The Quarterly Journal of Mathematics, Volume Advance Article (2) – Dec 14, 2017

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- Oxford University Press
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- 0033-5606
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- 10.1093/qmath/hax050
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Abstract The K-ring of symmetric vector bundles over a scheme X, the so-called Grothendieck–Witt ring of X, can be endowed with the structure of a (special) λ-ring. The associated γ-filtration generalizes the fundamental filtration on the (Grothendieck–)Witt ring of a field and is closely related to the ‘classical’ filtration by the kernels of the first two Stiefel–Whitney classes. 1. Introduction In this article, we establish a (special) λ-ring structure on the Grothendieck–Witt ring of symmetric vector bundles over a scheme, and some basic properties of the associated γ-filtration. As far as Witt rings of fields are concerned, there is an unchallenged natural candidate for a good filtration: the ‘fundamental filtration’, given by powers of the ‘fundamental ideal’. Its claim to fame is that the associated graded ring is isomorphic to the mod-2 étale cohomology ring, as predicted by Milnor [32] and verified by Voevodsky et al. [35, 43]. For the Witt ring of a more general variety X, there is no candidate filtration of equal renown. The two most frequently encountered filtrations are: A short filtration which we will refer to as the classical filtration Fclas*W(X), given by the whole ring, the kernel of the rank homomorphism, the kernels of the first two Stiefel–Whitney classes. This filtration is used, for example, in [16, 48]. The unramified filtration FK*W(X), given by the preimage of the fundamental filtration on the Witt ring of the function field K of X under the natural homomorphism W(X)→W(K). Said morphism is not generally injective (cf. [42]), at least not when dim(X)>3, and its kernel will clearly be contained in every piece of the filtration. Recent computations with this filtration include [19]. Clearly, the unramified filtration coincides with the fundamental filtration in the case of a field, and so does the classical filtration as far as it is defined. The same will be true of the γ-filtration introduced here. It may be thought of as an attempt to extend the classical filtration to higher degrees. In general, in order to define a ‘ γ-filtration’, we simply need to exhibit a pre- λ-structure on the ring in question. However, the natural candidates for λ-operations, the exterior powers, are not well-defined on the Witt ring W(X). We remedy this by passing to the Grothendieck–Witt ring GW(X). It is defined just like the Witt ring, except that we do not quotient out hyperbolic elements. Consequently, the two rings are related by an exact sequence K(X)→GW(X)→W(X)→0. The classical filtration and the unramified filtration on the Witt ring naturally extend to the Grothendieck–Witt ring GW(X) (see Section 5.2 for precise definitions). We will show that they are related to the γ-filtration on GW(X) as follows: Theorem 1.1 Let Xbe an integral scheme over a field kof characteristic not two. The γ-filtration on GW(k)is the fundamental filtration. The γ-filtration on GW(X)is related to the classical filtration as follows: Fγ1GW(X)=Fclas1GW(X)≔ker(GW(X)⟶rankZ)Fγ2GW(X)=Fclas2GW(X)≔ker(Fclas1GW(X)⟶w1Het1(X,Z/2))Fγ3GW(X)⊆Fclas3GW(X)≔ker(Fclas2GW(X)⟶w2Het2(X,Z/2)).However, the inclusion at the third step is not in general an equality. The γ-filtration on GW(X)is finer than the unramified filtration. We define the ‘ γ-filtration’ on the Witt ring as the image of the above filtration under the canonical projection GW(X)→W(X). Thus, each of the above statements easily implies an analogous statement for the Witt ring: the γ-filtration on the Witt ring of a field is the fundamental filtration, FγiW(X) agrees with FclasiW(X) for i<3, etc. The same example as for the Grothendieck–Witt ring (Example 6.5) will show that Fγ3W(X)≠Fclas3W(X) in general. Most statements of Theorem 1.1 also hold under weaker hypotheses—see (1) Proposition 