BASES OF TOTAL WITT GROUPS AND LAX-SIMILITUDE

PAUL BALMER AND BAPTISTE CALMÈS

Abstract. We explain how to work with total Witt groups, more specifically, how to circumvent the classical embarrassment of making choices for line bundles up to isomorphisms and up to squares.

Contents

Introduction 1 1. The category of quadratic alignments 3 2. Alignment isomorphisms on Witt groups 4 3. Lax pull-back and lax push-forward 6 4. Descending alignments 7 5. Relative alignments and lax module structure 10 6. Total bases of the total Witt group 13 7. Total bases in the localization long exact sequence 17 References 19

Introduction

After completing our computation of the total Witt group W^{tot}(Gr(d, n)) of Grassmannians in [3] and submitting that paper for publication, a somewhat rigid anonymous reader insisted on the problem that the total Witt group of a scheme Y

W^{tot}(Y) = ⨁_{[L]∈Pic(Y)/2} W^{∗}(Y, L)

did not formally exist, because the group W^{∗}(Y, L) depends on a choice of a representative L of [L] ∈ Pic(Y)/2 up to non-unique isomorphism. Although formally correct, it might have occurred to a less rigid or less anonymous human being that this problem had nothing to do with Grassmannians per se, neither much to do with triangular Witt groups, and exists since the foundations of Witt groups of schemes themselves, starting with Knebusch [5].

For us, two alternatives presented themselves, beyond cowardly withdrawing the paper. First, we could trace all twists in use in the special case of Grassmannians, basically turning a reasonably short and hopefully readable paper about Witt groups into an obscure mess about line bundles. Or, alternatively, we could write another paper in which we prove that the choices do not really matter, for Grassmannians and much more generally, providing a tool for seven generations of Witt groupists to use happily ever after. For some reason, we went for the second alternative. The outcome is what the reader holds in her electronic hands.

Date: April 29, 2011.

1991 Mathematics Subject Classification. 19G99, 11E81, 18E30.

Key words and phrases. total Witt group, triangulated category.

The first author is supported by NSF grant DMS-0969644.

Convention. Throughout the paper, X and Y stand for regular noetherian separated schemes over ℤ[\frac{1}{2}], of finite Krull dimension. (See however Remark 7.4.)

For i ∈ ℤ and for a line bundle L over Y, the Witt group W_{Z}^{i}(Y, L) is the triangular Witt group of the derived category D_{Z}^{b}(VB(Y)) of bounded complexes of vector bundles over Y with homology supported in a closed subset Z ⊂ Y, with respect to the usual duality derived from Hom_{O_{Y}} (−, L) and shifted i times. See [1, 2]. Of course, W^{i}(Y, L) stands for W_{Y}^{i} (Y, L). The line bundle L is often called the twist of the duality.

These Witt groups are periodic in two elementary ways. First they are 4-periodic in shift, i.e. there is an extremely natural isomorphism

W_{Z}^{i}(Y, L) ≅ W_{Z}^{i+4}(Y, L)

by [1, Proposition 2.14]. On the other hand, we have a product

W_{Z_{1}}^{i_{1}}(Y, L_{1}) × W_{Z_{2}}^{i_{2}}(Y, L_{2}) → W_{Z_{1}∩Z_{2}}^{i_{1}+i_{2}}(Y, L_{1} ⊗ L_{2})

by [4] that recovers the usual multiplication on the classical Witt group W(Y) = W^{0}(Y, O_{Y}) and that turns any W_{Z}^{i}(Y, L) into a W(Y)-module. For any line bundle M over Y, this product yields the other periodicity isomorphism, called the squareperiodicity isomorphism,

(1)  W_{Z}^{i}(Y, L) ⟶^{∼} W_{Z}^{i}(Y, M^{⊗2} ⊗ L)

and given by multiplication with the Witt class [M →^{∼} M^{∨} ⊗ M^{⊗2}] ∈ W^{0}(Y, M^{⊗2}), where M^{∨} denotes the dual of M. Finally, any isomorphism L →^{∼} L′ induces isomorphisms W_{Z}^{i}(Y, L) ⟶^{∼} W_{Z}^{i}(Y, L′) in the obvious way.

From these isomorphisms, it is clear that all the Witt groups of a scheme can be recovered once we know the Witt groups W_{Z}^{i}(Y, L) for L and i varying in a set of representatives of Pic(Y)/2 and ℤ/4 respectively. The direct sum of such groups is what is usually called “the” total Witt group of the scheme Y (with support in Z). However, this total group is not canonical since it involves the choice of a line bundle L for every class in Pic(Y)/2. Furthermore, if we want to turn this total Witt group into a ring, using the above product, we need to choose isomorphisms between L_{1} ⊗ L_{2} and the chosen line bundle representing [L_{1} ⊗ L_{2}] in Pic(Y)/2, including the choice of “square roots” (for the periodicity modulo 2), and so on. All these choices should further satisfy some compatibilities, of the highest sex appeal. Last not least, it is unclear how to make such choices in a functorial way, when varying the scheme, under pull-back and under push-forward.

To circumvent such technical obstacles, we propose a way of keeping the intuitive simplification allowed by the total Witt group, yet avoiding the unpleasant use of a non-canonical object. Of course, it is unfortunately not true in full generality that one can completely ignore choices. Here, we provide a large class S_{X} of schemes Y over a given base X (Definition 4.1), in which it makes rigorous sense to say that a collection of Witt classes over Y form what we usually call a “basis of W^{tot}(Y) over W^{tot}(X)”.

The initial concept is that of alignment A : L_{1} ⇝ L_{2} between line bundles (Section 1) and the alignment isomorphism

A^{⟲} : W^{∗}(Y, L_{1}) →^{∼} W^{∗}(Y, L_{2})

induced on Witt groups (Section 2). We show how these interact with the two functorialities of Witt theory : pull-back and conditional push-forward. This leads us to introduce lax pull-backs and lax push-forward (Section 3) which are heuristically pull-backs and push-forwards in which one only cares about twists in Pic /2. This mathematical peace of mind is formally provided by Theorem 4.4, where we show that a change of line bundle in some Ȳ over Y can be descended to Y, as long as both Ȳ and Y belong to our category S_{X}.

In Sections 5 and 6, we generalize the action of “W^{tot}(X)” on “W^{tot}(Y)” for Y ∈ S_{X} by again allowing alignments to move classes around. In this context, we discuss the notion of “total basis”.

We then show how the fundamental geometric results, localization long exact sequence, homotopy invariance and dévissage, behave with respect to the above flexibility (Section 7) and we explain how to trace a total basis of the total Witt group in such a localization sequence, without explicitly tracing the line bundles on the nose but only their classes in Pic /2. See Theorem 7.1.

We have already made use of this formalism in the revised version of [3] and we hope that other people computing total Witt groups will enjoy the therapy.

1. The category of quadratic alignments

1.1. Definition. Let L_{1} and L_{2} be line bundles on a scheme Y. We say that a pair A = (M, ϕ) consisting of a line bundle M on Y and an isomorphism

ϕ : M^{⊗2} ⊗ L_{1} ⟶^{∼} L_{2}

is a (quadratic) alignment from L_{1} to L_{2}. Such an alignment exists if and only if [L_{1}] = [L_{2}] in Pic(Y)/2. We use the following short notation for alignments :

A : L_{1} ⇝ L_{2} .

1.2. Definition. We say that two alignments A = (M, ϕ) and A′ = (M′, ϕ′) from L_{1} to L_{2} are isomorphic, denoted A ≃ A′, if there exists an isomorphism τ : M →^{∼} M′ such that the following diagram commutes :

[figure: M^{⊗2} ⊗ L_{1}; ϕ; τ^{⊗2}⊗1; L_{2}; M′^{⊗2} ⊗ L_{1}; ϕ′; L_{2} .]

There is only a set of isomorphism classes of alignments from L_{1} to L_{2} :

Al_{Y} (L_{1}, L_{2}) := {A : L_{1} ⇝ L_{2}}/≃ .

We denote by [A]_{≃} in Al(L_{1}, L_{2}) the isomorphism class of an alignment A : L_{1} ⇝ L_{2}. We define the alignment category, denoted by

Aℓ(Y) ,

to be the category of line bundles over Y with Al_{Y} (L_{1}, L_{2}) as morphism sets from L_{1} to L_{2}. The composition of L_{1} ⇝^{A_{1}} L_{2} ⇝^{A_{2}} L_{3} is defined as follows. If, say,

A_{i} = (M_{i}, ϕ_{i}) with ϕ_{i} : M_{i}^{⊗2} ⊗ L_{i} →^{∼} L_{i+1} for i = 1, 2, then

A_{2} ∘ A_{1} := (M_{2} ⊗ M_{1} , ϕ_{3}) : L_{1} ⇝ L_{3}

where ϕ_{3} is the obvious isomorphism (essentially ϕ_{2} ∘ ϕ_{1})

(M_{2} ⊗ M_{1})^{⊗2} ⊗ L_{1} ⟶^{∼}_{(23)} M_{2}^{⊗2} ⊗ M_{1}^{⊗2} ⊗ L_{1} ⟶^{∼}_{1⊗ϕ_{1}} M_{2}^{⊗2} ⊗ L_{2} ⟶^{∼}_{ϕ_{2}} L_{3} .

Composition is compatible with isomorphisms and is associative up to isomorphism hence turns Aℓ(Y) into a category, in which id_{L} is given by (O_{Y} , O_{Y}^{⊗2} ⊗ L ≅ L).

1.3. Lemma. The category Aℓ(Y) is a groupoid, i.e. for every alignment A : L_{1} ⇝ L_{2} there exists A′ : L_{2} ⇝ L_{1} with A ∘ A′ ≃ id_{L_{2}} and A′ ∘ A ≃ id_{L_{1}}, that is, [A′]_{≃} = [A]_{≃}^{−1}. In particular, given A_{1} and A_{2} with same source (resp. same target) there always exists an alignment B such that B ∘ A_{1} ≃ A_{2} (resp. A_{1} ∘ B ≃ A_{2}).

