem 8.13 says that for any regular scheme X of finite Krull dimension the total Witt ring W^{tot}(𝕌_{X}^{n}) is a graded W^{tot}(X)-algebra with only one generator ε which satisfies the relation ε^{2} = 0 if n ≥ 2 and ε^{2} = 1 if n = 1, so, this result clearly does not depend on any choice yielding a sign change in the definition of ε. A reader who prefers different sign conventions would maybe get from our construction −ε as generator which obviously satisfies the same relation as ε. Once again, these signs are not really relevant and we could have written everything “up to sign” using Lemma 3.3 extensively. We only carry the (hopefully) exact signs throughout the paper for sake of consistency.

4. Koszul cut in two

We want to “split” the Koszul complex of Section 3 into two pieces, dual to each other. This is easy to understand but a little tricky to write. Recall our running conventions of Section 1 that r + 4 q = n − 1, see (1), and that ℓ := [\frac{n}{2}], see (2). Now, more precisely, we want to define a symmetric r-pair ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ), such that there is an isometry

cone(C_{•}(A, \underline{T}), Ξ(A, \underline{T})) ≃ ( K_{•}(A, \underline{T}), Θ(A, \underline{T}) ) [−2q] .

We abbreviate the canonical form on K_{•} := K_{•}(A, \underline{T}) by Θ := Θ(A, \underline{T}), and set

S = S(A, \underline{T}) := Coker ( K_{ℓ+2}(A, \underline{T}) ⟶^{d_{ℓ+2}(A,\underline{T})} K_{ℓ+1}(A, \underline{T}) ) .  (4)

Let pr_{S} = pr_{S(A,\underline{T})} : K_{ℓ+1} ⟶ S = Coker d_{ℓ+2} be the projection. Since d_{ℓ+1}d_{ℓ+2} = 0 there exists a unique morphism d̄_{ℓ+1} = d̄_{ℓ+1}(A, \underline{T}) : S ⟶ K_{ℓ}, such that

d_{ℓ+1}(A, \underline{T}) = d̄_{ℓ+1}(A, \underline{T}) · pr_{S} .  (5)

For each j = 0, …, n, we have rank_{A}(K_{j}) = \binom{n}{j}. In particular, if n = 2ℓ + 1 is odd, we have rank_{A} K_{ℓ} = rank_{A} K_{ℓ+1} and life will be easy. When n = 2ℓ is even, K_{ℓ} has maximal (even) rank \binom{2ℓ}{ℓ} and we need some preparatory considerations. In this case, the symmetric n-form Θ_{•} : K_{•} ⟶^{≃} D_{A}(K_{•})[n] gives an isomorphism

Θ_{ℓ} : K_{ℓ} ⟶ K_{ℓ}^{∨} = Hom_{A}(K_{ℓ}, A)

which is symmetric if ℓ is even and skew-symmetric otherwise.

Lemma 4.1. If n = 2ℓ is even, there exists two totally isotropic subspaces L and M of (K_{ℓ}, Θ_{ℓ}), of same rank \frac{1}{2} \binom{2ℓ}{ℓ}, such that K_{ℓ} = L⊕M and such that Θ_{ℓ} becomes

Θ_{ℓ} = ( \begin{matrix}0 & (−1)^{ℓ}λ^{∨} can_{M} \\ λ & 0 \end{matrix}) : K_{ℓ} = L ⊕ M ⟶ L^{∨} ⊕ M^{∨} = K_{ℓ}^{∨} ,

where λ : L ⟶^{≃} M^{∨} is an isomorphism. Moreover, we have

(−1)^{ℓ}d_{ℓ+1}^{∨} · (pr_{L})^{∨} · λ^{∨} · can_{M} · pr_{M} ·d_{ℓ+1} + d_{ℓ+1}^{∨} · (pr_{M})^{∨} · λ · pr_{L} ·d_{ℓ+1} = 0 , (6)

where pr_{L} : K_{ℓ} ⟶ L and pr_{M} : K_{ℓ} ⟶ M denote the projections.

Proof. Let e_{1}, …, e_{n} be a basis of A^{n} and define L := e_{1} ∧ ⋀^{ℓ−1} A^{n}. The space

M := ⨁ ; 2 ≤ i_{1} < i_{2} < … < i_{ℓ} ≤ n A · e_{i_{1}} ∧ e_{i_{2}} ∧ … ∧ e_{i_{ℓ}}

is obviously a complement of L in K_{ℓ} = ⋀^{ℓ} A^{n} and both subspaces have rank \binom{2ℓ−1}{ℓ−1} = \binom{2ℓ−1}{ℓ} = \frac{1}{2} \binom{2ℓ}{ℓ}. Now use the description of Θ_{ℓ} given in Remark 3.2. Let

ω : ⋀^{n} A^{n} ⟶^{≃} A be the isomorphism which sends e_{1} ∧ … ∧ e_{n} to 1 ∈ A. Then Θ_{ℓ}(x)(y) = ±ω(x ∧ y). From this we easily get that both subspaces are totally isotropic : for L it is because e_{1} ∧ e_{1} = 0 and for M it is because two subsets with ℓ elements in {2, …, n} must intersect. Since Θ_{ℓ} is a (−1)^{ℓ}-symmetric isomorphism, its decomposition in L⊕M must be as claimed in the lemma. Equation (6) follows from the fact that Θ : K_{•} ⟶ D_{A}(K_{•})[n] is a morphism of complexes. □

Definition 4.2. As the above discussion shows, we will have to distinguish the cases where n is odd from those where n is even and the definition extends over 4.3 - 4.6 below. We shall consider a sign ϵ_{n} ∈ {−1, 1} which will be fixed later on, see 6.2.

We start with n = 2ℓ + 1 odd.

4.3. Case \underline{r = 0}.

Then ℓ = 2q is even and ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) is defined to be the following symmetric 0-pair :

\xymatrix{0 \ar[r] & K_{n} \ar[r]^{d_{n}} \ar[d] & ⋯ \ar[r] & K_{ℓ+2} \ar[r]^{d_{ℓ+2}} \ar[d] & K_{ℓ+1} \ar[r] \ar[d]_{ϵ_{n}Θ_{ℓ}·d_{ℓ+1}} & 0 \ar[r] \ar[d] & ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & K_{ℓ+1}^{∨} \ar[r]^{d_{ℓ+2}^{∨}} & K_{ℓ+2}^{∨} \ar[r] & ⋯ \ar[r]^{d_{n}^{∨}} & K_{n}^{∨} \ar[r] & 0 \\  & deg ℓ &  &  & deg 0 .}

If \underline{T} is a regular sequence then the Koszul complex K_{•} is exact and so we have the following quasi-isomorphism

[figure: C_{•}(A, \underline{T}) =; p=p(A,\underline{T}); :=; 0; K_{n}; d_{n}; ⋯; K_{ℓ+2}; d_{ℓ+2}; K_{ℓ+1}; _{pr_{S}}; 0; c_{0}(S) =; 0; 0; ⋯; 0; S; 0; deg ℓ; deg 0 .]

4.4. Case \underline{r = 2}.

Then ℓ = 2q + 1 is odd and ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) is defined to be the following symmetric 2-pair :

\xymatrix{0 \ar[r] & K_{n} \ar[r]^{d_{n}} \ar[d] & ⋯ \ar[r] & K_{ℓ+2} \ar[r]^{d_{ℓ+2}} \ar[d] & K_{ℓ+1} \ar[r] \ar[d]_{− ϵ_{n}Θ_{ℓ}·d_{ℓ+1}} & 0 \ar[r] \ar[d] & ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & K_{ℓ+1}^{∨} \ar[r]^{d_{ℓ+2}^{∨}} & K_{ℓ+2}^{∨} \ar[r] & ⋯ \ar[r]^{d_{n}^{∨}} & K_{n}^{∨} \ar[r] & 0 \\  & deg ℓ + 1 &  &  & deg 1 .}

As above if \underline{T} is a regular sequence the projection pr_{S} : K_{ℓ+1} ⟶ S induces a quasi-isomorphism of complexes p = p(A, \underline{T}) : C_{•}(A, \underline{T}) ⟶ c_{0}(S)[1].

Now let n = 2ℓ be even.

We fix two totally isotropic subspaces L and M of K_{ℓ} and an isomorphism λ : L ⟶ M^{∨} as in Lemma 4.1 and keep notations as there. We set

h := λ^{∨} · can_{M} · pr_{M} ·d_{ℓ+1} : K_{ℓ+1} ⟶ L^{∨} .

We now define the space ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) for n even. It follows from equation (6) in Lemma 4.1 that both squares in the middle of the two diagrams below commute and so the morphism Ξ(A, \underline{T}) is really a morphism of complexes.