5.1, (2) Propositions 5.4 and 5.8 and (3) Proposition 5.9. On the other hand, under some additional restrictions, the relation with the unramified filtration can be made more precise. For example, if X is a regular variety of dimension at most three and k is infinite, the unramified filtration on the Witt ring agrees with the global sections of the sheafified γ-filtration (Section 5.3). The crucial assertion is of course the equality of Fγ2GW(X) with the kernel of w1—all other statements would hold similarly for the naive filtration of GW(X) by the powers of the ‘fundamental ideal’ Fγ1GW(X). The equality follows from the fact that the exterior powers make GW(X) not only a pre- λ-ring, but even a λ-ring (In older terminology, pre- λ-rings are called ‘ λ-rings’, while λ-rings are referred to as ‘special λ-rings’. See also the introduction to [49].): Theorem 1.2 For any scheme Xover a field of characteristic not two, the exterior power operations give GW(X)the structure of a λ-ring. In the case when X is a field, this was established in [31]. The underlying pre- λ-structure for affine X has also recently been established independently in [46], where it is used to study sums-of-squares formulas. Although in this article the λ-structure is used mainly as a tool in proving Theorem 1.1, it should be noted that λ-rings and even pre- λ-rings have strong structural properties. Much of the general structure of Witt rings of fields could be (re)derived using the pre- λ-structure on their Grothendieck–Witt rings. As an example, we include a new proof of the well-known absence of odd torsion (see Corollary 4.4). Among the few results that generalize immediately to Grothendieck–Witt rings of schemes is the fact that torsion elements are nilpotent: this is true in any pre- λ-ring. For λ-rings, Clauwens has even found a sharp bound on the nilpotence degree [13]. In our situation, Clauwens result reads: Corollary Let Xbe as above. Suppose x∈GW(X)is an element satisfying pex=0for some prime pand some exponent e>0. Then xpe+pe−1=0. To put the corollary into context, recall that for a field k of characteristic not two, an element x∈GW(k) is nilpotent if and only if it is 2n-torsion for some n [29, VIII.8; 50]. This equivalence may be generalized at least to connected semi-local rings in which 2 is invertible, using the pre- λ-structure for one implication and [28, Example 3.11] for the other. See [31] for further applications of the λ-ring structure on Grothendieck–Witt rings of fields and [4] for nilpotence results for Witt rings of regular schemes. From the λ-theoretic point of view, the main complication in the Grothendieck–Witt ring of a general scheme as opposed to that of a field is that not all generators can be written as sums of line elements. In K-theory, this difficulty can often be overcome by embedding K(X) into the K-ring of some auxiliary scheme in which a given generator does have this property, but in our situation this is impossible: there is no splitting principle for Grothendieck–Witt rings (Section 4). 2. Generalities We understand a ring to be a ring with unit. 2.1. λ-rings We give a quick and informal introduction to λ-rings, treading medium ground between the traditional definition in terms of exterior power operations [38, Exposé V] and the abstract definition of λ-rings as coalgebras over a comonad [9, 1.17]. The main point we would like to get across is that a λ-ring is ‘a ring equipped with all possible symmetric operations’, not just ‘a ring with exterior powers’. This observation is not essential for anything that follows—we will later work exclusively with the traditional definition—but we hope that it provides some intrinsic motivation for considering this kind of structure. To make our statement more precise, let W be the ring of symmetric functions. That is, W consists of all formal power series ϕ(x1,x2,…) in countably many variables x1,x2,… with coefficients in Z such that ϕ has bounded degree and such that the image ϕ(x1,…,xn,0,0,…) of ϕ under the projection to Z[x1,…,xn] is a symmetric polynomial for all n. For example, W contains … elementary symmetric