Proof. If A = (M, ϕ) take A′ = (M^{−1}, ϕ′) : L_{2} ⇝ L_{1} where ϕ′ is (M^{−1})^{⊗2} ⊗ ϕ^{−1} followed by the canonical isomorphism (M^{−1})^{⊗2} ⊗ M^{⊗2} ⊗ L_{1} ≅ L_{1}. □

1.4. Definition. We can also tensor alignments : (L_{1} ⇝^{A_{1}} L′_{1}) ⊗ (L_{2} ⇝^{A_{2}} L′_{2}). If, say, A_{i} = (M_{i}, ϕ_{i}) with ϕ_{i} : M_{i}^{⊗2} ⊗ L_{i} →^{∼} L′_{i} for i = 1, 2, then their tensor product

A_{1} ⊗ A_{2} := (M_{1} ⊗ M_{2} , ϕ_{4}) : L_{1} ⊗ L_{2} ⇝ L′_{1} ⊗ L′_{2}

is given by the obvious isomorphism (ϕ_{4} is morally essentially ϕ_{1} ⊗ ϕ_{2})

(M_{1} ⊗ M_{2})^{⊗2} ⊗ L_{1} ⊗ L_{2} ⟶^{≅}_{(2453)} M_{1}^{⊗2} ⊗ L_{1} ⊗ M_{2}^{⊗2} ⊗ L_{2} ⟶^{∼}_{ϕ_{1}⊗ϕ_{2}} L′_{1} ⊗ L′_{2} .

The reader can verify that this tensor product preserves isomorphisms of alignments and turns Aℓ(Y) into a symmetric monoidal category.

1.5. Lemma. Every line bundle L over Y is invertible in Aℓ(Y) for this ⊗. In particular, the map L ⊗ − induces a bijection Al_{Y} (L_{1}, L_{2}) →^{∼} Al_{Y} (L ⊗ L_{1} , L ⊗ L_{2}) and similarly for − ⊗ L.

Proof. Indeed, L^{−1} is also the inverse of L in Aℓ(Y). □

Finally, some functoriality of alignments with respect to the scheme Y :

1.6. Remark. Given a morphism f : Ȳ → Y and an alignment A = (M, ϕ) : L_{1} ⇝ L_{2} on Y, there is an obvious alignment f^{∗}(A) := (f^{∗}M, f^{∗}ϕ) : f^{∗}L_{1} ⇝ f^{∗}L_{2}. The reader will verify functoriality of this construction : A ≃ A′ ⟹ f^{∗}A ≃ f^{∗}A′ and f^{∗}(A_{2} ∘ A_{1}) ≃ f^{∗}(A_{2}) ∘ f^{∗}(A_{1}) and f^{∗}(A_{1} ⊗ A_{2}) ≃ f^{∗}A_{1} ⊗ f^{∗}A_{2} and (gf)^{∗}(A) ≃ f^{∗}(g^{∗}(A)). So we get a well-defined ⊗-functor

f^{∗} : Aℓ(Y) ⟶ Aℓ(Ȳ)

such that (gf)^{∗} ≃ f^{∗}g^{∗}.

If Y and Ȳ are regular and f : Ȳ → Y is proper, with relative canonical bundle ω_{f} , and if A = (M, ϕ) : L_{1} ⇝ L_{2} is an alignment on Y, we define an alignment f^{!}(A) : ω_{f} ⊗ f^{∗}L_{1} ⇝ ω_{f} ⊗ f^{∗}L_{2} on Ȳ by

(2)  f^{!}(A) = (f^{∗}M , f^{!}ϕ_{2})

where f^{!}ϕ_{2} is the canonical isomorphism

f^{∗}M^{⊗2}⊗ω_{f} ⊗f^{∗}L_{1} ⟶^{∼}_{(123)} ω_{f} ⊗f^{∗}M^{⊗2}⊗f^{∗}L_{1} ≅ ω_{f} ⊗f^{∗}(M^{⊗2}⊗L_{1}) ⟶^{∼}_{1⊗f^{∗}ϕ} ω_{f} ⊗f^{∗}L_{2} .

Using the monoidal structure, this can be stated as f^{!}(A) = id_{ω_{f}} ⊗f^{∗}(A). In particular, f^{!} is as functorial as f^{∗} was, except that f^{!} is not monoidal.

2. Alignment isomorphisms on Witt groups

2.1. Notation. Let ϕ : L →^{∼} L′ be an isomorphism of line bundles on a scheme Y. The isomorphism induced on Witt groups is denoted by

\underline{ϕ} : W_{Z}^{i}(Y, L) →^{∼} W_{Z}^{i}(Y, L′) .

The square periodicity isomorphism associated to a line bundle M is denoted by

per_{M} : W_{Z}^{i}(Y, L) →^{∼} W_{Z}^{i}(Y, M^{⊗2} ⊗ L).

2.2. Remark. Here are some easy compatibilities between those isomorphisms, that we leave to the reader. We use the obvious notation.

(i) \underline{ϕ′} ∘ \underline{ϕ} = \underline{ϕ′ ∘ ϕ}.

(ii) per_{M_{2}} ∘per_{M_{1}} = \underline{(23)} ∘per_{M_{2}⊗M_{1}} for (M_{2}⊗M_{1})^{⊗2}⊗L →_{(23)}^{≅} M_{2}^{⊗2}⊗M_{1}^{⊗2}⊗L.

(iii) per_{M} ∘ \underline{ϕ} = (\underline{1 ⊗ ϕ}) ∘ per_{M} where 1 ⊗ ϕ : M^{⊗2} ⊗ L_{1} →^{∼} M^{⊗2} ⊗ L_{2}.

(iv) per_{M′} = (\underline{τ^{⊗2} ⊗ 1}) ∘ per_{M}, for every isomorphism τ : M →^{∼} M′.

It follows that any composition of per_{M_{i}} and \underline{ϕ_{j}} (in any order) can always be reduced to one composition of the form \underline{ϕ} ∘ per_{M}. This is the true reason for the notion of alignment introduced in Section 1 and yields naturally :

2.3. Definition. For every alignment A = (M, ϕ) : L_{1} ⇝ L_{2} (Definition 1.1), i.e. ϕ : M^{⊗2} ⊗ L_{1} →^{∼} L_{2}, we define an isomorphism A^{⟲} := \underline{ϕ} ∘ per_{M}

A^{⟲} : W_{Z}^{∗}(Y, L_{1}) ⟶^{∼} W_{Z}^{∗}(Y, L_{2})

that we call the alignment isomorphism (on Witt groups) associated to the alignment A : L_{1} ⇝ L_{2}.

2.4. Example. To a unit u ∈ O_{Y} (Y)^{×} we can associate two things : An isomorphism u· : L →^{∼} L for any line bundle L and a Witt class ⟨u⟩ in W(Y) = W^{0}(Y, O_{Y}). It is an easy exercise to verify that multiplication by ⟨u⟩ on Witt groups is the same as the alignment isomorphism \underline{(u·)} = A_{u}^{⟲}, where A_{u} = (O_{Y} , (u·)).

With this example in mind, it seems reasonable to use the following terminology :

2.5. Definition. Let us say that two Witt classes w_{1} ∈ W_{Z}^{j}(Y, L_{1}) and w_{2} ∈ W_{Z}^{j}(Y, L_{2}) are lax-similar if there exists an alignment A : L_{1} ⇝ L_{2} such that A^{⟲}(w_{1}) = w_{2}. This is an equivalence relation, that we denote by

w_{1} ↭ w_{2} .

2.6. Remark. We have w ↭ 0 if and only w = 0. However, Witt classes up to lax-similitude do not form a group, as we already know from ordinary similitude.

2.7. Proposition. The assignment A ↦ A^{⟲} respects the structures of Section 1 :

(a) Isomorphic alignments A ≃ A′ induce the same alignment isomorphism A^{⟲} = A′^{⟲} on Witt groups. Hence (−)^{⟲} is well-defined on Aℓ(Y).

(b) Given two alignments L_{1} ⇝^{A_{1}} L_{2} ⇝^{A_{2}} L_{3}, we have

(A_{2} ∘ A_{1})^{⟲} = A_{2}^{⟲} ∘ A_{1}^{⟲}

on Witt groups. Hence (−)^{⟲} is functorial.

(c) Given two alignments A_{i} : L_{i} ⇝ L′_{i} and two Witt classes w_{i} ∈ W_{Z_{i}}^{j_{i}}(Y, L_{i}), for i = 1, 2, we have

(A_{1} ⊗ A_{2})^{⟲} (w_{1} · w_{2}) = (A_{1}^{⟲}(w_{1})) · (A_{2}^{⟲}(w_{2})) .

in W_{Z_{1}∩Z_{2}}^{j_{1}+j_{2}}(Y, L′_{1} ⊗ L′_{2}). Hence (−)^{⟲} is (somewhat) monoidal.

Proof. Part (a) follows from Remark 2.2 (i) and (iv). For (b), contemplate

\xymatrix{W_{Z}^{∗}(Y, L_{1}) \ar[r]^{^{per^{M1}}} \ar[d]^{_{per_{M2⊗M1}}} & W_{Z}^{∗}(Y, M_{1}^{⊗2} ⊗ L_{1}) \ar[r]^{\underline{ϕ_{1}}} \ar[d]^{_{per_{M2}}} & W_{Z}^{∗}(Y, L_{2}) \ar[d]^{_{per_{M2}}} \\ W_{Z}^{∗}(Y, (M_{2} ⊗ M_{1})^{⊗2} ⊗ L_{1}) \ar[r]_{\underline{(23)}} & W_{Z}^{∗}(Y, M_{2}^{⊗2} ⊗ M_{1}^{⊗2} ⊗ L_{1}) \ar[r]_{\underline{1⊗ϕ_{1}}} & W_{Z}^{∗}(Y, M_{2}^{⊗2} ⊗ L_{2})}

This diagram commutes by Remark 2.2 (ii) and (iii). Post-composing the two outside compositions with \underline{ϕ_{2}} : W_{Z}^{∗}(Y, M_{2}^{⊗2} ⊗ L_{2}) →^{∼} W_{Z}^{∗}(Y, L_{3}) yields the result. Part (c) is straightforward from the definition of the Witt group product [4]. □

2.8. Remark. Since Aℓ(Y) is a groupoid (Lemma 1.3), Proposition 2.7 (b) gives us that any zig-zag of alignment isomorphisms (on a given scheme) can be realized by one single alignment isomorphism. Note however that there might be non-isomorphic alignments from L_{1} to L_{2}, hence possibly non-equal alignment isomorphisms on Witt groups.