4.5. Case \underline{r = −1}.

Then ℓ = 2q is even and ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) is defined to be the following symmetric (−1)-pair :

\xymatrix{0 \ar[r] & K_{n} \ar[r]^{d_{n}} \ar[d] & ⋯ \ar[r] & K_{ℓ+2} \ar[r]^{d_{ℓ+2}} \ar[d] & K_{ℓ+1} \ar[r]^{pr_{L} d_{ℓ+1}} \ar[d]_{ϵ_{n}h} & L \ar[r] \ar[d]^{ϵ_{n}h^{∨} can_{L}} & 0 ⋯ \ar[r] \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & L^{∨} \ar[r]_{−(pr_{L} d_{ℓ+1})^{∨}} & K_{ℓ+1}^{∨} \ar[r]_{−d_{ℓ+2}^{∨}} & ⋯ \ar[r]_{−d_{n}^{∨}} & K_{n}^{∨} \ar[r] & 0 \\  & deg ℓ − 1 &  &  & deg 0 & deg −1 .}

If the sequence \underline{T} is regular the homology of C_{•}(A, \underline{T}) is not concentrated in one degree (as in the case n odd) but there exists a “short” complex F_{•}(A, \underline{T}) defined as follows and which is quasi-isomorphic to C_{•}(A, \underline{T}) :

\xymatrix{C_{•}(A, \underline{T}) = \ar[d]_{p=p(A,\underline{T})}^{:=} & ⋯ \ar[r] & K_{ℓ+2} \ar[r]^{d_{ℓ+2}} \ar[d] & K_{ℓ+1} \ar[r]^{pr_{L} ·d_{ℓ+1}} \ar[d]^{_{pr_{S}}} & L \ar[r] \ar[d]^{=} & 0 \ar[r] \ar[d] & ⋯ \\ F_{•}(A, \underline{T}) := & ⋯ \ar[r] & 0 \ar[r] & S \ar[r]_{pr_{L} ·d̄_{ℓ+1}} & L \ar[r] & 0 \ar[r] & ⋯ \\  &  &  & deg 0 & deg −1 .}

4.6. Case \underline{r = 1}.

Then ℓ = 2q + 1 is odd and ( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) is defined to be the following symmetric 1-pair :

\xymatrix{0 \ar[r] & K_{n} \ar[r]^{d_{n}} \ar[d] & ⋯ \ar[r] & K_{ℓ+2} \ar[r]^{d_{ℓ+2}} \ar[d] & K_{ℓ+1} \ar[r]^{pr_{L} d_{ℓ+1}} \ar[d]_{ϵ_{n}h} & L \ar[r] \ar[d]^{−ϵ_{n}h^{∨} can_{L}} & 0 ⋯ \ar[r] & 0 \ar[r] \ar[d] & 0 \\ 0 \ar[r] & 0 \ar[r] & ⋯ \ar[r] & 0 \ar[r] & L^{∨} \ar[r]_{−(pr_{L} d_{ℓ+1})^{∨}} & K_{ℓ+1}^{∨} \ar[r]_{−d_{ℓ+2}^{∨}} & ⋯ \ar[r]_{−d_{n}^{∨}} & K_{n}^{∨} \ar[r] & 0 \\  & deg ℓ &  &  & deg 1 & deg 0 .}

As in the case r = −1, when \underline{T} is a regular sequence, we have a quasi-isomorphism p = p(A, \underline{T}) : C_{•}(A, \underline{T}) ⟶ F_{•}(A, \underline{T}), where F_{•}(A, \underline{T}) is now the complex

\xymatrix{⋯ \ar[r] & 0 \ar[r] & S \ar[r]^{pr_{L} ·d̄_{ℓ+1}} & L \ar[r] & 0 \ar[r] & ⋯ \\  &  & deg 1 & deg 0 .}

Remark 4.7. Let f : A ⟶ B be a morphism of rings. The natural isomorphism

f^{∗}(K_{•}(A, \underline{T})) = K_{•}(A, \underline{T}) ⊗_{A} B ⟶^{≃} K_{•}(B, f(\underline{T}))

induces a natural isometry

f^{∗}( K_{•}(A, \underline{T}), Θ(A, \underline{T}) ) ≃ ( K_{•}(B, f(\underline{T})), Θ(B, f(\underline{T})) ) .

Restricting this isomorphism above to the subcomplex C_{•}(A, \underline{T}) we get a natural isometry f^{∗}( C_{•}(A, \underline{T}) , Ξ(A, \underline{T}) ) ≃ ( C_{•}(B, f(\underline{T})) , Ξ(B, f(\underline{T})) ).

Lemma 4.8. The mapping cone of the morphism Ξ(A, \underline{T}) is isomorphic (as a complex) to K_{•}(A, \underline{T})[−2q].

Proof. This is an easy direct computation, which we leave to the reader. It is clear in the cases where n is odd and it requires Lemma 4.1 for n even. In all four cases, we use the isomorphism Θ to replace the K_{j}^{∨} by K_{n−j} for j ≥ ℓ + 1. □

5. The Koszul symmetric space KX^{(n)} over 𝔸_{X}^{n}

Let R be a ring. We apply the constructions of Section 3 to A := R[T_{1}, …, T_{n}], the polynomial ring in n variables over R, and to the sequence \underline{T} := (T_{1}, …, T_{n}). The reader can think of R = ℤ or R = ℤ[1/2], since these are the important cases, from which the rest will be deduced.

Definition 5.1. The Koszul symmetric n-space KR^{(n)} = (KR^{(n)} , ΘR^{(n)}) over 𝔸_{R}^{n} is the symmetric n-space where KR^{(n)} := K_{•}(A, \underline{T}) will be called the Koszul complex over 𝔸_{R}^{n} and where the symmetric n-form ΘR^{(n)} := Θ(A, \underline{T}) is the one of Definition 3.1.

Remark 5.2. Pay attention : KR^{(n)} is a symmetric n-space defined over the ring A = R [T_{1}, …, T_{n}] and not over the ring R, as the notation might suggest.

It is clear that the Koszul symmetric n-space behaves well with respect to basechange. More precise, let f : R ⟶ R′ be a morphism of rings and let

α_{f} : R [T_{1}, …, T_{n}] ⟶ R′ [T_{1}, …, T_{n}]

be the obvious induced morphism. Then, by Remark 4.7 there is a natural isometry

α_{f}^{∗}(KR^{(n)}) ⟶^{≃} KR′^{(n)} .

In particular, KR^{(n)} is extended from Kℤ^{(n)}. This justifies the following extension of Definition 5.1.

Definition 5.3. Let X be a scheme. We define the symmetric n-space

KX^{(n)} := α_{X}^{∗}(Kℤ^{(n)})

where α_{X} : 𝔸_{X}^{n} ⟶ 𝔸_{ℤ}^{n} is the base-change morphism, see 1.4. We call KX^{(n)} the Koszul symmetric n-space over 𝔸_{X}^{n}. Like before, we denote the underlying complex of free O_{𝔸_{X}^{n}} -modules and its symmetric n-form by

KX^{(n)} = α_{X}^{∗}(Kℤ^{(n)})  and  ΘX^{(n)} = α_{X}^{∗}(Θℤ^{(n)}) .

Remark 5.4. It is obvious from the definition that for any morphism of schemes f : Y ⟶ X we have an isometry α_{f}^{∗}(KX^{(n)}) ≃ KY^{(n)} over 𝔸_{Y}^{n}

Definition 5.5. By Remark 3.4, the complex KX^{(n)} has support in the closed subset 𝔸_{X}^{n} ∖ 𝕌_{X}^{n} of 𝔸_{X}^{n} which we identify with X in the following via the zero section of the bundle 𝔸_{X}^{n} ⟶ X. Therefore, the symmetric n-space KX^{(n)} represents a Witt class

κX^{(n)} := [KX^{(n)}] ∈ W_{𝔸_{X}^{n}∖𝕌_{X}^{n}}^{n} (𝔸_{X}^{n}) = W_{X}^{n}(𝔸_{X}^{n}) .

6. The half-Koszul symmetric space EX^{(n)} over 𝕌_{X}^{n}

Definition 6.1. Let R be a ring. We now apply the splitting of Section 4 to the space KR^{(n)} of Section 5. As above, we put A := R [T_{1}, …, T_{n}] and \underline{T} := (T_{1}, …, T_{n}). We define

CR^{(n)} := C_{•}(A, \underline{T}) and ΞR^{(n)} := Ξ(A, \underline{T})

as defined in 4.3 to 4.6. For any scheme X we define

CX^{(n)} := α_{X}^{∗}(Cℤ^{(n)})  and  ΞX^{(n)} := α_{X}^{∗}(Ξℤ^{(n)})

where α_{X} : 𝔸_{X}^{n} ⟶ 𝔸_{ℤ}^{n} is the base-change morphism. This coincides with the above in the affine case by Remark 4.7. For all n ∈ ℕ, the symmetric r-pair (CX^{(n)}, ΞX^{(n)}) on 𝔸_{X}^{n} will be denoted by CX^{(n)}.