functions λk≔∑i1<⋯<ikxi1·…·xik, complete symmetric functions σk≔∑i1≤⋯≤ikxi1·…·xik, Adams symmetric functions ψk≔∑ixik, Witt symmetric functions θk [10, 4.5]. The elementary symmetric functions, the complete symmetric functions and the Witt symmetric functions each define a set of algebraically independent generators of W over Z, so they can be used to identify W with a polynomial ring in countably many variables. The Adams symmetric functions are also algebraically independent, but they only generate W⊗ZQ over Q. In any case, we have no need to choose any specific set of generators just now. Given a commutative ring A, we write WA for the universal λ-ring (This ring is also known as the big Witt ring with coefficients in A, or as the big ring of Witt vectors over A. We avoid this terminology here. While the ‘Witt’ in ‘Witt vectors’ and the ‘Witt’ in ‘Witt ring of quadratic forms’ refer to the same Ernst Witt, these are otherwise fairly independent concepts.) over A. As a set, it consists of all ring homomorphisms from W to A: WA=Rings(W,A). In particular, for every symmetric function ϕ∈W, we have an evaluation map evϕ:WA→A. The universal λ-ring WA becomes a commutative ring via a coproduct Δ+ and a comultiplication Δ× on W. (These cooperations are determined by the equations Δ+(ψn)=1⊗ψn+ψn⊗1 and Δ×(ψn)=ψn⊗ψn for all n [10, Introduction].) Given a ring homomorphism f:A→B, we obtain an induced ring homomorphism WA→WB via postcomposition. Thus, W defines an endofunctor on the category of commutative rings. Definition 2.1 A pre- λ-structure on a commutative ring A is a group homomorphism θA:A→WA such that commutes. A pre- λ-ring (A,θA) is a commutative ring A equipped with a fixed such structure θA. A morphism of pre- λ-rings (A,θA)→(B,θB) is a ring homomorphism f:A→B such that W(f)θA=θBf. We refer to such a morphism as a λ-morphism. It would, of course, appear more natural to ask for the map θA to be a ring homomorphism. But this requirement is only one of the two additional requirements reserved for λ-rings. The second additional requirement takes into account that the universal λ-ring WA can itself be equipped with a canonical pre- λ-structure for any ring A. Definition 2.2 A pre- λ-structure θA:A→WA is a λ-structure if it is a λ-morphism. A λ-ring (A,θA) is a commutative ring A equipped with a λ-structure θA. It turns out that the canonical pre- λ-structure does make the universal λ-ring WA a λ-ring, so the terminology is the same. The observation alluded to at the beginning of this section is that any symmetric function ϕ∈W defines an ‘operation’ on any (pre-) λ-ring A, that is a map A→A: the composition of evϕ with θA: In particular, we have families of operations λk, σk, ψk and θk corresponding to the symmetric functions specified above. They are referred to as exterior power operations, symmetric power operations, Adams operations and Witt operations. The underlying additive group of the universal λ-ring WA is isomorphic to the multiplicative group (1+tA〚t〛)× inside the ring of invertible power series over A, and the isomorphism can be chosen such that the projection onto the coefficient of ti corresponds to evλi (cf. [23, Proposition 1.14, Remark 1.22]). Thus, a pre- λ-structure is completely determined by the operations λi, and conversely, any family of operations λi for which the map A⟶λt(1+tA〚t〛)×a↦1+λ1(a)t+λ2(a)t2+⋯ is a group homomorphism, and for which λ1(a)=a, defines a λ-structure. This recovers the traditional definition of a pre- λ-structure as a family of operations λi (with λ0=1 and λ1=id) satisfying the relation λk(x+y)=∑i+j=kλi(x)λj(y) for all k≥0 and all x,y∈A. The question whether the resulting pre- λ-structure is a λ-structure can similarly be reduced to certain polynomial identities, though these are more difficult to state and often also more difficult to verify in practice. However, for pre- λ-rings with some additional structure, there are certain standard criteria that make life easier. Definition 2.3 An augmented (pre-) λ-ring is a (pre-) λ-ring A together with a λ-morphism d:A→Z, where the (pre-) λ-structure on Z is defined by λi(n)≔(ni). A (pre-) λ-ring with positive structure is an augmented (pre-) λ-ring A together with a specified subset A>0⊂A on which d is positive and which generates A in the strong sense that any element of A can be written as a difference of elements in A>0; it is moreover required to satisfy a list of axioms for which we refer to [49, Section 3]. For example, one of the axioms for a positive structure is that for an element e∈A>0, the exterior powers λke vanish for all k>d(e). We will refer to elements of A>0 as positive elements, and to positive elements l of augmentation d(l)=1 as line elements. The motivating example, the K-ring K(X) of a connected scheme X, is augmented by the rank homomorphism, and a set of positive elements is given by the classes of non-zero vector bundles. The Grothendieck–Witt ring GW(X) of a connected scheme is likewise augmented by the rank homomorphism, and a set of positive elements is given by the classes of non-zero symmetric vector bundles. Here are two simple criteria for showing that a pre- λ-ring with positive structure is a λ-ring: Splitting Criterion If all positive elements of A decompose into sums of line elements, then A is a λ-ring. Detection Criterion If for any pair of positive elements e1,e2∈A>0 we can find a λ-ring A′ and a λ-morphism A′→A with both e1 and e2 in its image, then A is a λ-ring. We again refer to [49] for details. 2.2. The γ-filtration The γ-operations on a pre- λ-ring A can be defined as γn(x)≔λn(x+n−1). They again satisfy the identity γk(x+y)=∑i+j=kγi(x)γj(y). Definition 2.4 The γ-filtration on an augmented pre- λ-ring A is defined as follows: Fγ0A≔AFγ1A≔ker(A⟶dZ)FγiA≔(subgroupgeneratedbyallfiniteproducts∏jγij(aj)withaj∈Fγ1Aand∑jij≥i)fori>1. This is in fact a filtration by ideals, multiplicative in the sense that FγiA·FγjA⊂Fγi+jA, hence we have an associated graded ring grγ*A≔⨁iFγiA/Fγi+1A. See [2, Section 4] or [18, III Section 1] for details. The following lemma is sometimes useful for concrete computations. Lemma 2.5 If Ais a pre- λ-ring with positive structure such that every positive element in Acan be written as a sum of line elements, then FγkA=(Fγ1A)k. More generally, suppose that Ais an augmented pre- λ-ring, and let E⊂Abe some set of additive generators of Fγ1A. Then FγkAis additively generated by finite products of the form ∏jγij(ej)with ej∈Eand ∑jij≥k. Proof The first assertion may be found in [18, III Section 1]. It also follows from the second, which we now prove. As each x∈Fγ1A can be written as a linear combination of elements of E, we can write any γi(x) as a linear combination of products of the form ∏jγij(±ej) with ej∈E and ∑jij=i. Thus, FγkA can be generated by finite products of the form ∏jγij(±ej), with ej∈E and ∑jij≥k. Moreover, γi(−e) is a linear combination of products of the form ∏jγij(e) with ∑jij=i: this follows from the above identity for γk(x+y). Thus, FγkA is already generated by products of the form described.□ For λ-rings with positive structure, we also have the following general fact: Lemma 2.6 [18, III, Theorem 1.7] For any λ-ring Awith positive structure, the additive group gr1A=Fγ1A/Fγ2Ais isomorphic to the multiplicative group of line elements in A. 3. The λ-structure on the Grothendieck–Witt ring Given a scheme X, we denote by GW(X) and W(X) the Grothendieck–Witt and the Witt ring of the exact category with duality of vector bundles over X. Precise definitions may be found in [27, Section 4] or [37, Section 2]. No assumption on the invertibility of 2 is required in these definitions. For a ring R, the Witt ring W(Spec(R)) is the classical Witt ring W(R) of symmetric bilinear forms over R. When 2 is invertible, it may equivalently be defined in terms of quadratic forms. 