Compatibility of A^{⟲} with morphisms f : Ȳ → Y is the subject of Section 3.

2.9. Remark. It is easy to verify that for every alignment A : L_{1} ⇝ L_{2} the alignment isomorphism A^{⟲} : W_{Z}^{∗}(Y, L_{1}) →^{∼} W_{Z}^{∗}(Y, L_{2}) commutes with the periodicity isomorphism W^{∗} →^{∼} W^{∗+4}, the latter being simply induced on the underlying triangulated category by the square of the shift Σ^{2}. This automorphism Σ^{2} commutes with all operations involved in A^{⟲}. Note that Σ^{2} has no hidden sign either.

2.10. Remark. Alignment isomorphisms are compatible with localization long exact sequences. Compatibility of A^{⟲} with restriction to an open subscheme is a special case of Corollary 3.2 below. Let us discuss the other two types of homomorphisms.

(1) Let Z ⊂ Z′ be closed subsets of Y. Then extension-of-support e : W_{Z}^{∗}(Y, L) → W_{Z′}^{∗}(Y, L) obviously commutes with alignment isomorphism A^{⟲}.

(2) For the connecting homomorphism, let U = Y ∖ Z and consider A : L ⇝ L′ over Y and its obvious restriction A_{|U} : L_{|U} ⇝ L′_{|U}. Then the following diagram

\xymatrix{W^{j}(U, L_{|U}) \ar[r]^{∂} \ar[d]_{_{A_{|U}_{⟲}}} & W_{Z}^{j+1}(Y, L) \ar[d]^{A^{⟲}} \\ W^{j}(U, L′_{|U}) \ar[r]^{∂} & W_{Z}^{j+1}(Y, L′)}

commutes. Indeed, if A = (M, ϕ), then A^{⟲} = \underline{ϕ} ∘ per_{M} and ∂ clearly commutes with \underline{ϕ}. It is proved in [4] that ∂ is W^{∗}(Y)-linear, which explains why it commutes with per_{M}.

3. Lax pull-back and lax push-forward

We assume for simplicity (in the treatment of the push-forward) that all schemes Y, Ȳ, etc., are smooth over our regular base X.

Given a morphism f : Ȳ → Y and a closed subset Z of Y, we have a pull-back

f^{∗} : W_{Z}^{∗}(Y, L) → W_{f^{−1}(Z)}^{∗}(Ȳ , f^{∗}L) .

On the other hand, when f is proper and has constant relative dimension dim(f) (the latter is automatic if Y is connected) and has relative canonical bundle ω_{f}, we have a push-forward homomorphism for every closed subset Z̄ ⊂ Ȳ

f_{∗} : W∗+dim(f) ; Z̄ (Ȳ , ω_{f} ⊗ f^{∗}L) ⟶ W_{f(Z̄)}^{∗}(Y, L) .

3.1. Remark. Continuing Remark 2.2, for every morphism of schemes f : Ȳ → Y, we have naturality :

(v) f^{∗} ∘ \underline{ϕ} = \underline{f^{∗}ϕ} ∘ f^{∗}.

(vi) f^{∗} ∘ per_{M} = per_{f^{∗}M} ∘ f^{∗}. Finally, if f : Ȳ → Y is proper with relative canonical bundle ω_{f}, we have :

(vii) \underline{ϕ} ∘ f_{∗} = f_{∗} ∘ (\underline{1 ⊗ f^{∗}ϕ}) for the canonical 1 ⊗ f^{∗}ϕ : ω_{f} ⊗ f^{∗}L_{1} →^{∼} ω_{f} ⊗ f^{∗}L_{2}.

(viii) per_{M} ∘ f_{∗} = f_{∗} ∘ \underline{(123)} ∘ per_{f^{∗}M} by the projection formula, where (123) :

(f^{∗}M)^{⊗2}⊗ω_{f} ⊗f^{∗}L_{1} ⟶^{≅} ω_{f} ⊗f^{∗}(M^{⊗2}⊗L_{1}) is the canonical isomorphism.

3.2. Corollary. Recall Remark 1.6 for f^{∗} and f^{!} on alignments. For f : Ȳ → Y and for an alignment A = (M, ϕ) : L_{1} ⇝ L_{2} on Y, we have on Witt groups

f^{∗} ∘ A^{⟲} = (f^{∗}A)^{⟲} ∘ f^{∗} .

When f is proper, we have on Witt groups

A^{⟲} ∘ f_{∗} = f_{∗} ∘ (f^{!}A)^{⟲} .

Proof. This is a compact form of Remark 3.1. □

3.3. Remark. Given a morphism f : Ȳ → Y, we can compose the pull-back f^{∗} with alignment isomorphisms on Y and on Ȳ. By Corollary 3.2, any such composition can be brought down to one of the form Ā^{⟲} ∘ f^{∗} for some alignment Ā on Ȳ. Similarly, one can combine the push-forward f_{∗} with alignment isomorphisms on Y and Ȳ and again the alignment isomorphisms on Y are redundant, i.e. such a composition always boils down to one of the form f_{∗} ∘ Ā^{⟲} for some alignment Ā on Ȳ. Let us give names to those generalized pull-back and push-forward.

3.4. Definition. Let f : Ȳ → Y be a morphism. Let L and L̄ be line bundles on Y and Ȳ respectively and let Ā : f^{∗}L ⇝ L̄ be an alignment on Ȳ (Definition 1.1). We define the lax pull-back homomorphism from W_{Z}^{∗}(Y, L) to W_{f^{−1}Z}^{∗}(Ȳ , L̄) by

Pull_{Ā,f} := Ā^{⟲} ∘ f^{∗} : W_{Z}^{∗}(Y, L) ⟶ W_{f^{−1}Z}^{∗}(Ȳ , f^{∗}L) ⟶^{∼} W_{f^{−1}Z}^{∗}(Ȳ , L̄) .

3.5. Definition. Let f : Ȳ → Y be a proper morphism with relative canonical bundle ω_{f} and relative dimension d = dim(f). Let L and L̄ be line bundles on Y and Ȳ respectively, and let Ā : L̄ ⇝ ω_{f} ⊗f^{∗}L be an alignment on Ȳ (note ω_{f} here). We define the lax push-forward homomorphism from W∗+d ; Z̄(Ȳ , L̄) to W_{fZ̄}^{∗}(Y, L) by

Push_{f,Ā} := f_{∗} ∘ Ā^{⟲} : W∗+d ; Z̄(Ȳ , L̄) ⟶^{∼} W∗+d ; Z̄(Ȳ , ω_{f} ⊗ f^{∗}L) ⟶ W_{fZ̄}^{∗}(Y, L) .

By Proposition 2.7 (a), both Pull_{Ā,f} and Push_{f,Ā} only depend on the isomorphism class of Ā.

3.6. Proposition. For composable Ỹ →^{g} Ȳ →^{f} Y and for alignments Ā on Ȳ and Ã on Ỹ such that Ã and g^{∗}Ā are composable, we have

PullÃ, g ∘ PullĀ, f = PullÃ ∘ (g^{∗}Ā) , fg .

If instead g^{!}Ā and Ã are composable and if f and g are moreover proper, then

Pushf, Ā ∘ Pushg, Ã = Pushfg , (g^{!}Ā) ∘ Ã

Proof. Direct by Corollary 3.2, Definitions 3.4 and 3.5 and Proposition 2.7 (b). □

4. Descending alignments

Given a morphism f : Ȳ → Y, it might happen that two line bundles L_{1} and L_{2} on Y have f^{∗}L_{1} and f^{∗}L_{2} aligned over Ȳ, in the sense of Definition 1.1, without necessarily being aligned over Y. This can cause sorrow in the taverns. We propose a simple solution, which will be convenient in applications.

4.1. Definition. Recall our separated, noetherian regular base scheme X over ℤ[\frac{1}{2}]. Let Y be a scheme over X. We denote by π_{Y} : Y → X the structure morphism and by Pic_{X}(Y) the cokernel of π_{Y}^{∗} : Pic(X) → Pic(Y). Consider the full subcategory

S_{X}

of smooth X-schemes π_{Y} : Y → X (with morphisms over X of course) such that :

(I) The map π_{Y}^{∗} : Pic(X) → Pic(Y) is injective.

(II) The abelian group Pic_{X}(Y) = Pic(Y)/π_{Y}^{∗} ( Pic(X)) has no 2-torsion.

(III) The map π_{Y}^{∗} : 𝔾_{m}(X) → 𝔾_{m}(Y)/𝔾_{m}(Y)^{2} is surjective.

4.2. Remark. Note that X itself is in S_{X}. Projective spaces over X, Grassmannians and various flag varieties over X are in S_{X}, as explained in [3]. If X is local (e.g. the spectrum of a field) and thus has trivial Picard group, then projective varieties over X with no 2-torsion in their Picard group are in S_{X}.

Assumptions (I) and (II) allow an easy chase :

4.3. Lemma. Let f : Ȳ → Y be a morphism in S_{X}. Then :

(a) The homomorphism f^{∗} : Pic(Y) → Pic(Ȳ) induces an isomorphism on 2torsion subgroups _{2}Pic(Y) →^{∼} _{2}Pic(Ȳ).

(b) The sequence 0 → Pic(X)/2 → Pic(Y)/2 → Pic_{X}(Y)/2 → 0 is exact.

(c) The homomorphism Pic(Y)/2 ⟶ Pic_{X}(Y)/2 ⊕ Pic(Ȳ)/2 is injective.

(d) If a line bundle L̄ on Ȳ is such that [L̄] = [f^{∗}L] in Pic_{X}(Ȳ)/2 for some L over Y, then there exists L′ over Y with [L′] = [L] in Pic_{X}(Y)/2 and [L̄] = [f^{∗}L′] in Pic(Ȳ)/2 already.