6.2. The symmetric cone of CX^{(n)}.

Instead of calculating cone(CX^{(n)}) directly (which is possible, but cumbersome) we take full advantage of Lemma 3.3. More precisely, we use the fact that any quasi-isomorphism K_{•}(ℤ, \underline{T}) ⟶ D_{ℤ[T_{1},...,T_{n}]}(K_{•}(ℤ, \underline{T}))[n] is equal to the symmetric

n-form ±Θℤ^{(n)} in 𝔻^{b}(VB_{ℤ[T_{1},...,T_{n}]}).

So let for a moment R = ℤ and A = ℤ[T_{1}, …, T_{n}]. We abbreviate K_{•} := K_{•}(A, \underline{T}) and Θ = Θ(A, \underline{T}) . We get from Lemma 4.8 the following commutative diagram (where D = D_{A})

\xymatrix{Cℤ^{(n)} \ar[r]^{Ξ_{ℤ}^{(n)}} \ar[d]^{(−1)^{\frac{r(r+1)}{2}} ϖ_{C{(n) \atop ℤ}}} & D(Cℤ^{(n)})[r] \ar[r]^{u} \ar[d]^{=} & K_{•}[−2q] \ar[r]^{v} & Cℤ^{(n)}[1] \ar[d]_{(−1)^{\frac{r(r+1)}{2}} ϖ_{C{(n) \atop ℤ}} [1]} \\ DD(Cℤ^{(n)}) \ar[r]_{D(Ξ_{ℤ}^{(n)})[r]} & D(Cℤ^{(n)})[r] \ar[r]_{−D(v)[r+1]} & D(K_{•}[−2q])[r + 1] \ar[r]_{(−1)^{r}D(u)[r+1]} & DD(Cℤ^{(n)})[1]}

(7)

whose rows are exact triangles for all n ∈ ℕ (the bottom row is the dual of the upper one, shifted r times). By the very basic properties of triangulated categories there exists an isomorphism

ς : K_{•}[−2q] ⟶ D(K_{•}[−2q])[r + 1] = (D(K_{•})[n])[−2q] ,

in 𝔻^{b}(VB_{A}) such that diagram (7) commutes. By Lemma 3.3 the isomorphism ς is equal to ±Θ[−2q]. Replacing if necessary Ξℤ^{(n)} by −Ξℤ^{(n)}, i.e. replacing ϵ_{n} by −ϵ_{n} in the definition of CR^{(n)}, we can assume that ς = Θ[−2q] for all n ∈ ℕ, i.e.

(K_{•}[−2q], ς) = Kℤ^{(n)}[−2q] for all n ∈ ℕ.

We fix ϵ_{n} as explained above, namely ϵ_{n} is the unique element in {−1, 1}, such that cone(Cℤ^{(n)}) = Kℤ^{(n)}[−2q].

Remark 6.3. A straightforward but slightly cumbersome calculation shows that

ϵ_{n} = (−1)^{\frac{n(n+1)}{2}} .  (8)

Note that this depends on the sign choices made in 4.3–4.6. The latter have been made such that Formula (8) is true.

However as already said in Remark 3.7 these signs are not important for our work. We only need the fact that there exists a symmetric r-pair Cℤ^{(n)} over 𝔸_{ℤ}^{n}, such that cone Cℤ^{(n)} = Kℤ^{(n)}[−2q].

We can now calculate cone(CX^{(n)}) for any scheme X and any n ∈ ℕ. The pull-back via the base-change morphism α_{X} : 𝔸_{X}^{n} ⟶ 𝔸_{ℤ}^{n} of diagram (7) above is a cone diagram for the symmetric r-form α_{X}^{∗}(Cℤ^{(n)}). We have an isometry CX^{(n)} ≃ α_{X}^{∗}(Cℤ^{(n)}) (cf. Remark 4.7) and so we get cone(CX^{(n)}) ≃ α_{X}^{∗}(Kℤ^{(n)}[−2q]) ≃ α_{X}^{∗}(Kℤ^{(n)})[−2q] (cf. Lemma B.1 for the later isometry). We have proven :

Theorem 6.4. With this choice of ϵ_{n}, the cone of the symmetric pair CX^{(n)} is the Koszul symmetric space shifted as follows :

cone(CX^{(n)}) = KX^{(n)}[−2q] .

□

In particular, ΞX^{(n)}|_{𝕌_{X}^{n}} is an isomorphism in 𝔻^{b}(VB_{𝕌_{X}^{n}}) because the homology of K_{•}|_{𝕌_{X}^{n}} vanishes.

Definition 6.5. Let X be a scheme. The symmetric r-space

EX^{(n)} := CX^{(n)}|_{𝕌_{X}^{n}}

will be called the half-Koszul space over the scheme X. Its Witt class is denoted by

εX^{(n)} := [EX^{(n)}] ∈ W^{r}(𝕌_{X}^{n}) .

The following is obvious (cf. Remark 4.7).

Lemma 6.6. Let f : Y ⟶ X be a morphism of schemes. Then there is a natural

isometry υ_{f}^{∗}(εX^{(n)}) ⟶^{≃} εY^{(n)}. □

7. The short symmetric space FX^{(n)} over 𝕌_{X}^{n}

By the main result of [3] we know that EX^{(n)} is Witt equivalent to a space living on a short complex, see 2.4. In fact, we will now see that EX^{(n)} is not only Witt equivalent, but isometric to such a “short symmetric space”.

We use the notation of 4.3–4.6 with R = ℤ, i.e. A = ℤ[T_{1}, …, T_{n}] is the polynomial ring in n variables over ℤ, \underline{T} = (T_{1}, …, T_{n}), and K_{•} = K_{•}(A, \underline{T}) is the Koszul complex of the sequence \underline{T} over A. As in 4.3–4.6 we denote the differential of this Koszul complex by d_{•} and set

S = S(A, \underline{T}) = Coker ( K_{ℓ+2} ⟶^{d_{ℓ+2}} K_{ℓ+1} ) .

Note that \underline{T} is a regular sequence and so K_{•} is a finite free resolution of ℤ ≃ A/I, where I is the ideal generated by \underline{T}. It follows that K_{•}(A, \underline{T})|_{Spec A[T_{i}^{−1}]} is a split exact sequence and so

Eℤ^{(n)} := Coker ( K_{ℓ+2} ⟶^{d_{ℓ+2}} K_{ℓ+1} )|_{𝕌_{ℤ}^{n}} = S|_{𝕌_{ℤ}^{n}} ≃ Ker d_{ℓ}|_{𝕌_{ℤ}^{n}}  (9)

is a locally free O_{𝕌_{ℤ}^{n}}-module of rank ∑_{i=0}^{ℓ}(−1)^{i}\binom{n}{ℓ−i} = \binom{n−1}{n−ℓ−1}. Clearly the same

is true for the pull-back

EX^{(n)} := υ_{X}^{∗}(Eℤ^{(n)}) ,  (10)

where υ_{X} : 𝕌_{X}^{n} ⟶ 𝕌_{ℤ}^{n} is induced by base change, see (3). Note that we have

EX^{(n)} = Coker ( KX,ℓ+2^{(n)} ⟶ KX,ℓ+1^{(n)} )|_{𝕌_{X}^{n}} ≃ Ker ( KX,ℓ^{(n)} ⟶ KX,ℓ−1^{(n)} )|_{𝕌_{X}^{n}} ,

where KX^{(n)} = KX,•^{(n)} = α_{X}^{∗}(Kℤ^{(n)}) (see Definition 5.3).

We consider now the case n odd and n even separately.