Proof. Since π_{Y} f = πȲ it suffices to prove (a) for f = π_{Y} : Y → X. Multiplication by 2 yields an endomorphism of the following short sequence of abelian groups

0 → Pic(X) ⟶^{π_{Y}^{∗}} Pic(Y) ⟶ Pic_{X}(Y) → 0,

which is exact by (I) above. The Snake Lemma and Assumption (II) give (a) and (b). For (c), it suffices to compare the exact sequences (b) for Y and Ȳ via f^{∗}:

[figure: 0; Pic(X)/2; Pic(Y)/2; f^{∗}; Pic_{X}(Y)/2; f^{∗}; 0; 0; Pic(X)/2; Pic(Ȳ)/2; Pic_{X}(Ȳ)/2; 0]

to chase the wanted injectivity (in fact, the right-hand square is cartesian). For (d), if there exists a line bundle K on X with [L̄] = [f^{∗}L ⊗ π∗ ; Ȳ K] = [f^{∗}(L ⊗ π_{Y}^{∗} K)] in Pic(Ȳ)/2, take L′ := L⊗π_{Y}^{∗} K, which is in the same class as L in Pic_{X}(Y)/2. □

Here is the announced descent of alignments along morphisms Ȳ → Y in S_{X}.

4.4. Theorem. Let f : Ȳ → Y be a morphism in S_{X}. Let L_{1} and L_{2} be line bundles on Y and let Ā : f^{∗}L_{1} ⇝ f^{∗}L_{2} be an alignment on Ȳ (Definition 1.1). Suppose that [L_{1}] = [L_{2}] in Pic_{X}(Y)/2. Then there exists an alignment A : L_{1} ⇝ L_{2} on Y and an isomorphism f^{∗}A ≃ Ā on Ȳ (see Definition 1.2 and 1.6).

Proof. Write Ā = (M̄, ϕ̄) so that ϕ̄ is an isomorphism M̄^{⊗2} ⊗ f^{∗}L_{1} →^{∼} f^{∗}L_{2} on Ȳ. In particular, in the group Pic(Ȳ)/2, we have [f^{∗}L_{1}] = [f^{∗}L_{2}]. Since we also assume [L_{1}] = [L_{2}] in Pic_{X}(Y)/2, we get by Lemma 4.3 (c) that [L_{1}] = [L_{2}] in Pic(Y)/2. Hence there exists a line bundle M′ over Y such that M′^{⊗2} ⊗ L_{1} ≃ L_{2}. Pulling-back to Ȳ, we get (f^{∗}M′)^{⊗2} ≃ M̄^{⊗2} over Ȳ. Hence [(f^{∗}M′) ⊗ M̄^{−1}] ∈ 2 Pic(Ȳ) ←^{∼} _{2} Pic(Y) by Lemma 4.3 (a). Hence there exists [M′′] ∈ _{2} Pic(Y) such that f^{∗}(M′ ⊗ M′′) ≃ M̄. Defining M := M′ ⊗ M′′ we now have

f^{∗}M ≃ M̄  and also  M^{⊗2} ⊗ L_{1} ≃ L_{2}

since M′^{⊗2} ⊗ L_{1} ≃ L_{2} already and M′′^{⊗2} is trivial.

Now choose two isomorphisms τ̄_{0} : f^{∗}M →^{∼} M̄ and ϕ_{0} : M^{⊗2} ⊗ L_{1} →^{∼} L_{2}, which exist by the above construction, and consider the following diagram over Ȳ

(3)  [figure: (f^{∗}M)^{⊗2} ⊗ f^{∗}L_{1}; τ̄_{0}^{⊗2}⊗1; M̄^{⊗2} ⊗ f^{∗}L_{1}; ϕ̄; f^{∗}(M^{⊗2} ⊗ L_{1}); f^{∗}ϕ_{0}; f^{∗}L_{2} .]

A priori, this only commutes up to a unit t ∈ 𝔾_{m}(Ȳ), like every diagram of isomorphisms of line bundles. By Assumption (III), t = π∗ ; Ȳ (u) · v^{2} for some u ∈ 𝔾_{m}(X) and v ∈ 𝔾_{m}(Ȳ). Hence we can define ϕ = π_{Y}^{∗} (u) · ϕ_{0} and τ̄ = v^{−1} · τ̄_{0}, so that ϕ and τ̄ make Diagram (3) strictly commute (in place of ϕ_{0} and τ̄_{0} respectively, of course).

Consider the alignment A := (M, ϕ) : L_{1} ⇝ L_{2} on Y. The commutativity of (3) precisely means f^{∗}A ≃ Ā (see Remark 1.6). □

4.5. Corollary. Let f : Ȳ → Y be a proper morphism in S_{X}. Let L_{1} and L_{2} be line bundles on Y and Ā : ω_{f} ⊗ f^{∗}L_{1} ⇝ ω_{f} ⊗ f^{∗}L_{2} be an alignment on Ȳ. Suppose that [L_{1}] = [L_{2}] in Pic_{X}(Y)/2. Then there exists an alignment A : L_{1} ⇝ L_{2} on Y and an isomorphism f^{!}A ≃ Ā on Ȳ (Remark 1.6).

Proof. Immediate from Theorem 4.4 since f^{!} = ω_{f} ⊗ f^{∗} : Aℓ(Y) → Aℓ(Ȳ) and since B̄ ↦ ω_{f} ⊗ B̄ is a bijection by Lemma 1.5. □

The following two results turn the above descent of line bundle alignments into descent of alignment isomorphisms on the level of Witt groups.

4.6. Proposition. Let f : Ȳ → Y be a morphism of schemes in S_{X}. Let L_{1} and L_{2} be line bundles on Y with [L_{1}] = [L_{2}] ∈ Pic_{X}(Y)/2. Let Ā_{1} : f^{∗}L_{1} ⇝ L̄ and Ā_{2} : f^{∗}L_{2} ⇝ L̄ be two alignments with same target L̄ on Ȳ. Then there exists an alignment A : L_{1} ⇝ L_{2} on Y such that the following diagram commutes

[figure: W_{Z}^{∗}(Y, L_{1}); _{Pull_{Ā_{1},f}}; A^{⟲}; W_{Z}^{∗}(Y, L_{2}); _{Pull_{Ā_{2},f}}; W_{Z}^{∗}(Ȳ , L̄)]

Proof. By Lemma 1.3, there exists an alignment Ā : f^{∗}L_{1} ⇝ f^{∗}L_{2} on Ȳ such that Ā_{2} ∘ Ā ≃ Ā_{1}. By Theorem 4.4, there exists A : L_{1} ⇝ L_{2} such that f^{∗}A ≃ Ā. So, Ā_{2} ∘ f^{∗}A ≃ Ā_{1}. By Propositions 3.6 and 2.7 (a), we have Pull_{Ā_{2},f} ∘A^{⟲} = Pull_{Ā_{2}∘f^{∗}A,f} = Pull_{Ā_{1},f}. □

4.7. Proposition. Let f : Ȳ → Y be a proper morphism in S_{X}. Let L_{1} and L_{2} be line bundles on Y such that [L_{1}] = [L_{2}] ∈ Pic_{X}(Y)/2. Let Ā_{1} : L̄ ⇝ ω_{f} ⊗ f^{∗}L_{1} and Ā_{2} : L̄ ⇝ ω_{f} ⊗ f^{∗}L_{2} be two alignments from the same source L̄ on Ȳ. Then there exists an alignment A : L_{1} ⇝ L_{2} on Y such that the following diagram commutes

[figure: W_{Z}^{∗}(Ȳ , L̄); _{Push_{f,Ā_{1}}}; _{Push_{f,Ā_{2}}}; W_{Z}^{∗}(Y, L_{1}); A^{⟲}; W_{Z}^{∗}(Y, L_{2}) .]

Proof. By Corollary 4.5 for Ā = Ā_{2} ∘ Ā_{1}^{−1}, there exists A : L_{1} ⇝ L_{2} such that f^{!}A ∘ Ā_{1} ≃ Ā_{2}. By Proposition 3.6, A^{⟲} ∘ Push_{f,Ā_{1}} = Push_{f,f^{!}A∘Ā_{1}} = Push_{f,Ā_{2}}. □

4.8. Remark. Combined with Corollary 3.2, the last two propositions tell us that, for a morphism f : Ȳ → Y in S_{X}, it is not so important to know where a lax pull-back Pull_{Ā,f} or a lax push-forward Push_{f,Ā} exactly lands, as long as we keep track of classes in Pic_{X}(?)/2. Different choices can always be “realigned”.

5. Relative alignments and lax module structure

Now that we have a stable understanding of alignments, we introduce a relative version of this notion, allowing a line bundle on the base X to intervene.

5.1. Definition. Let π_{Y} : Y → X be a scheme over X. We say that two line bundles L_{1} and L_{2} over Y are (quadratically) X-aligned if [L_{1}] = [L_{2}] ∈ Pic_{X}(Y) = coker(π_{Y}^{∗} : Pic(X) → Pic(Y)/2). This amounts to the existence of a line bundle K over X and an alignment π_{Y}^{∗} K ⊗ L_{1} ⇝ L_{2} as in Definition 1.1.

By extension, it will be very convenient to say that a Witt class w ∈ W_{Z}^{∗}(Y, L_{1}) is X-aligned with L_{2} when the line bundle L_{1} is X-aligned with L_{2}.

Specifying a line bundle K over X, we say that L_{1} and L_{2} are K-aligned if π_{Y}^{∗} K ⊗L_{1} is aligned with L_{2}. Unfolding everything, this means that there exists an alignment A = (M, ϕ) : π_{Y}^{∗} K⊗L_{1} ⇝ L_{2}, i.e. an isomorphism ϕ : M^{⊗2}⊗π_{Y}^{∗} K⊗L_{1} →^{∼} L_{2}. We call A a K-alignment of L_{1} with L_{2} and use the condensed notation

A : L_{1}⇝_{K}L_{2} .

5.2. Definition. Let A : L_{1}⇝_{K}L_{2} be a K-alignment in Y. We are going to define a lax product or product realigned under A

− ·_{A} − : W^{i}(X, K) × W_{Z}^{j}(Y, L_{1}) ⟶ W_{Z}^{i+j}(Y, L_{2}) .

By Definition 2.3, there exists an alignment isomorphism A^{⟲} : W_{Z}^{∗}(Y, π_{Y}^{∗} K ⊗ L_{1}) ⟶^{∼} W_{Z}^{∗}(Y, L_{2}). Then for every Witt class λ ∈ W^{i}(X, K) on the base and every Witt class w ∈ W_{Z}^{j}(Y, L_{1}) on Y, we define

λ ·_{A} w := A^{⟲}(π_{Y}^{∗} (λ) · w)

for the image in W_{Z}^{i+j}(Y, L_{2}) of the product π_{Y}^{∗} (λ)·w, under the alignment isomorphism A^{⟲}. We call this the lax-structure of W^{tot}(X)-module on W_{Z}^{tot}(Y).

5.3. Remark. Taking λ = 1 ∈ W(X), we see that λ ·_{A} − = A^{⟲}(−). So the above homomorphisms λ ·_{A} − generalize the alignment isomorphisms.

This action behaves nicely with respect to all possible alignment isomorphisms :

5.4. Lemma. Let A : L_{1}⇝_{K}L_{2} be a K-alignment so that we have the product ·_{A} of Definition 5.2. Let B : L_{2} ⇝ L_{3}, C : J ⇝ K and D : L_{0} ⇝ L_{1} be (plain) alignments over Y, X and Y respectively. Then E := B ∘ A ∘ ((π_{Y}^{∗} C) ⊗ D) is a J-alignment L_{0}⇝_{J}L_{3} on Y. For every λ ∈ W^{i}(X, J) and w ∈ W_{Z}^{j}(Y, L_{0}) we have

B^{⟲}(C^{⟲}(λ) ·_{A} D^{⟲}(w)) = λ ·_{E} w

in W_{Z}^{i+j}(Y, L_{3}). In words, the lax product commutes with lax-similitude.

Proof. A direct computation :

B^{⟲}(C^{⟲}(λ) ·_{A} D^{⟲}(w)) =^{def} B^{⟲} ∘ A^{⟲}(π_{Y}^{∗} (C^{⟲}(λ)) · D^{⟲}(w))

=^{3.2} B^{⟲} ∘ A^{⟲}(((π_{Y}^{∗} C)^{⟲}(π_{Y}^{∗} (λ))) · D^{⟲}(w))

=^{2.7(c)} B^{⟲} ∘ A^{⟲} ∘ ((π_{Y}^{∗} C) ⊗ D)^{⟲} (π_{Y}^{∗} (λ) · w)

=^{2.7(b)} (B ∘ A ∘ ((π_{Y}^{∗} C) ⊗ D))^{⟲} (π_{Y}^{∗} (λ) · w) =^{def} λ ·_{E} w . □

The real question is whether this product λ ·_{A} w depends significantly on the K-alignment A : L_{1} ⇝ L_{2}, for L_{1} and L_{2} fixed. A priori, this might be the case. However, our class of schemes S_{X} (Definition 4.1) turns out to be well-behaved.

5.5. Lemma. Let Y be a scheme in S_{X}. Let A_{i} : L_{1}⇝_{K_{i}}L_{2} be K_{i}-alignments over Y (for the same L_{1} and L_{2}), for i = 1, 2. Then there exits an alignment C : K_{1} ⇝ K_{2} on X such that, for every λ_{1} ∈ W^{∗}(X, K_{1}) and every w ∈ W_{Z}^{∗}(Y, L_{1}), we have

λ_{1} ·_{A_{1}} w = λ_{2} ·_{A_{2}} w

in W_{Z}^{∗}(Y, L_{2}), where λ_{2} = C^{⟲}(λ_{1}) ∈ W^{∗}(X, K_{2}).

Proof. In terms of plain alignments over Y, note that A_{1} : π_{Y}^{∗} K_{1} ⊗ L_{1} ⇝ L_{2} and A_{2} : π_{Y}^{∗} K_{2} ⊗ L_{1} ⇝ L_{2} have the same target L_{2}. By Lemma 1.3, there exists an alignment A′ : π_{Y}^{∗} K_{1} ⊗ L_{1} ⇝ π_{Y}^{∗} K_{2} ⊗ L_{1} such that A_{2} ∘ A′ ≃ A_{1}. Note that L_{1} appears at the two ends of A′. So, by Lemma 1.5, there exists an alignment A′′ : π_{Y}^{∗} K_{1} ⇝ π_{Y}^{∗} K_{2} such that A′ ≃ A′′ ⊗L_{1}. Finally, by Proposition 4.6 applied to f = π_{Y}, there exists an alignment C : K_{1} ⇝ K_{2} such that A′′ ≃ π_{Y}^{∗} C. The result follows from Lemma 5.4, applied to our C, and to B̄ := id_{L_{2}}, D̄ := id_{L_{1}}, A := A_{2}, checking that E is here A_{2} ∘ ((π_{Y}^{∗} C) ⊗ L_{1}) ≃ A_{2} ∘ (A′′ ⊗ L_{1}) ≃ A_{2} ∘ A′ ≃ A_{1}. □

Let us a say a word about associativity of the lax-action.

5.6. Lemma. Let Y be an X-scheme and K_{1} and K_{2} be line bundles on X. Set K_{3} := K_{2} ⊗ K_{1}. Consider X-alignments A_{1} : L_{0}⇝_{K_{1}}L_{1} and A_{2} : L_{1}⇝_{K_{2}}L_{2} and A_{3} : L_{0}⇝_{K_{3}}L_{2} over Y. Then for any choice of two out of A_{1}, A_{2} and A_{3}, the third can be constructed such that the following diagram commutes in Aℓ(Y) :

(4)  [figure: π_{Y}^{∗} (K_{2} ⊗ K_{1}) ⊗ L_{0}; 1⊗A_{1}; A_{3}; π_{Y}^{∗} K_{2} ⊗ L_{1}; A_{2}; L_{2}]

Then, for every w ∈ W_{Z}^{j}(Y, L_{0}), λ_{1} ∈ W^{i_{1}}(X, K_{1}) and λ_{2} ∈ W^{i_{2}}(X, K_{2}), we have

λ_{2} ·_{A_{2}} (λ_{1} ·_{A_{1}} w) = (λ_{2} · λ_{1}) ·_{A_{3}} w

in W_{Z}^{i_{1}+i_{2}+j}(Y, L_{2}).

Proof. The first part follows from Lemmas 1.3 and 1.5. The rest is direct :

λ_{2} ·_{A_{2}} (λ_{1} ·_{A_{1}} w) = A_{2}^{⟲}(π_{Y}^{∗} λ_{2} · A_{1}^{⟲}(π_{Y}^{∗} λ_{1} · w))  by definition
= A_{2}^{⟲} ∘ (1 ⊗ A_{1})^{⟲}(π_{Y}^{∗} λ_{2} · π_{Y}^{∗} λ_{1} · w) by Proposition 2.7 (c)
= A_{3}^{⟲}(π_{Y}^{∗} λ_{2} · π_{Y}^{∗} λ_{1} · w)  by (4) and Prop. 2.7
= A_{3}^{⟲}(π_{Y}^{∗} (λ_{2} · λ_{1}) · w)  π^{∗} is a ring morphism
= (λ_{2} · λ_{1}) ·_{A_{3}} w  by definition. □

Let us now discuss the lax-linearity of lax pull-back and lax push-forward.

5.7. Lemma. Let f : Ȳ → Y be a morphism in S_{X} and Z ⊂ Y be closed. Consider two lax pull-backs (Definition 3.4) :

Pull_{Ā,f} : W_{Z}^{∗}(Y, L) → W_{f^{−1}Z}^{∗}(Ȳ , L̄) and Pull_{B̄,f} : W_{Z}^{∗}(Y, M) → W_{f^{−1}Z}^{∗}(Ȳ , M̄)

for two alignments Ā : f^{∗}L ⇝ L̄ and B̄ : f^{∗}M ⇝ M̄ over Ȳ. Suppose that the line bundles L and M over Y are X-aligned, i.e. [L] = [M] in Pic_{X}(Y)/2.

(a) For every K-alignment C : L⇝_{K}M over Y, there exists a K-alignment C̄ : L̄⇝_{K}M̄ over Ȳ such that for all λ ∈ W^{∗}(X, K) and all w ∈ W_{Z}^{∗}(Y, L)

Pull_{B̄,f} (λ ·_{C} w) = λ ·_{C̄} ( Pull_{Ā,f}(w)) in W_{f^{−1}Z}^{∗}(Ȳ , M̄) .

(b) For every K-alignment C̄ : L̄⇝_{K}M̄ over Ȳ, there exists a K-alignment C : L⇝_{K}M over Y such that the very same equation holds (maybe better read from right to left this time).

Proof. For (a), use Lemma 1.3 to construct C̄ such that B̄ ∘ f^{∗}C ≃ C̄ ∘ (id ⊗Ā), i.e. solve the following left-hand square :

[figure: π∗ ; Ȳ K ⊗ f^{∗}L; f^{∗}C; id ⊗Ā; f^{∗}M; B̄; π∗ ; Ȳ K ⊗ L̄; ∃ C̄; M̄; π∗ ; Ȳ K ⊗ f^{∗}L; ∃ D̄; id ⊗Ā; f^{∗}M; B̄; π∗ ; Ȳ K ⊗ L̄; C̄; M̄]

For (b), first solve the above right-hand square to find D̄ and use Theorem 4.4 to find C : π_{Y}^{∗} K ⊗ L ⇝ M such that D̄ ≃ f^{∗}C. In both cases we have

(5)  B̄ ∘ f^{∗}C ≃ C̄ ∘ (id ⊗Ā) .