7.1. The space EX^{(n)} if n = 2ℓ + 1 is odd, i.e. r = 0 or r = 2.

Since the functor (−)^{∨} = Hom_{A}( − , A) is left exact we have S^{∨} = Ker ( K_{ℓ+1}^{∨} ⟶^{d_{ℓ+2}^{∨}} K_{ℓ+2}^{∨} ) and hence a well defined homomorphism

φℤ^{(n)} := (−1)^{ℓ} ϵ_{n} · θ_{ℓ} · d̄_{ℓ+1}|_{𝕌_{ℤ}^{n}} : Eℤ^{(n)} ⟶ Hom_{O_{𝕌^{n}_{ℤ}}} (Eℤ^{(n)}, O_{𝕌_{ℤ}^{n}}) = S^{∨}|_{𝕌_{ℤ}^{n}}

which is (−1)^{ℓ}-symmetric, where d̄_{ℓ+1} is the unique morphism K_{ℓ+1} ⟶ S, such that d_{ℓ+1} = d̄_{ℓ+1} · pr_{S} (cf. (5)). We set

(Fℤ^{(n)}, ϕℤ^{(n)}) := \{ \begin{matrix}c_{0}(Eℤ^{(n)}, φℤ^{(n)}) & if ℓ is even \\ c_{0}(Eℤ^{(n)}, φℤ^{(n)})[1] & if ℓ is odd. \end{matrix}

This is a symmetric r-pair. Recall now from 4.3–4.6 that the projection pr_{S} : K_{ℓ+1} ⟶ S induces a quasi-isomorphism (hence an isomorphism in 𝔻^{b}(VB_{A}))

p|_{𝕌_{ℤ}^{n}} = p(ℤ, \underline{T})|_{𝕌_{ℤ}^{n}} : Cℤ^{(n)} ⟶^{≃} c_{0}(Eℤ^{(n)}) (respectively Cℤ^{(n)} ⟶^{≃} c_{0}(Eℤ^{(n)})[1]),

which is easily seen to be an isometry Eℤ^{(n)} ⟶^{≃} (Fℤ^{(n)}, ϕℤ^{(n)}). It follows that ϕℤ^{(n)} is an isomorphism and so (Fℤ^{(n)}, ϕℤ^{(n)}) a symmetric r-space. In particular (Eℤ^{(n)}, φℤ^{(n)}) is a (−1)^{ℓ}-symmetric space over 𝕌_{ℤ}^{n}. Applying the pull-back υ_{X}^{∗} we get :

(i) The pair

(EX^{(n)}, φX^{(n)}) := υ_{X}^{∗}(Eℤ^{(n)}, φℤ^{(n)})

is a (−1)^{ℓ}-symmetric space over 𝕌_{X}^{n}. (ii) The half Koszul space EX^{(n)} is isometric to the short symmetric r-space:

FX^{(n)} := (FX^{(n)}, ϕX^{(n)}) := υ_{X}^{∗}(Fℤ^{(n)}, ϕℤ^{(n)}) = \{ \begin{matrix}c_{0}(EX^{(n)}, φX^{(n)}) & if ℓ is even \\ c_{0}(EX^{(n)}, φX^{(n)})[1] & if ℓ is odd. \end{matrix}

7.2. The space EX^{(n)} if n = 2ℓ is even, i.e. r = −1 or r = 1.

We fix L, M ⊂ K_{ℓ} and λ : L ⟶^{≃} M^{∨} as in Lemma 4.1 (with R = ℤ), and let pr_{L} : K_{ℓ} ⟶ L and pr_{M} : K_{ℓ} ⟶ M be the respective projections. We denote L := L|_{𝕌_{ℤ}^{n}} and pr_{L} := pr_{L} |_{𝕌_{ℤ}^{n}} : K_{ℓ}|_{𝕌_{ℤ}^{n}} ⟶ L.

On the complex Fℤ^{(n)} := F_{•}(A, \underline{T})|_{𝕌_{ℤ}^{n}} we have the following symmetric r-form:

[figure: Fℤ^{(n)} =; ϕ_{ℤ}^{(n)}; :=; ⋯; 0; Eℤ^{(n)}; pr_{L} ·d̄_{ℓ+1}|_{𝕌^{n}_{ℤ}}; ϵ_{n}h̄|_{𝕌^{n}_{ℤ}}; L; (−1)^{ℓ}ϵ_{n}(h̄|_{𝕌^{n}_{ℤ}})^{∨} can_{L}; D_{𝕌_{ℤ}^{n}}(Fℤ^{(n)})[(−1)^{ℓ+1}] =; ⋯; 0; L^{∨}; (pr_{L} ·d̄_{ℓ+1}|_{𝕌^{n}_{ℤ}})^{∨}; (Eℤ^{(n)})^{∨}; if r = −1; deg 0; deg −1; if r = 1; deg 1; deg 0]

where h̄ = λ · can_{M} · pr_{M} ·d̄_{ℓ+1}. Since d_{ℓ+1}|_{𝕌_{ℤ}^{n}} = (d̄_{ℓ+1} · pr_{S})|_{𝕌_{ℤ}^{n}} we see that the quasi-isomorphism p|_{𝕌_{ℤ}^{n}} : Fℤ^{(n)} ⟶^{≃} Cℤ^{(n)} is an isometry Eℤ^{(n)} ⟶^{≃} (Fℤ^{(n)}, ϕℤ^{(n)}), and so ϕℤ^{(n)} is an isomorphism in 𝔻^{b}(VB_{ℤ}). Therefore (Fℤ^{(n)}, ϕℤ^{(n)}) is a symmetric r-space over 𝕌_{ℤ}^{n}. Applying the pull-back υ_{X}^{∗} we get :

The half Koszul space EX^{(n)} is isometric to the short symmetric r-space:

FX^{(n)} := (FX^{(n)}, ϕX^{(n)}) := υ_{X}^{∗}(Fℤ^{(n)}, ϕℤ^{(n)}) .

8. Witt groups of the punctured affine space

Recall the notations of Section 1, like formula (3), defining r ∈ {−1, 0, 1, 2} by n = 4 q + r + 1. We begin with an easy application of triangular Witt theory :

Theorem 8.1. Let X be a regular scheme containing \frac{1}{2}. There exists a split short exact sequence

\xymatrix{0 \ar[r] & W^{i}(X) \ar[r]^{σ_{X}^{∗}} & W^{i}(𝕌_{X}^{n}) \ar[r]^{∂} & W_{X}^{i+1}(𝔸_{X}^{n}) \ar[r] & 0 ,}

for all i ∈ ℤ, where ∂ is the connecting homomorphism of the localization 𝕌_{X}^{n} ⊂ 𝔸_{X}^{n}. This sequence is natural in X in the obvious way. Moreover, a left inverse to σ_{X}^{∗} is given by γ^{∗} : W^{i}(𝕌_{X}^{n}) ⟶ W^{i}(X) for any X-point γ : X → 𝕌_{X}^{n}, i.e. any morphism γ : X → 𝕌_{X}^{n} such that σ_{X} ∘ γ = id_{X}.

Proof. Follows from the localization sequence [2] and homotopy invariance [4]. □

We want to apply “dévissage” to the relative groups W_{X}^{i+1}(𝔸_{X}^{n}) and we will need

Theorem 8.2. Let X be a regular ℤ[1/2]-scheme of finite Krull dimension. Consider the structure morphism π_{X} : 𝔸_{X}^{n} ⟶ X. Then, the homomorphism

ϑX^{(n)} : W^{i−n}(X) ⟶ W_{X}^{i}(𝔸_{X}^{n}) , w ⟼ π_{X}^{∗}(w) ⋆ κX^{(n)}

is an isomorphism for all i ∈ ℤ.

Proof. The affine case X = Spec R is [10, Thm. 9.3] and the global case follows from this using the Mayer-Vietoris exact sequence [4, Thm. 2.6]. □

Remark 8.3. We do not know whether ϑX^{(n)} is an isomorphism for more general schemes like e.g. regular schemes of infinite Krull dimension. The proof of [10, Thm. 9.3] uses coherent Witt theory and therefore only applies to regular rings of finite Krull dimension.

Theorem 8.4. Let X be a ℤ[1/2]-scheme. Let 1 ≤ i ≤ n be an integer and consider the X-point γ_{i} : X → 𝕌_{X}^{n} ⊂ 𝔸_{X}^{n} corresponding to T_{i} = 1 and T_{j} = 0 for all j ≠ i. If n ≥ 2, then the evaluation at this point of the Witt class εX^{(n)} ∈ W^{r}(𝕌_{X}^{n}) of the half-Koszul space is zero

γ_{i}^{∗}(εX^{(n)}) = 0 in W^{r}(X) .

Proof. Consider the “same” point γ_{i} = (0, …, 0, 1, 0, …, 0) but over ℤ[1/2] instead of X, that is, γ_{i} : Spec(ℤ[1/2]) ⟶ 𝕌_{ℤ[1/2]}^{n}. We have a commutative diagram

\xymatrix{W^{r}(𝕌_{ℤ[1/2]}^{n}) \ar[r]^{γ_{i}^{∗}} \ar[d]_{υ_{X}^{∗}} & W^{r}(ℤ[1/2]) \ar[d] \\ W^{r}(𝕌_{X}^{n}) \ar[r]_{γ_{i}^{∗}} & W^{r}(X)}

with the obvious morphisms and we know from 6.6 that υ_{X}^{∗}(εℤ[1/2]^{(n)}) = εX^{(n)}. Therefore, it suffices to prove the result for ℤ[1/2].