Then compute

Pull_{B̄,f} (λ ·_{C} w) = B̄^{⟲} ∘ f^{∗} ∘ C^{⟲}(π_{Y}^{∗} (λ) · w)  by definition
= B̄^{⟲} ∘ (f^{∗}C)^{⟲} ∘ f^{∗}(π_{Y}^{∗} (λ) · w)  by Corollary 3.2
= B̄^{⟲} ∘ (f^{∗}C)^{⟲}(π∗ ; Ȳ (λ) · f^{∗}(w)) f^{∗} is a ring homomorphism
= C̄^{⟲} ∘ (id ⊗Ā)^{⟲}(π∗ ; Ȳ (λ) · f^{∗}(w)) by (5) and Proposition 2.7
= C̄^{⟲}(π∗ ; Ȳ (λ) · (Ā^{⟲} ∘ f^{∗}(w)))  by Proposition 2.7 (c)
= λ ·C̄ ( Pull_{Ā,f}(w)) .  by definition.  □

5.8. Lemma. Let f : Ȳ → Y be a proper morphism in S_{X}, of constant relative dimension, and Z̄ ⊂ Ȳ be closed. Consider two lax push-forwards (Definition 3.5) :

Push_{f,Ā} : W⋆ ; Z̄(Ȳ , L̄) → W_{fZ̄}^{∗}(Y, L) and Push_{f,B̄} : W⋆ ; Z̄(Ȳ , M̄) → W_{fZ̄}^{∗}(Y, M)

where ⋆ = ∗ + dim f for two alignments Ā : L̄ ⇝ ω_{f} ⊗ f^{∗}L and B̄ : M̄ ⇝ ω_{f} ⊗ f^{∗}M over Ȳ. Suppose that the line bundles L and M over Y are X-aligned, i.e. [L] = [M] in Pic_{X}(Y)/2.

(a) For every K-alignment C : L⇝_{K}M over Y, there exists a K-alignment C̄ : L̄⇝_{K}M̄ over Ȳ such that for all w̄ ∈ W⋆ ; Z̄(Ȳ , L̄) and λ ∈ W^{∗}(X, K) we have

λ ·_{C} ( Push_{f,Ā}(w̄)) = Push_{f,B̄} (λ ·_{C̄} w̄) in W_{fZ̄}^{∗}(Y, M) .

(b) For every K-alignment C̄ : L̄⇝_{K}M̄ over Ȳ, there exists a K-alignment C : L⇝_{K}M over Y such that the very same property holds (read backwards).

Proof. For (a), use Lemma 1.3 to construct C̄ such that the following left-hand square commutes :

[figure: π∗ ; Ȳ K ⊗ L̄; ∃ C̄; Ā′:=(12)∘(id ⊗Ā); M̄; B̄; ω_{f} ⊗ π∗ ; Ȳ K ⊗ f^{∗}L; f^{!}C; ω_{f} ⊗ f^{∗}M; π∗ ; Ȳ K ⊗ L̄; C̄; Ā′; M̄; B̄; ω_{f} ⊗ π∗ ; Ȳ K ⊗ f^{∗}L; ∃ D̄; ω_{f} ⊗ f^{∗}M]

For (b), first solve the above right-hand square to find D̄ and use Corollary 4.5 to find C : π_{Y}^{∗} K ⊗ L ⇝ M such that D̄ ≃ f^{!}C̄. In both cases we have

(6)  B̄ ∘ C̄ ≃ f^{!}C ∘ Ā′ .

Then compute

Push_{f,B̄} (λ ·C̄ w̄) = f_{∗} ∘ B̄^{⟲} ∘ C̄^{⟲}(π∗ ; Ȳ (λ) · w̄)  by definition
= f_{∗} ∘ (f^{!}C)^{⟲} ∘ Ā′^{⟲}(π∗ ; Ȳ (λ) · w̄) by (6) and Proposition 2.7
= C^{⟲} ∘ f_{∗} ∘ Ā′^{⟲}(π∗ ; Ȳ (λ) · w̄)  by Corollary 3.2
= C^{⟲} ∘ f_{∗}(\underline{(12)}(π∗ ; Ȳ (λ) · Ā^{⟲}(w̄))) by Proposition 2.7 (c)
= C^{⟲}(π_{Y}^{∗} (λ) · f_{∗}(Ā^{⟲}(w̄)))  by projection formula for f_{∗}
= λ ·_{C} ( Push_{f,Ā}(w̄)) .  by definition.

The permutation of line bundles \underline{(12)} in the fourth equation is usually dropped but actually is the precise way to state the projection formula. □

5.9. Remark. Following up on Remark 2.10, it is easy to verify that the lax module structure is compatible with the localization long exact sequence.

6. Total bases of the total Witt group

We want to define what should be a basis of the non-existent total Witt group of Y with support in Z, over the similarly evanescent total Witt group of X. The intuitive meaning is simple. We want every Witt class of W^{∗}(Y, L) to be a sum of lax-products of elements of the basis by Witt classes over X (the coefficients) and we want no linear relation among the Witt classes in the basis, with coefficients over X.

6.1. Setup. We will repeatedly use the following situation : Let L_{1}, …, L_{n} be line bundles over an X-scheme Y, let j_{1}, …, j_{n} be integers and Z ⊂ Y a closed subset. We consider Witt classes w_{1}, …, w_{n} where each w_{i} ∈ W_{Z}^{j_{i}}(Y, L_{i}) lives in its own Witt group of Y with support in the common closed subset Z ⊂ Y. We want to make sense of linear combinations of w_{1}, …, w_{n} with coefficients in the Witt groups of X. With the lax module structure of Section 5, there are many ways to multiply each w_{i} by a coefficient λ_{i} over X. We clarify this first.

6.2. Definition. Let w_{1}, …, w_{n} be Witt classes over Y as in 6.1 and assume they are X-aligned in the sense of Definition 5.1, i.e. [L_{1}] = ⋯ = [L_{n}] in Pic_{X}(Y)/2.

A set of compatible coefficients for w_{1} , …, w_{n} consists of two ingredients :

• Witt classes λ_{1} ∈ W^{i_{1}}(X, K_{1}), …, λ_{n} ∈ W^{i_{n}}(X, K_{n}) over X, with the

property that i_{1} + j_{1} = i_{2} + j_{2} = ⋯ = i_{n} + j_{n}; call this number k ∈ ℤ.

• A K_{i}-alignment C_{i} : L_{i}⇝_{K_{i}}L (Definition 5.1), for every i = 1, …, n, for a common line bundle L over Y; that is, a pair C_{i} = (M_{i}, ϕ_{i}) where M_{i} is a

line bundle on Y and ϕ_{i} : M_{i}^{⊗2} ⊗ π_{Y}^{∗} K_{i} ⊗ L_{i} ⟶^{∼} L is an isomorphism.

When k ∈ ℤ and the line bundle L over Y are specified in advance, we speak of (k, L)-compatible coefficients. Naturally, we abbreviate all this by writing that the “λ_{1}, …, λ_{n} are compatible coefficients for w_{1}, …, w_{n}”. We also use the mildly abusive notation λ_{i}w_{i} for λ_{i} ·_{C_{i}} w_{i} when there is no risk of confusion, but we insist that the alignments C_{i} come with the coefficients λ_{i}’s in any case, possibly implicitly. This lax product λ_{i} ·_{C_{i}} w_{i} belongs to W^{k}(Y, L). We then define the lax linear combination of the w_{1}, …, w_{n} with coefficients λ_{1}, …, λ_{n} as the following element in W_{Z}^{k}(Y, L) :

∑ λ_{i}w_{i} := ∑ λ_{i} ·_{C_{i}} w_{i} .

6.3. Definition. Let Z ⊂ Y be closed and let I be a set. A family (w_{i})_{i∈I} of Witt classes w_{i} ∈ W_{Z}^{j(i)}(Y, L_{i}) is called totally independent over X if for every finite subset J of I such that the (w_{i})_{i∈J} are X-aligned and every compatible coefficients (λ_{i})_{i∈J}, the relation ∑_{J} λ_{i}w_{i} = 0 forces all λ_{i} to be zero.

6.4. Remark. In the following definitions, we are going to use a subset P of Pic_{X}(Y)/2. The reader might want to assume at first that P is the whole Pic_{X}(Y)/2 for this will often be the case. Allowing other P ’s will only become relevant when dealing with the functorial behavior of these notions, and only in “fringe cases”. If [L] ∈ P, we say that L is X-aligned with P and we also say that every Witt class w ∈ W_{Z}^{∗}(Y, L) is X-aligned with P. These conditions are empty for P = Pic_{X}(Y).

6.5. Definition. Let Z ⊂ Y be closed and P be a subset of Pic_{X}(Y)/2. Let (w_{i})_{i∈I} be a family of Witt classes over Y with support in Z, which are all Xaligned with P. We say that (w_{i})_{i∈I} totally generates the P -part of the Witt groups of Y with support in Z, over X, if for every line bundle L over Y such that [L] ∈ P, every integer k and every y ∈ W_{Z}^{k}(Y, L), there exists a finite subset J of I such that (w_{i})_{i∈J} are aligned with L, and (k, L)-compatible coefficients (λ_{i})_{i∈J} over X such that y = ∑_{i∈J} λ_{i}w_{i} as in Definition 6.2.

6.6. Definition. Let P ⊂ Pic_{X}(Y)/2 and Z ⊂ Y closed. We say that a family (w_{i})_{i∈I} of Witt classes X-aligned with P forms a total basis of the P -part of the Witt groups of Y with support in Z, over X, if it is totally independent (Definition 6.3) and totally generates (Definition 6.5).

6.7. Example. For Z = Y = X, the unit 1 ∈ W^{0}(X, O_{X}) is a total basis over X.

6.8. Remark. Unlike the classical notion, total independence does not strictly imply uniqueness of coefficients in a linear combination; given totally independent classes w_{1}, …, w_{n}, all X-aligned with some line bundle L, we could have ∑ λ_{i} ·_{C_{i}} w_{i} = ∑ λ′_{i} ·_{C′_{i}} w_{i} without λ_{i} = λ′_{i} for all i. Equality only follows from independence if the alignments C_{i} and C′_{i} are the same. However, Lemma 5.5 tells us that if Y is in S_{X}, we can find an alignment A_{i} : K_{i} ⇝ K′_{i} over X for every i such that λ_{i} ·_{C_{i}} w_{i} = λ′′_{i} ·_{C_{i}} w_{i} with λ′′_{i} = A_{i}^{⟲}(λ′_{i}). Then, we must have λ_{i} = λ′′_{i} by total independence.

Anyway, for X-schemes in our class S_{X} (Definition 4.1), we have the following “classical” interpretation of a total basis :

6.9. Proposition. Let Y ∈ S_{X}. Let P ⊂ Pic_{X}(Y)/2 be a subset, Z ⊂ Y closed and let (w_{i} ∈ W_{Z}^{j_{i}}(Y, L_{i}))_{i∈I} be a set of Witt classes on Y with support in Z, such that

each [L_{i}] ∈ P. For each p ∈ P set I_{p} = {i ∈ I | [L_{i}] = p in Pic_{X}(Y)/2}. Then the following properties are equivalent :

(i) The family (w_{i})_{i∈I} is a total basis of the P -part of the Witt groups of Y with support in Z, over X (Definition 6.6).

(ii) For every line bundle L with [L] ∈ P, every k ∈ ℤ and for every choice, for those i ∈ I_{[L]}, of a line bundle K_{i} over X and a K_{i}-alignment C_{i} : L_{i}⇝_{K_{i}}L, the following map is an isomorphism

(7)  \begin{matrix}θ = θ((C_{i})_{i}) : ⨁_{i∈I_{[L]}}W^{k−j_{i}}(X, K_{i}) & ⟶^{∼} & W_{Z}^{k}(Y, L) \\ (x_{i})_{i∈I_{[L]}} & ⟼ & ∑ x_{i} ·_{C_{i}} w_{i} . \end{matrix}

(iii) For every class p ∈ P and every k ∈ ℤ, there exists a choice of L ∈ p and there exists a choice, for each i ∈ I_{p}, of a line bundle K_{i} over X and a K_{i}-alignment C_{i} : L_{i}⇝_{K_{i}}L for which (7) is an isomorphism.

Note that θ as in (7) is always a homomorphism of W(X)-modules by Lemma 5.6.

Proof. (i) ⇒ (ii) : Injectivity is straightforward from total independence (Definition 6.3). For surjectivity, let y ∈ W_{Z}^{k}(Y, L) and use total generation (Definition 6.5) to write y as ∑_{i∈J} λ_{i} ·_{C_{i}} w_{i} for some finite subset J ⊂ I, some coefficients λ_{i} ∈ W^{k−j_{i}}(X, J_{i}) and some J_{i}-alignments D_{i} : L_{i}⇝_{J_{i}}L, for i ∈ J. A priori, J_{i} might differ from K_{i} and D_{i} might differ from C_{i}. But Lemma 5.5 tells us that each λ_{i} ·_{D_{i}} w_{i} = x_{i} ·_{C_{i}} w_{i} for a suitable x_{i} ∈ W^{∗}(X, K_{i}) lax-similar to λ_{i} . Hence y = ∑_{i∈J} x_{i} ·_{C_{i}} w_{i} ∈ im(θ). So, θ is surjective.

(ii) ⇒ (iii) : Do choose L ∈ p and X-alignments C_{i} : L_{i}⇝_{K_{i}}L for i ∈ I_{p}.

(iii) ⇒ (i) : Total generation is immediate from surjectivity of θ and Lemma 5.5. For total independence, let (w_{i})_{i∈J} be X-aligned (as in 6.1) and let (λ_{i})_{i∈J} be (L′, k)-compatible coefficients for some L′, such that ∑_{i∈J} λ_{i} ·_{D_{i}} w_{i} = 0 for suitable alignments D_{i}. Note that [L′] ∈ P. Choose L, K_{i} and C_{i} as in (iii) for p = [L′]. Choose also A : L′ ⇝ L. Then A^{⟲}( ∑_{i∈J} λ_{i} ·_{D_{i}} w_{i}) = 0 as well. By Lemmas 5.4 and 5.5 again, each A^{⟲}(λ_{i} ·_{D_{i}} w_{i}) = x_{i} ·_{C_{i}} w_{i} for some x_{i} lax-similar to λ_{i}. We then get θ((x_{i})_{i∈J}) = 0 which forces all x_{i} = 0 by injectivity of θ. But then λ_{i} = 0 as well since alignment isomorphisms are... isomorphisms. □

6.10. Lemma. Let P and P′ be subsets of Pic_{X}(Y)/2 and let (w_{i})_{i∈I} (resp. (w_{i})_{i∈I′}) be a totally generating family of the P -part (resp. the P′-part) of the Witt groups of Y with support in Z, over X. Then the union family (w_{i})_{i∈I∪I′} is a totally generating family of the P ∪ P′-part of the Witt groups of Y over X. If P and P′ are disjoint and if the families (w_{i})_{i∈I} and (w_{i})_{i∈I′} are both totally independent, then their union is totally independent.

Proof. Clear. □

6.11. Remark. Given i ≡ j modulo 4, there is a canonical isomorphism W^{i} →^{∼} W^{j}, given by (Σ^{2})^{\frac{j−i}{4}} , which involves no choice and no sign. Moreover, this isomorphism commutes with every pull-back, push-forward, alignment isomorphism and products (still no sign because \frac{j−i}{2} is even). In other words, a Witt class w ∈ W^{i} corresponds to a unique Witt class of W^{j}.

If one has an X-scheme Y and a family of Witt classes on Y, one can wonder whether the notions of total independence and total generation (Definitions 6.3 and 6.5) would be different if one identified every Witt class w ∈ W^{i} with its image in W^{j}, for j ≡ i modulo 4. The answer is no, as long as one does the same on X.

Indeed, Σ^{2}(λ ·_{A} w) = (Σ^{2}(λ)) ·_{A} w. Hence every occurrence of Σ^{2} on Y can be “absorbed” in the coefficients.

The following analogy might help the puzzled reader. If R = ⊕_{i∈ℤ}R^{i} is a ℤgraded ring and M = ⊕_{i∈ℤ}M^{i} is a graded R-module (or R-algebra), and if there exists s ∈ R^{4} invertible and central (nothing special about 4, of course), then one can consider the ℤ/4-graded ring R̄ = ⊕_{[i]∈ℤ/4}R^{i} with 0 ≤ i ≤ 3 and the graded R̄-module M̄ = ⊕_{[i]∈ℤ/4}M^{i} with 0 ≤ i ≤ 3, where a product taking values outside of the range 0 ≤ i ≤ 3 is brought back in that range by using the unique power of s which does the job. The point is that a collection M ⊂ ∪_{i∈ℤ}M^{i} of homogeneous elements in M form an R-basis of M if and only if the very same collection forms an R̄-basis of M̄ (once brought back in the range 0 ≤ i ≤ 3). In particular M̄ has the same dimension over R̄ as M had over R. The only possible confusion would come from the perverse contemplation of M̄ as an R-module.

Let us now examine how these notions behave under pull-backs or push-forwards. The assumption about injectivity of f_{|P}^{∗} below is the very reason we allow the flexibility of those subsets P ⊂ Pic_{X}(Y), see Remark 6.17.

6.12. Proposition. Let f : Ȳ → Y be a morphism of schemes in S_{X}. Let P be a subset of Pic_{X}(Y)/2 and Z ⊂ Y closed. Suppose that the pull-back map f^{∗}|_{P} : P → Pic_{X}(Ȳ)/2 is injective, as a map of sets. Suppose also that the pullback f^{∗} : W_{Z}^{k}(Y, L) → W_{f^{−1}(Z)}^{k}(Ȳ , f^{∗}L) is an isomorphism for all L with [L] ∈ P and all k ∈ ℤ.

Let (w_{i})_{i∈I} be a set of Witt classes w_{i} ∈ W_{Z}^{j_{i}}(Y, L_{i}) over Y with support in Z, with all [L_{i}] ∈ P. Choose for every i an alignment Ā_{i} : f^{∗}L_{i} ⇝ L̄_{i} over Ȳ, hence a lax pull-back Pull_{Ā_{i},f} : W_{Z}^{j_{i}}(Y, L_{i}) → W_{f^{−1}(Z)}^{j_{i}}(Ȳ , L̄_{i}). Let w̄_{i} := Pull_{Ā_{i},f}(w_{i}).

Then (w_{i})_{i∈I} is a total basis of the P -part of the Witt group of Y with support in Z, over X, if and only if (w̄_{i})_{i∈I} is a total basis of the f^{∗}P -part of the Witt group of Ȳ with support in f^{−1}Z, over X.

Proof. We simply use Proposition 6.9, and its notation, both for Y and for Ȳ, in the following commutative diagram (for line bundles and alignments to be specified) :

(8)  [figure: W^{k−j_{i}}(X, K_{i}); θ; W_{Z}^{k}(Y, L); _{Pull_{B̄,f}}; ≃; W^{k−j_{i}}(X, K_{i}); θ̄; W_{f^{−1}Z}^{k}(Ȳ , L̄)] ⨁_{i∈I_{[L]}} ⨁_{i∈I_{[L̄]}}

Let k ∈ ℤ. Given a line bundle L on Y, we can set L̄ = f^{∗}L and B̄ = id. Conversely, given L̄ on Ȳ with [L̄] ∈ f^{∗}(P) ⊂ Pic_{X}(Y)/2, Lemma 4.3 (d) provides an L over Y with [L] ∈ P such that f^{∗}[L] = [L̄] in Pic(Ȳ)/2 already. The latter allows us to choose B̄ : f^{∗}L ⇝ L̄ a (plain) alignment over Ȳ, hence to use the lax pull-back Pull_{B̄,f} as on the right-hand side of (8).

Of course, every w̄_{i} is X-aligned with f^{∗}P. Furthermore, our assumption about the Picard-group f^{∗} being injective on P implies that w_{i} is X-aligned with L if and only if w̄_{i} is X-aligned with L̄ (use part (c) of 4.3), thus f^{∗} : I_{[L]} →^{∼} I_{[L̄]} is a bijection and the left hand side of (8) also makes sense.