If r ≠ 0 this is trivially true because in this case W^{r}(ℤ[1/2]) = 0. In fact, since ℤ[1/2] is a Dedekind domain, we have by [6, Thm. 10.1] that W^{−1}(ℤ[1/2]) =

W^{2}(ℤ[1/2]) = 0 and W^{1}(ℤ[1/2]) ≃ Coker ( W(ℤ[1/2]) \underrightarrow{∑ ∂_{p}} ⨁_{p≠2} W(ℤ/ℤp)), where

∂_{p} is a second residue homomorphism associated with the prime number p. But this cokernel is also zero by the classical calculation of the Witt group of ℚ, cf. e.g. [17, Thm. VI.6.11].

If r = 0, i.e. n = 2ℓ + 1 with ℓ ≠ 0 even, this follows from the next lemma. □

Lemma 8.5. Let R be a ring and n = 2ℓ + 1 with ℓ ≥ 2 even. Then γ_{i}^{∗}(ER^{(n)}) ≃ γ_{i}^{∗}(ER^{(n)}, φR^{(n)}) is trivial in W(R) ≃ W^{0}(R).

Proof. After renumbering we may assume i = 1. Let K_{•} = K_{•}(A, \underline{T}) be the Koszul complex of the regular sequence \underline{T} = (T_{1}, …, T_{n}) over A = R[T_{1}, …, T_{n}], and Θ = Θ(A, \underline{T}) : K_{•} ⟶^{≃} D_{A}(K_{•})[n] the canonical symmetric n-form. Recall that the differential d_{s} : K_{s} ⟶ K_{s−1} is then given by

e_{i_{1}} ∧ … ∧ e_{i_{s}} ⟼ ∑_{j=1}^{s}(−1)^{j−1}T_{i_{j}} · e_{i_{1}} ∧ … ∧ e_{i_{j−1}} ∧ e_{i_{j+1}} ∧ … ∧ e_{i_{s}} ,

where e_{1}, …, e_{n} constitute a basis of A^{n}. We denote (cf. 1.4) ι_{R} the open immer-

sion 𝕌_{R}^{n} ↪ 𝔸_{R}^{n}. Then we have γ_{1}^{∗}(ER^{(n)}) = γ_{1}^{∗}ι_{R}^{∗}(CR^{(n)}) and K′_{•} = γ_{1}^{∗}ι_{R}^{∗}(K_{•}) is the

Koszul complex K_{•}(R, \underline{t}) for the sequence \underline{t} = (1, 0, …, 0) ⊂ R. We denote the differential of this Koszul complex by d′. Note that this complex is split exact. The isomorphism of complexes Θ′ := γ_{1}^{∗}ι_{R}^{∗}(Θ) is a symmetric n-form on K′_{•} and

(C′, Ξ′) := γ_{1}^{∗}ι_{R}^{∗}(CR^{(n)})

is the symmetric 0-pair ( C_{•}(R, \underline{t}) , Ξ(R, \underline{t}) ), (see 4.3) which is in fact a symmetric 0-space since K′_{•} = cone Ξ(R, \underline{t}) is split exact.

We give now a direct summand S′ of K′_{ℓ+1}, such that the projection K′_{ℓ+1} ⟶ S′ induces a quasi-isomorphism C′_{•} ⟶^{≃} c_{0}(S′), where c_{0} : VB_{R} ⟶ 𝔻^{b}(VB_{R}) is the natural embedding.

The elements v_{i} = 1 ⊗ e_{i} (i = 1, …, n) are a basis of R ⊗_{A} A^{n} = γ_{1}^{∗}ι_{R}^{∗}(A^{n}), and so the exterior products v_{i_{1}} ∧ … ∧ v_{i_{s}} (1 ≤ i_{1} < … < i_{s} ≤ n) are free generators of K′_{s} ≃ ∧^{s}R^{n}. The differential d′_{s} acts on them as follows :

d′_{s}(v_{i_{1}} ∧ … ∧ v_{i_{s}}) = \{ \begin{matrix}0 & 2 ≤ i_{1} < … < i_{s} ≤ n \\ v_{i_{2}} ∧ … ∧ v_{i_{s}} & 1 = i_{1} < i_{2} < … < i_{s} ≤ n , \end{matrix}

therefore the R-module S′ := v_{1} ∧ ( ⋀^{ℓ} R^{n} ) ⊂ K′_{ℓ+1} is isomorphic to Coker d′_{ℓ+2}. Hence c_{0}(S′) ≃ C′_{•} because C′_{•} has non vanishing homology only in degree 0, C′_{0} = K′_{ℓ+1} and C′_{i} = 0 for i < 0. We get an isometry c_{0}(S′, φ′) ≃ (C′, Ξ′), where φ′ := ϵ_{n}(Θ′_{l} · d′_{ℓ+1})|_{S′}.

Consider now the following free submodule M′ := (v_{1} ∧ v_{2}) ∧ ( ⋀^{ℓ−1} R^{n} ) of S′. We claim that M′ is a totally isotropic subspace of (S′, φ′). From this the lemma follows because rank M′ = \frac{1}{2} rank S′ and so (S′, φ′) is hyperbolic by [1, I Thm. 4.6].

To see this we use the description of Θ given in Remark 3.2. Let ω : ∧^{n}A^{n} ⟶^{≃} A be as in this remark and ω′ := id_{R} ⊗ω. Then ω′(v_{1} ∧ … ∧ v_{n}) = 1 and Θ′_{ℓ}(x)(y) = ±ω′(x ∧ y) for all x ∈ K′_{ℓ} and y ∈ K′_{ℓ+1}. If now x, y ∈ M′ then y = v_{1} ∧ v_{2} ∧ y′ and d′_{ℓ+1}(x) = v_{2} ∧ x′ for some x′, y′ ∈ ∧^{ℓ−1}R^{n}, and so ±φ′(x)(y) = ω′(d′_{ℓ+1}(x) ∧ y) = 0 since v_{2} ∧ v_{2} = 0. □

Theorem 8.6. Let X be a regular ℤ[1/2]-scheme. The composition of the connecting homomorphism with the 4-periodicity isomorphism :

\xymatrix{W^{r}(𝕌_{X}^{n}) \ar[r]^{∂} & W_{X}^{r+1}(𝔸_{X}^{n}) \ar[r]^{τ^{q}}_{≃} & W_{X}^{n}(𝔸_{X}^{n}) ,}

maps the Witt class εX^{(n)} ∈ W^{r}(𝕌_{X}^{n}) of the half-Koszul space to the Witt class κX^{(n)} ∈ W_{X}^{n}(𝔸_{X}^{n}) of the Koszul space over 𝔸_{X}^{n}.

Proof. Recall that we always have n = 4 q + r + 1. The statement is a direct consequence of Theorem 6.4, using the definition of the connecting homomorphism ∂ via the symmetric cone 2.10 and the fact that εX^{(n)} = [CX^{(n)}|_{𝕌_{X}^{n}} ] by Definition 6.5. □

Theorem 8.7. Let X be a regular ℤ[1/2]-scheme. For all i ∈ ℤ, define the following homomorphism :

ρX^{(n)} : W^{i−r}(X) ⟶ W^{i}(𝕌_{X}^{n}) , w ⟼ σ_{X}^{∗}(w) ⋆ εX^{(n)} .

Then the following diagram commutes :

\xymatrix{W^{i−r}(X) \ar[d]_{ρ_{X}^{(n)}} & W^{i+1−n}(X) \ar[d]^{ϑ_{X}^{(n)}}_{≃} \ar[l]_{τ^{q}}^{≃} \\ W^{i}(𝕌_{X}^{n}) \ar[r]^{∂} & W_{X}^{i+1}(𝔸_{X}^{n})}

for all i ∈ ℤ, where the isomorphism ϑX^{(n)} is the one of Theorem 8.2 and where τ is the 4-periodicity isomorphism.

Proof. Recall of course that r = n − 4q − 1 by (1). We have to show that

∂ρX^{(n)}([x]) = ϑX^{(n)}([ x[−2q] ])  (11)

for all [x] ∈ W^{i−r}(X). Using the fact that σ_{X} : 𝕌_{X}^{n} ⟶ Spec X factors as

𝕌_{X}^{n} ⟶^{ι_{X}} 𝔸_{X}^{n} ⟶^{π_{X}} Spec X ,

we get:

∂ρX^{(n)}([x]) = ∂( σ_{X}^{∗}(π_{X}^{∗}([x])) ⋆ εX^{(n)})
= π_{X}^{∗}([x]) ⋆ ∂(εX^{(n)})  by [11, Thm. 2.11]
= π_{X}^{∗}([x]) ⋆ τ^{−q}(κX^{(n)})  by Theorem 6.4.

But this is equal to the right hand side of (11) because:

ϑX^{(n)}([ x[−2q] ]) = π_{X}^{∗}([x])[−2q] ⋆ κX^{(n)}  by Lemma B.1
= π_{X}^{∗}([x]) ⋆ τ^{−q}(κX^{(n)})  by Lemma B.3.
□

Corollary 8.8. Let X be a regular ℤ[1/2]-scheme. We have an isomorphism

(σ_{X}^{∗}, ρX^{(n)}) : W^{i}(X) ⊕ W^{i−r}(X) ⟶^{≃} W^{i}(𝕌_{X}^{n}) .

for all i ∈ ℤ.

Proof. Comparing the split exact sequence of Theorem 8.1 and the obvious split exact sequence 0 → W^{i}(X) ⟶ W^{i}(X) ⊕ W^{i−r}(X) ⟶ W^{i−r}(X) → 0, it suffices to note that ∂ · ρX^{(n)} is an isomorphism, as follows from Theorem 8.7. □

To understand the ring structure on W^{tot}(𝕌_{X}^{n}), we need some properties of the symmetric spaces KX^{(n)} and EX^{(n)}, which can be proven for non necessarily regular schemes as well. The case n = 1, i.e. the “Laurent scheme” case, is well-known, so we have to deal with n ≥ 2.

Theorem 8.9. Let X be a ℤ[1/2]-scheme. If n ≥ 2 then the symmetric r-space EX^{(n)} is locally trivial, i.e. for any x ∈ 𝕌_{X}^{n} we have [(EX^{(n)})_{x}] = 0 in W^{r}(O_{𝕌_{X}^{n},x}).

Proof. Define for all i ∈ {1, …, n} the principal open 𝕍_{X}^{n}(i) of 𝔸_{X}^{n} given by the equation T_{i} ≠ 0. Let J ⊆ {1, …, n} ⊂ ℕ. We define

𝕍_{X}^{n}(J) := ⋃_{j∈J} 𝕍_{X}^{n}(j) ⊆ 𝕌_{X}^{n} = 𝕍_{X}^{n}({1, …, n})

and denote ι_{J} : 𝕍_{X}^{n}(J) ⟶ 𝕌_{X}^{n} the corresponding open immersion. Since n ≥ 2, we can cover 𝕌_{X}^{n} with the open subschemes 𝕍_{X}^{n}(J) with |J| ≤ n − 1. So it suffices to prove the following stronger result. □

Theorem 8.10. With the above notations, if |J| ≤ (n − 1) then [EX^{(n)}|_{𝕍_{X}^{n}(J)}] = 0 in W^{r}(𝕍_{X}^{n}(J)).

Proof. We easily reduce to the case X = Spec ℤ[1/2]. In this case we argue as follows. For brevity we set R := ℤ[1/2].

For J empty, the result is trivial since 𝕍_{R}^{n}(J) = ∅ and its Witt group is zero. So we assume that J ≠ ∅. Consider the closed complement 𝕐_{R}(J) := 𝔸_{R}^{n} ∖ 𝕍_{R}^{n}(J) of 𝕍_{R}^{n}(J) ⊂ 𝕌_{R}^{n}. Note that 𝔸_{R}^{n} ∖𝕌_{R}^{n} ⊂ 𝕐_{R}(J). By Theorem 8.1, we have the following commutative diagram with exact rows :

\xymatrix{0 \ar[r] & W^{r}(R) \ar[r]^{σ_{R}^{∗}} \ar[d]^{=} & W^{r}(𝕌_{R}^{n}) \ar[r]^{∂} \ar[d]^{ι_{J}^{∗}} & W_{𝔸_{R}^{n}∖𝕌_{R}^{n}}^{r+1}(𝔸_{R}^{n}) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & W^{r}(R) \ar[r]^{ι_{J}^{∗}σ_{R}^{∗}} & W^{r}(𝕍_{R}^{n}(J)) \ar[r]^{∂} & W_{𝕐_{R}(J)}^{r+1}(𝔸_{R}^{n}) \ar[r] & 0 .}

We get from the right-hand commutative square, from Theorem 8.6, and from Proposition 3.5 that ∂(ι_{J}^{∗}(εR^{(n)})) = 0. Therefore, by exactness of the above second row, there exists a unique class w ∈ W^{r}(R) such that ι_{J}^{∗}(εR^{(n)}) = ι_{J}^{∗}(σ_{R}^{∗}(w)). In fact, w = γ^{∗}(ι_{J}^{∗}(εR^{(n)})) for any R-point γ : Spec(R) → 𝕍_{R}^{n}(J), which exists by assumption J ≠ ∅. Choose j ∈ J and define the R-point γ : Spec(R) → 𝕍_{R}^{n}(J) to be given by T_{j} = 1 and T_{i} = 0 for i ≠ j. Since w = γ^{∗}(ι_{J}^{∗}(εR^{(n)})) = (ι_{J} · γ)^{∗}(εR^{(n)})) and since ι_{J} · γ : Spec(R) → 𝕌_{R}^{n} is simply the R-point γ_{j} of Theorem 8.4, we conclude from it that w = 0. Hence ι_{J}^{∗}(εR^{(n)}) = 0 as wanted. □

Remark 8.11. The statement of Theorem 8.9 is obviously not true for n = 1. The proof fails for n = 1 because then 𝕍_{X}^{n}(J) = ∅ for any J such that |J| ≤ n − 1 = 0 and hence we cannot cover 𝕌_{X}^{n} with these.

It follows from this theorem above and [5, Thm. 4.2] that if n ≥ 2 the space εX^{(n)} is nilpotent in W^{tot}(𝕌_{X}^{n}). We prove a more precise result.

Theorem 8.12. Let X be a ℤ[\frac{1}{2}]-scheme. Assume that n ≥ 2 then

(εX^{(n)})^{2} = εX^{(n)} ⋆ εX^{(n)} = 0

in W^{tot}(𝕌_{X}^{n}). If n = 1 then (εX^{(n)})^{2} = 1 in W^{tot}(𝕌_{X}^{n}).

Proof. The case n = 1 is classical, so let n ≥ 2. Since α_{X}^{∗} : W^{tot}(ℤ[1/2]) ⟶ W^{tot}(X) is a morphism of graded rings (cf. [11, Thm. 3.2]) and α_{X}^{∗}(εℤ[1/2]^{(n)}) = εX^{(n)} (cf. Lemma 6.6) it is enough to prove this for the affine scheme X = Spec ℤ[1/2].

Because we assume n ≥ 2 there exists non-empty subsets J_{1}, J_{2} ⊂ { 1, …, n } with J_{1} ≠ J_{2} and J_{1} ∪ J_{2} = {1, …, n}. We define 𝕍_{ℤ[1/2]}^{n}(J_{i}) ⊆ 𝕌_{ℤ[1/2]}^{n} as in the proof of Theorem 8.9 above and let 𝕐_{ℤ[1/2]}^{n}(J_{i}) := 𝕌_{ℤ[1/2]}^{n}\𝕍_{ℤ[1/2]}^{n}(J_{i}) be the complement (i = 1, 2). Note that J_{1} ∪J_{2} = {1, …, n} implies 𝕐_{ℤ[1/2]}^{n}(J_{1})∩𝕐_{ℤ[1/2]}^{n}(J_{2}) = ∅.

By Theorem 8.10 we know that [Eℤ[1/2]^{(n)}|_{𝕍_{ℤ[1/2]}^{n}(J_{i})}] = 0 for i = 1, 2. Therefore by the localization sequence there exists x_{i} ∈ W_{𝕐_{ℤ[1/2]}^{n}(J_{i})}^{r}(𝕌_{ℤ[1/2]}^{n}) with x_{i} = εℤ[1/2]^{(n)}

in W^{r}(𝕌_{ℤ[1/2]}^{n}) for i = 1, 2, and so x_{1} ⋆ x_{2} = (εℤ[1/2]^{(n)})^{2} in W^{2r}(𝕌_{ℤ[1/2]}^{n}). But the

space x_{1} ⋆ x_{2} lives on a complex with support in 𝕐_{ℤ[1/2]}^{n}(J_{1}) ∩ 𝕐_{ℤ[1/2]}^{n}(J_{2}) = ∅ and

so (εℤ[1/2]^{(n)})^{2} = 0. □

Denote W^{tot}(X)[ε] the graded skew polynomial ring in one variable ε of degree r over the graded ring W^{tot}(X). Recall that this means that c · ε = (−1)^{r deg c}(ε · c) for a homogeneous element c ∈ W^{tot}(X). We have a homogeneous homomorphism of graded rings given by

W^{tot}(X)[ε] ⟶ W^{tot}(𝕌_{X}^{n})  ∑_{i=0}^{m} c_{i}ε^{i} ⟼ ∑_{i=0}^{m} σ_{X}^{∗}(c_{i}) ⋆ (εX^{(n)})^{i} .