Now choose for every i ∈ I_{[L]} a K_{i}-alignment C_{i} : L_{i}⇝_{K_{i}}L, so that we can create θ : (x_{i}) ↦ ∑_{i∈I} x_{i} ·_{C_{i}} w_{i} in (8) as we did in (7). By Lemma 5.7 (a), there exists K_{i}-alignments C̄_{i} : f^{∗}L_{i}⇝_{K_{i}}L̄ over Ȳ such that

Pull_{B̄,f}(x ·_{C_{i}} w_{i}) = x ·_{C̄_{i}} Pull_{Ā_{i},f}(w_{i})

for all x ∈ W^{∗}(X, K_{i}) and all i ∈ I_{[L]}. So we can define θ̄ by (x_{i}) ↦ ∑ x_{i} ·C̄_{i} w̄_{i}, to make (8) commutative. Consequently, θ and θ̄ are simultaneously isomorphisms. By Proposition 6.9, (w_{i})_{i∈I} and (w̄_{i})_{i∈I} are simultaneously bases. □

6.13. Corollary. Hypotheses of Proposition 6.12 hold when f : Ȳ → Y is an affine bundle. So, in that case, a family is a total basis over X of the P -part of the Witt groups of Y with support in Z, if and only if, it is pulled-back to a total basis over X of the f^{∗}(P)-part of the Witt groups of Ȳ with support in f^{−1}Z. □

6.14. Corollary. For Y ∈ S_{X}, the notions of total independence, total generation, and total basis are stable under alignment isomorphisms (Definition 2.3). For instance, if (w_{i})_{i∈I} is a total basis of the P -part of the Witt group of Y with support in Z, over X, the family (A_{i}^{⟲}(w_{i}))_{i∈I} is still such a basis for any family of alignment isomorphisms (A_{i}^{⟲})_{i∈I} (e.g. multiplications by a unit of Y, see Example 2.4).

Proof. Apply Proposition 6.12 to f = id_{Y}. □

6.15. Proposition. Let f be a proper morphism of schemes in S_{X} with constant relative dimension d. Let P be a subset of Pic_{X}(Y)/2 and Z̄ ⊂ Ȳ. Let f^{!}P := [ω_{f}] · f^{∗}P ⊆ Pic_{X}(Ȳ)/2. Suppose the function f_{|P}^{∗} : P → Pic_{X}(Ȳ)/2 injective.

Suppose also that for any line bundle L such that [L] ∈ P, the push-forward map f_{∗} : Wk+d ; Z̄(Ȳ , ω_{f} ⊗ f^{∗}L) → W_{fZ̄}^{k}(Y, L) is an isomorphism for all k ∈ ℤ.

Then a family (w̄_{i})_{i∈I} of Witt classes on Ȳ with support in Z̄, X-aligned with f^{!}P, is a total basis of the f^{!}P -part of the Witt group of Ȳ with support in Z̄ if and only if the image family ( Push_{f,Ā_{i}}(w̄_{i}))_{i∈I} under any family of lax push-forwards corresponding to alignments (Ā_{i})_{i∈I} is a total basis of the P -part of the Witt group of Y with support in f(Z).

Proof. The proof is similar to that of Proposition 6.12, mutatis mutandis. One compares two θ homomorphisms that are “arranged” via Lemma 5.8 this time. □

In particular, the dévissage isomorphism for Witt groups yields the following.

6.16. Corollary. Let ι : Z ↪ Y be a closed immersion of constant codimension, with Z and Y in S_{X}. Let P be a subset of Pic_{X}(Y)/2 such that the map of sets ι_{|P}^{∗} : P → Pic_{X}(Z)/2 is injective. Let ι^{!}P = [ω_{ι}] · ι^{∗}P ⊆ Pic_{X}(Z)/2. Let (w_{i})_{i∈I} be elements of the P -part of the total Witt groups of Y with support in Z, and for each i ∈ I, let v_{i} be a Witt class in the ι^{!}P -part of the Witt groups of Z over X such that w_{i} = ι_{∗}(v_{i}) (this is always possible by dévissage). The family (v_{i})_{i∈I} is a total basis of the ι^{!}P -part of the Witt groups of Z, over X, if and only if the family (w_{i})_{i∈I} is a total basis of the P -part of the Witt groups of Y with support in Z, over X.

6.17. Remark. The injectivity condition on f^{∗} : P → Pic_{X}(Ȳ)/2 is not really harmful because one can always split Pic_{X}(Y)/2 in smaller P -chunks to ensure that the condition holds for each of them, and then use Lemma 6.10 to obtain a total basis for the whole P = Pic_{X}(Y)/2. This happens in “fringe” cases, in the cellular decomposition of the Grassmannians, for instance, see [3].

7. Total bases in the localization long exact sequence

Let U be the open complement of a closed subset Z ⊂ Y, and let υ : U ↪ Y be the corresponding open embedding. Assume both Y and U are in S_{X}. Let e : W_{Z}^{∗}(Y, L) → W^{∗}(Y, L) be the extension of support map. Recall that in this situation, there is a long exact sequence of localization

⋯ ⟶ W_{Z}^{i}(Y, L) ⟶^{e} W^{i}(Y, L) ⟶^{υ^{∗}} W^{i}(U, υ^{∗}L) ⟶^{∂} W_{Z}^{i+1}(Y, L) → ⋯

7.1. Theorem. Let P be a subset of Pic_{X}(Y)/2. Assume that the restriction υ_{|P}^{∗} : P → Pic_{X}(U)/2 is injective and let P_{U} = υ^{∗}(P) ⊂ Pic_{X}(U)/2.

Let I, J and K be sets and let (w′_{i})_{i∈I} and (w_{j})_{j∈J} be elements in Witt groups of Y, let (v_{i})_{i∈I} and (v′_{k})_{k∈K} be elements in Witt groups of Y with support in Z and let (u_{k})_{k∈K} and (u′_{j})_{j∈J} be elements in Witt groups of U, whose line bundles are restricted from Y. (Recall lax-similitude ↭ from Definition 2.5.) Suppose the following conditions hold (see Figure 1) :

(a) for every i ∈ I, we have e(v_{i}) ↭ w′_{i}

(b) for every j ∈ J, we have υ^{∗}(w_{j}) ↭ u′_{j}

(c) for every k ∈ K, we have ∂(u_{k}) ↭ v′_{k}.

Then, the following properties are satisfied :

(1) for every i ∈ I, we have υ^{∗}(w′_{i}) = 0;

(2) for every j ∈ J, we have ∂(u′_{j}) = 0;

(3) for every k ∈ K, we have e(v′_{k}) = 0.

(4) If, furthermore, out of the three following statements:

(i) the (v_{i})_{i∈I} and (v′_{k})_{k∈K} form a total basis of the P -part of the Witt groups of Y with support in Z, over X,

(ii) the (w′_{i})_{i∈I} and (w_{j})_{j∈J} form a total basis of the P -part of the Witt

groups of Y, over X, (iii) the (u_{k})_{k∈K} and (u′_{j})_{j∈J} form a total basis of the P_{U}-part of the

Witt groups of U, over X, two are true, then the remaining one is also true.

\begin{matrix}v_{i} & w′_{i} & u_{k} \\ v′_{k} & w_{j} & u′_{j} \end{matrix} eυ^{∗} ∂ P -part of W_{Z}(Y) P -part of W(Y) P_{U}-part of W(U)

Figure 1. Families mapping to each other up to lax-similitude in Theorem 7.1. No arrow means mapped to zero.

Proof. Parts (1)–(3) follow immediately from Remark 2.10. The proof of (4) goes through as for classical modules over a ring. Choose a class p ∈ P over Y, which by hypothesis is the same thing as choosing its image f^{∗}(p) ∈ f^{∗}P, i.e. a class in f^{∗}P over U. Up to replacing the u′_{j}, v′_{k} and w′_{i} up to lax-similitude, which does not change their total-basis qualities by Corollary 6.14, we can assume that relations (a)–(c) are equalities. Now, choosing X-alignments as in Proposition 6.9 for those u_{k}, u′_{j}, v_{i}, v′_{k}, w_{j} and w′_{i} which are X-aligned with L, one can define three homomorphisms θ as in that proposition. One can construct a split exact sequence at the level of Witt groups of X, and compare it with the localization long exact sequence for Y and L via the various θ. This is a morphism of long exact sequences by the compatibilities of the lax product structure with extension of support, restriction to an open and connecting homomorphism, see Remark 5.9. The Five Lemma then gives the result. We leave the details to the reader. □

7.2. Remark. The same theorem holds with support, i.e. replacing W(Y) by W_{Z′}(Y), and consequently W_{Z}(Y) by W_{Z∩Z′}(Y) and W(U) by W_{Z′∩U}(U).

7.3. Remark. The benefit of this theorem, together with the one on dévissage, is that we can build a total basis on Y out of smaller ones on Y with support in Z and on U. As in Remark 6.17 and for the same reasons, the injectivity assumption on P → Pic_{X}(U)/2 is not really restrictive in actual computations.

7.4. Remark. All this “total” formalism still holds in the non-necessarily regular case with the following modifications. All schemes should be noetherian and separated. The category S_{X} should be replaced by the category of X-schemes Y that have a dualizing complex (not necessarily injectively bounded), with the conditions on Picard groups left unchanged : the important point is that two dualizing complexes always differ by tensoring by a line bundle (and a shift). The Witt groups considered for such schemes Y should be the coherent Witt groups, and the formalism will mimic how they behave as a (total) module over the locally free Witt groups of X. In particular, the X-alignment of definition 5.1 should be replaced by an isomorphism ϕ : M^{⊗2} ⊗ π_{Y}^{∗} N ⊗ K′ →^{∼} K where M and N are line bundles, as before, but K and K′ are dualizing complexes. Pull-backs of coherent Witt groups should only be considered along flat morphisms preserving dualizing complexes (e.g. open embeddings in the localization sequence). Morphisms involving pull-backs of locally free Witt groups can be considered without restriction (e.g. pull-backs from the base X).

References

1. P. Balmer, Triangular Witt groups Part I: The 12-term localization exact sequence, K-Theory 4 (2000), no. 19, 311–363.

2. , Triangular Witt groups Part II: From usual to derived, Math. Z. 236 (2001), no. 2, 351–382.

3. P. Balmer and B. Calmès, Witt groups of Grassmann varieties, arXiv:0807.3296, 2007.

4. S. Gille and A. Nenashev, Pairings in triangular Witt theory, J. Algebra 261 (2003), no. 2, 292–309.

5. M. Knebusch, Symmetric bilinear forms over algebraic varieties, Queen’s Papers in Pure and Appl. Math., No. 46, Queen’s Univ., Kingston, Ont., 1977, pp. 103–283.

Paul Balmer, Department of Mathematics, UCLA, Los Angeles, CA 90095-1555, USA

   URL: http://www.math.ucla.edu/~balmer

Baptiste Calmès, Université d’Artois, Laboratoire de Mathématiques de Lens, France

   URL: http://www.math.uni-bielefeld.de/~bcalmes