Using this morphism we can restate Corollary 8.8 and Theorem 8.12 as follows

Theorem 8.13. Let X be a regular scheme of finite Krull dimension over ℤ[1/2]. Then we have an isomorphism of graded rings

\begin{matrix}if n ≥ 2 : & W^{tot}(X)[ε] / (ε^{2}) \\ if n = 1 : & W^{tot}(X)[ε] / (ε^{2} − 1) \end{matrix} \} ⟶^{≃} W^{tot}(𝕌_{X}^{n}) .

Example 8.14. Let R be a regular ring of finite Krull dimension and

Σ_{R}^{2n−1} := Spec R[T_{1}, …, T_{n}, Y_{1}, …, Y_{n}]/( 1 − ∑_{i=1}^{n} T_{i}Y_{i} )

the hyperbolic (2n − 1)-sphere over R. The R-algebra morphism

R[T_{1}, …, T_{n}] ⟶ R[T_{1}, …, T_{n}, Y_{1}, …, Y_{n}]/( 1 − ∑_{i=1}^{n} T_{i}Y_{i} )

which maps T_{i} to T_{i} mod 1 − ∑ T_{i}Y_{i} induces a flat morphism f : Σ_{R}^{2n−1} ⟶ 𝕌_{R}^{n} whose fibers are affine spaces over the appropriate residue fields. Therefore by strong homotopy invariance [9, Cor. 4.2] we get an isomorphism of graded Witt rings f^{∗} : W^{tot}(𝕌_{R}^{n}) ⟶^{≃} W^{tot}(Σ_{R}^{2n−1}). In particular W^{tot}(Σ_{R}^{2n−1}) is also a free W^{tot}(R)-module with two generators. This has been proven by Karoubi [13] if R is the field of complex numbers.

Note that if R is a regular domain and R(Σ_{R}^{2n−1}) is the function field of the hyperbolic sphere then Theorem 8.9 implies that the natural morphism

W(Σ_{R}^{2n−1}) ⟶ W(R(Σ_{R}^{2n−1}))

is not injective.

9. Witt non-triviality of the (half) Koszul spaces

Theorem 9.1. Let X be a scheme which is not of equicharacteristic 2. Then the Witt class of the symmetric n-space KX^{(n)} is non-trivial in the Witt group with support W_{X}^{n}(𝔸_{X}^{n}).

Proof. By assumption, there is a point x ∈ X whose residue field k(x) has characteristic different from 2. By specialization at x (see Remark 5.4 for naturality), it suffices to prove the result for the regular ℤ[1/2]-scheme X := Spec(k(x)). Here, we apply Theorem 8.2 with i := n and w := 1 ∈ W^{0}(X), the unit of the Witt ring. □

Theorem 9.2. Let X be any scheme which is not of equicharacteristic 2. Then the Witt class εX^{(n)} of the symmetric r-space EX^{(n)} is not in the image of the natural homomorphism W^{r}(𝔸_{X}^{n}) ⟶ W^{r}(𝕌_{X}^{n}). In particular, EX^{(n)} cannot be extended to the whole affine space 𝔸_{X}^{n}.

Proof. Again, by specialization at a point x with char(k(x)) ≠ 2, we are reduced to prove the result for the ℤ[1/2]-regular scheme X := Spec(k(x)). In this case,

the following composition vanishes : \xymatrix{W^{r}(𝔸_{X}^{n}) \ar[r]^{ι_{X}^{∗}} & W^{r}(𝕌_{X}^{n}) \ar[r]^{∂} & W_{X}^{r+1}(𝔸_{X}^{n})} .

Here, the connecting homomorphism ∂ is, for instance, as in Theorem 8.6, where we proved that ∂(εX^{(n)}) coincides with [KX^{(n)}], up to 4-periodicity. So, εX^{(n)} can not be extended to 𝔸_{X}^{n} since [KX^{(n)}] ≠ 0 ∈ W_{X}^{n}(𝔸_{X}^{n}) by Theorem 9.1. Note that we pass via the regular case to use the connecting homomorphism ∂. □

Appendix A. The locally free module EX^{(n)}

We use the notation of the main part of the text. We want to prove:

Theorem A.1. Let X be a noetherian scheme and n ≥ 4. Then there does not exist a locally free O_{𝔸_{X}^{n}} -module F, such that F|_{𝕌_{X}^{n}} ≃ EX^{(n)}. In particular EX^{(n)} is not

a free O_{𝕌_{X}^{n}} -module.

Let x ∈ X and Spec k(x) ⟶^{f} X be the corresponding point. If there exists a locally free O_{𝔸_{X}^{n}} -module F, such that F|_{𝕌_{X}^{n}} ≃ EX^{(n)}, then

Ek(x)^{(n)} ≃ υ_{f}^{∗}(EX^{(n)}) ≃ υ_{f}^{∗}(F|_{𝕌_{X}^{n}}) ≃ α_{f}^{∗}(F)|_{𝕌_{k(x)}^{n}} ,

and so it is enough to show the theorem for X = Spec R with R a field. Similarly, localizing R[T_{1}, …, T_{n}] at the origin, we are reduced to the local case which follows from the following result of commutative algebra.

Theorem A.2. Let (A, 𝔪) be a regular local ring, \underline{T} = (T_{1}, …, T_{n}) a regular

system of parameters (see [7, Def. 2.2.1]), and 𝕌 = ⋃_{i=1}^{n} Spec A_{T_{i}} = Spec A \ {𝔪} the

punctured spectrum of A. Assume that dim A = n ≥ 3. Then

S_{j} := Ker ( K_{j}(A, \underline{T}) ⟶^{d_{j}(A,\underline{T})} K_{j−1}(A, \underline{T}) )|_{𝕌}

cannot be extended to a free A-module if n > j ≥ 2.

Let in the following I_{j} = Ker d_{j}(A, \underline{T}), i.e. S_{j} = I_{j}|_{𝕌}. Recall also that (−)^{∨} = Hom_{A}( − , A). For the proof we need:

Proposition A.3. (i) Let j ≥ 2. Then the A-module I_{j} is reflexive, i.e. the

natural morphism can : I_{j} ⟶ I_{j}^{∨∨} is an isomorphism. (ii) If M and N are finitely generated A-modules, such that M|_{𝕌} ≃ N|_{𝕌}, and

both M|_{𝕌} and N|_{𝕌} are locally free, then M^{∨} ≃ N^{∨}.

Proof. By assumption I_{j} is a second syzygy and so (i) is a consequence of [8, Thm. 3.6]. For (ii), by [12, Thm. 6.9.17] there exists c ≥ 0, such that the given isomorphism M|_{𝕌} ⟶^{≃} N|_{𝕌} is the restriction of a morphism 𝔪^{c}M ⟶ N. Therefore we can assume that there exists g : M ⟶ N, such that g|_{𝕌} is an isomorphism, i.e. Ker g and Coker g have finite length.

Now we use the following fact (see [7, Thm. 1.2.8]). Since dim A ≥ 2 and A is regular (hence in particular Cohen-Macaulay) we have Ext_{A}^{i}(G, A) = 0 for any finite length module G and i = 0, 1.

This and the exact sequences \xymatrix{0 \ar[r] & Ker g \ar[r] & M \ar[r] & Im g \ar[r] & 0} and

\xymatrix{0 \ar[r] & Im g \ar[r] & N \ar[r] & Coker g \ar[r] & 0} give M^{∨} ≃ (Im g)^{∨} ≃ N^{∨}. □

Proof. (of Theorem A.2) Assume that P is a free A-module, such that P |_{𝕌} ≃ S_{j}. We have I_{j}|_{𝕌} ≃ S_{j}, too, and so I_{j}^{∨} ≃ P^{∨} by Proposition A.3 (ii). Part (i) of this proposition tells us that I_{j} is reflexive and hence I_{j} ≃ I_{j}^{∨∨} ≃ P^{∨∨} is free. But this is impossible, because Tor_{n−j}^{A}(I_{j}, A/𝔪) ≃ Tor_{n}^{A}(A/𝔪, A/𝔪) ≃ A/𝔪 ≠ 0 (note that we need here j < n = dim A). We are done. □

Appendix B. Product and 4-periodicity

We have used in this work the fact that the product commutes with the translation. This has not been established in [11]. For the sake of completeness we give here a proof but refer to loc. cit. for unexplained notations and definitions.

To start with let A^{(0)} = (A, D_{A}, δ_{A}, ϖ^{A}) and B^{(0)} = (B, D_{B}, δ_{B}, ϖ^{B}) be triangulated categories with δ_{A}- respectively δ_{B}-exact duality (like e.g. 𝔻^{b}(VB_{X}) with the usual 1-exact duality as in the main part of this work). We denote the shift functor in these triangulated categories by Σ_{A} respectively Σ_{B} (to distinguish we do not use X ↦ X[1]). A symmetric i-space in A^{(0)} is a pair (X, ψ) consisting of an object X ∈ A and a symmetric i-form X ⟶^{ψ} Σ_{A}^{i}D_{A}X which is an isomorphism, the symmetry of an i-form reads Σ_{A}^{i}D_{A}(ψ) · ϖ_{X}^{A} = (−1)^{\frac{i(i+1)}{2}} δ_{A}^{i} · ψ. As in the case of derived categories if (X, ψ) is a symmetric i-form then (Σ_{A}^{2}X, Σ_{A}^{2}(ψ)) is a symmetric (i + 4)-form.

Let (F, ρ) : A^{(0)} ⟶ B^{(0)} be a duality preserving functor, i.e. ρ : F D_{A} ⟶^{≃} D_{B}F is an isomorphism of functors satisfying some compatibility axioms. We will only use the following. Since F is a covariant exact functor between triangulated categories there exists a family of isomorphisms of functors θ^{(i)} : F Σ_{A}^{i} ⟶^{≃} Σ_{B}^{i}F (i ∈ ℤ) which are related by the following formulas :

θ^{(i+j)} = Σ_{B}^{i}(θ^{(j)}) · θ_{Σ_{A}^{j}}^{(i)}  (12)

(i, j ∈ ℤ). Then we have

D_{B}Σ_{B}^{−1}(θ_{Σ_{A}^{−1}}^{(1)}) · ρ_{Σ_{A}^{−1}} = (δ_{A}δ_{B}) · Σ_{B}(ρ) · θ_{D_{A}}^{(1)}  (13)

(cf. loc. cit. Definition 1.8). This axioms are made such that if (X, ψ) is a symmetric i-space in A^{(0)} then (F, ρ)_{∗}(X, ψ) := ( F X , (δ_{A}δ_{B})^{i} Σ_{B}^{i}(ρ_{X}) · θ_{D_{A}X}^{(i)} · F (ψ) ) is a symmetric i-space in B^{(0)}.

Lemma B.1. Let (X, ψ) be a symmetric i-space in A. Then there is an isometry

(F, ρ)_{∗}(Σ_{A}^{2}X, Σ_{A}^{2}(ψ)) ⟶^{≃} Σ_{B}^{2}((F, ρ)_{∗}(X, ψ)) .

Proof. We claim that θ_{X}^{(2)} : F Σ_{A}^{2}X ⟶^{≃} Σ_{B}^{2}F X is an isometry, i.e. we have to show

(δ_{A}δ_{B})^{i+4} Σ_{B}^{i+4}(ρ_{Σ_{A}^{2}X}) · θ_{D_{A}Σ_{A}^{2}X}^{(i+4)} · F Σ_{A}^{2}(ψ)

= (δ_{A}δ_{B})^{i} Σ_{B}^{i+4}D_{B}(θ_{X}^{(2)}) · Σ_{B}^{i+2}(ρ_{X}) · Σ_{B}^{2}(θ_{D_{A}X}^{(i)}) · Σ_{B}^{2}F (ψ) · θ_{X}^{(2)}

(14)

We observe first that Σ_{B}^{2}F (ψ) · θ_{X}^{(2)} = θ_{Σ_{A}^{i}D_{A}X}^{(2)} · F Σ_{A}^{2}(ψ) since θ^{(2)} is a natural transformation. By using (12) three times we get (recall that by definition Σ_{A}D_{A} = D_{A}Σ_{A}^{−1}):

θ_{D_{A}Σ_{A}^{2}X}^{(i+4)} = Σ_{B}^{i+3}(θ_{D_{A}Σ_{A}^{2}X}^{(1)}) · Σ_{B}^{i+2}(θ_{D_{A}Σ_{A}X}^{(1)}) · Σ_{B}^{2}(θ_{D_{A}X}^{(i)}) · θ_{Σ_{A}^{i}D_{A}X}^{(2)}

and so (14) is equivalent to

Σ_{B}^{i+4}(ρ_{Σ_{A}^{2}X}) · Σ_{B}^{i+3}(θ_{D_{A}Σ_{A}^{2}X}^{(1)}) · Σ_{B}^{i+2}(θ_{D_{A}Σ_{A}X}^{(1)}) = Σ_{B}^{i+4}D_{B}(θ_{X}^{(2)}) · Σ_{B}^{i+2}(ρ_{X}) .

Hence the result follows from the following calculation:

Σ_{B}^{i+3}(Σ_{B}(ρ_{Σ_{A}^{2}X}) · θ_{D_{A}Σ_{A}^{2}X}^{(1)}) · Σ_{B}^{i+2}(θ_{D_{A}Σ_{A}X}^{(1)})

= (δ_{A}δ_{A}) Σ_{B}^{i+3}(D_{B}Σ_{B}^{−1}(θ_{Σ_{A}X}^{(1)}) · ρ_{Σ_{A}X}) · Σ_{B}^{i+2}(θ_{D_{A}Σ_{A}X}^{(1)}) by Eq. (13)
= (δ_{A}δ_{A}) Σ_{B}^{i+4}(D_{B}θ_{Σ_{A}X}^{(1)}) · Σ_{B}^{i+2}(Σ_{B}(ρ_{Σ_{A}X}) · θ_{D_{A}Σ_{A}X}^{(1)})
= Σ_{B}^{i+4}D_{B}(Σ_{B}θ_{X}^{(1)} · θ_{Σ_{A}X}^{(1)}) · Σ_{B}^{i+2}(ρ_{X})  by Eq. (13)
= Σ_{B}^{i+4}D_{B}(θ_{X}^{(2)}) · Σ_{B}^{i+2}(ρ_{X})

since θ_{X}^{(2)} = Σ_{B}θ_{X}^{(1)} · θ_{Σ_{A}X}^{(1)} by Eq. (12). □

Assume now we have a third triangulated category with duality, say C^{(0)} = (C, D_{C}, δ_{C}, ϖ^{C}), and a dualizing pairing (loc. cit. Definition 1.11)

⊠ : A^{(0)} × B^{(0)} ⟶ C^{(0)} .

Example B.2. Let X be a scheme and Z ⊆ X a closed subset. Then the (derived) tensor product

⊗_{O_{X}} : 𝔻^{b}(VB_{X}) × 𝔻_{Z}^{b}(VB_{X}) ⟶ 𝔻_{Z}^{b}(VB_{X})

is a dualizing pairing. Note that in this case δ_{A} = δ_{B} = δ_{C} = 1.

Let (X, ψ) be a symmetric i-space in A^{(0)} and (Y, ϕ) a symmetric j-space in B^{(0)}. The left product (X, ψ) ⋆_{l} (Y, ϕ) is then defined by considering X ⊠ − as duality preserving functor with the aid of a duality transformation L(ψ) which depends on ψ, i.e. the left product (X, ψ) ⋆_{l} (Y, ϕ) of these spaces is by definition (X ⊠ − , L(ψ))_{∗}(Y, ϕ). The right product ⋆_{r} is defined analogous by making the functor − ⊠ Y duality preserving using the symmetric j-form ϕ. Both products are related by the following isometry:

(X, ψ) ⋆_{l} (Y, ϕ) ≃ (δ_{A}δ_{C})^{j} · (δ_{B}δ_{C})^{i} · (−1)^{ij} · ( (X, ψ) ⋆_{r} (Y, ϕ) ) .  (15)

(loc. cit. Theorem 2.9). From this we easily deduce

Lemma B.3. There is an isometry

(Σ_{A}^{2}X, Σ_{A}^{2}(ψ)) ⋆_{l} (Y, ϕ) ≃ (X, ψ) ⋆_{l} (Σ_{B}^{2}Y, Σ_{B}^{2}(ϕ)) ,

and the same is true for the right product.

Proof. From Lemma B.1 we get isometries (X, ψ) ⋆_{l} (Σ_{B}^{2}Y, Σ_{B}^{2}(ϕ)) ≃ Σ_{C}^{2}((X, ψ) ⋆_{l} (Y, ϕ)) and (Σ_{A}^{2}X, Σ_{A}^{2}(ψ))⋆_{r} (Y, ϕ) ≃ Σ_{C}^{2}((X, ψ)⋆_{r} (Y, ϕ)). Hence the lemma follows by applying (15) twice. □

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Paul Balmer, D-Math, ETH Zentrum, 8092 Zürich, Switzerland

  E-mail address: balmer@math.ethz.ch
  URL: http://www.math.ethz.ch/∼balmer

Stefan Gille, D-Math, ETH Zentrum, 8092 Zürich, Switzerland

  E-mail address: gille@math.ethz.ch
  URL: http://www.math.ethz.ch/∼gille