3.1. The pre- λ-structure Proposition 3.1 Let Xbe a scheme. The exterior power operations λk:(M,μ)↦(ΛkM,Λkμ)induce well-defined maps on GW(X)which provide GW(X)with the structure of a pre- λ-ring. Our proof of the existence of a pre- λ-structure will follow the same pattern as the proof for symmetric representation rings in [49]: Step 1. The assignment λi(M,μ)≔(ΛiM,Λiμ) is well defined on the set of isometry classes of symmetric vector bundles over X, so that we have an induced map λt:{isometryclassesofsymmetricvectorbundlesoverX}⟶(1+tGW(X)〚t〛)×. We extend it linearly to a group homomorphism ⨁Z(M,μ)⟶(1+tGW(X)〚t〛)×, where the sum on the left is over all isometry classes of symmetric vector bundles over X. Step 2. The map λt is additive in the sense that λt((M,μ)⊥(N,ν))=λt(M,μ)λt(N,ν). Thus, it factors through the quotient of ⨁Z(M,μ) by the ideal generated by the relations ((M,μ)⊥(N,ν))=(M,μ)+(N,ν). Step 3. The homomorphism λt respects the relation (M,μ)=H(L) for every metabolic vector bundle (M,μ) with Lagrangian L. Thus, we obtain the desired factorization λt:GW(X)→(1+tGW(X)〚t〛)×. To carry out these steps, we only need to replace all arguments on the level of vector spaces of [49] with local arguments. We formulate the key lemma in detail and then sketch the remaining part of the proof. Filtration Lemma 3.2 (c.f. [38, Exposé V]) Fix a natural number n and vector bundles Land Nover a scheme X. With any extension e=(0→L→Me→N→0)of vector bundles, we can associate a filtration Me•of Λn(Me)by sub-vector bundles Λn(Me)=Me0⊃Me1⊃Me2⊃⋯together with isomorphisms πei:Mei/Mei+1≅ΛiL⊗Λn−iN.More precisely, we can do so in a unique way that is functorial with respect to restrictions to open subsets and isomorphisms of extensions, in the sense detailed in condition (1) below, and moreover normalized in the sense of condition (2) below. To describe the functoriality, we first define appropriate categories. Keep X, L, N and n fixed as above. The source category of our functor will be a category ext whose objects are pairs (U,e) consisting of an open subscheme U⊂X and an extension of vector bundles e=(0→L∣U→Me→N∣U→0) over U. A morphism (U,e)→(V,f) in ext is a pair (U⊂V,ϕ) consisting of an inclusion U⊂V and an isomorphism of extensions ϕ:e≅f∣U: As target category, we consider a category filtext whose objects are quadruples (U,e,Me•,πe•) with (U,e)∈ext as above, Me• a descending filtration of Λn(Me) and πe• a sequence of isomorphisms πei:Mei/Mei+1≅(ΛiL⊗Λn−iN)∣U as in the Filtration Lemma 3.2. A morphism (U,e,Me•,πe•)→(V,f,Mf•,πf•) in filtext is a morphism (U⊂V,ϕ):(U,e)→(V,f) in ext with the additional property that Λnϕ restricts to isomorphisms ϕi:Mei⟶≅(Mfi)∣U (1filt) that are compatible with the isomorphisms πei and πfi in the sense that the following triangle commutes for all i: (1π) The conditions of the Filtration Lemma 3.2 are: The association of a filtration Me• and isomorphisms πei with an extension of vector bundles defines a functor: ext→filtext(U,e)↦(U,e,Me•,πe•). This functor is normalized in the sense that, for any open U⊂X, the trivial extension e:(0→L∣U⟶(id0)(L⊕N)∣U⟶(0id)N∣U→0) is sent to the filtration of Λn(L⊕N)∣U≅⨁j(ΛjL⊗Λn−jN)∣U by the sub-vector bundles Mei≔⨁j≥i(ΛjL⊗Λn−jN)∣U, together with the isomorphisms πei:Mei/Mei+1⟶≅(ΛiL⊗Λn−iN)∣U induced by the canonical projections. Proof of the Filtration Lemma 3.2 We first show that a functor as in (1) satisfying condition (2) is uniquely determined. So suppose such a functor exists, pick an arbitrary extension (V,f)∈ext and let (V,f,Mf•,πf•) be its image under this functor. Choose a cover of V by open subsets U over which the extension f splits. For each such open subset U, we then have a morphism (U,e)→(V,f) in ext, where e denotes the trivial extension over U. The image of (U,e) under our functor is determined by (2), and thus the restrictions of the sub-vector bundles Mfi and of the isomorphisms πfi to each U are uniquely determined by (1filt) and (1π). As the open subsets U cover V, this determines the filtration Mf• and the isomorphisms πfi completely. To prove the existence of such a functor, we give an explicit construction. To simplify notation, we describe the functor only on objects of the form (X,e), that is, on extensions defined over X itself, but it will be clear that the same construction makes sense for extensions over open subsets. So let e=(0→L⟶ιM⟶πN→0) be such an extension over X. Consider the morphism ΛiL⊗Λn−iM→ΛnM induced by ι. Let Mi be its kernel and Mi its image, so that we have a short exact sequence of quasi-coherent sheaves: We claim (a) that the sheaves Mi and Mi are again vector bundles, (b) that the morphism ΛiL⊗Λn−iM→ΛiL⊗Λn−iN induced by π factors through Mi and induces an isomorphism πMi:Mi/Mi+1→ΛiL⊗Λn−iN, (c1) that this construction of subbundles Mi and isomorphisms πMi is functorial in the sense of condition (1) above, and (c2) that it satisfies the normalization property (2). These claims can be checked in the following order. First, verify claim (c2): the above construction yields the desired filtration and isomorphisms for split extensions. Next, verify half of claim (c1): the above filtration is functorial in the sense that, given a morphism (U⊂V,ϕ) in ext, the morphism Λnϕ induces an isomorphism of filtrations as in (1filt). Then claim (a) follows because locally the extension splits, and hence Mi and Mi are locally isomorphic to the vector bundles ⨁j<iΛjL⊗Λn−jN and ⨁j≥iΛjL⊗Λn−jN, respectively. Claim (b) can similarly be checked by restricting to open subsets over which the extension splits. Finally, to complete the verification of claim (c1), we need to verify the commutativity of the triangles (1π). Again, it suffices to do so locally.□ Proof of Proposition 3.1 For Step 1, we note that the exterior power operation Λi:Vec(X)→Vec(X) is a duality functor in the sense that we have an isomorphism ηM:Λi(M∨)→≅(ΛiM)∨ for any vector bundle M. Indeed, we can define ηM on sections by ϕ1∧⋯∧ϕi↦(m1∧⋯∧mi↦det(ϕα(mβ)). The fact that this is an isomorphism can be checked locally and follows from [11, Chapter 3, Section 11.5, (30 bis)]. We therefore obtain a well-defined operation on the set of isometry classes of symmetric vector bundles over X by defining λi(M,μ)≔(ΛiM,ηM◦Λi(μ)). Step 2 is completely analogous to the argument in [49]. For Step 3, let (M,μ) be metabolic with Lagrangian L, so that we have a short exact sequence 0→L⟶iM⟶i∨μL∨→0. (3.1) When n is odd, say n=2k−1, we claim that Mk is a Lagrangian of Λn(M,μ). When n is even, say n=2k, we claim that Mk+1 is an admissible sub-Lagrangian of Λn(M,μ) with (Mk+1)⊥=Mk, and that the composition of isomorphisms Mk/Mk+1≅ΛkL⊗Λk(L∨)≅H(L)k/H(L)k+1 of the Filtration Lemma 3.2 is even an isometry between (Mk/Mk+1,μ) and (H(L)k/H(L)k+1,(0110¯)). All of these claims can be checked locally. Both the local arguments and the conclusions are analogous to those of [49].□ Remark 3.3 In many cases, Step 3 can be simplified. If X is affine, then Step 3 is redundant because any short exact sequence of vector bundles splits. If X is a regular quasi-projective variety over a field, we can reduce the argument to the affine case using Jouanolou’s trick and homotopy invariance. Indeed, Jouanolou’s trick yields an affine vector bundle torsor π:E→X [45, Proposition 4.3], and as X is regular, π* is an isomorphism: it is an isomorphism on K(−) by [45, below Example 4.7], an isomorphism on W(−) by [20, Corollary 4.2], and hence an isomorphism on GW(−) by Karoubi induction. 3.2. The pre- λ-structure is a λ-structure We would now like to show that the pre- λ-structure on GW(X) discussed above is an actual λ-structure. In some cases, this is easy: Proposition 3.4 For any connected semi-local commutative ring Rin which 2 is invertible, the Grothendieck–Witt ring GW(R)is a λ-ring. Proof Over such a ring, any symmetric space decomposes into a sum of line elements [8, Proposition I.3.4/3.5], so the result follows from the splitting criterion (Section 2.1).□ The following result is more interesting: Theorem 3.5 For any connected scheme Xover a field of characteristic not two, the pre- λ-structure on GW(X)introduced above is a λ-structure. The usual strategy for proving the analogous result in K-theory is via a ‘geometric splitting principle’: see Section 4. However, as we will see there, no such principle is available for the Grothendieck–Witt ring. So instead, we follow an alternative strategy, which we recall from [38, Exposé VI] (The result of Serre invoked at the end of the proof in [38] to verify (K1) is [40, Theorem 4].). Let G be a linear algebraic group scheme over k. The principal components of this alternative strategy are: (K1) The representation ring K(Rep(G)) is a λ-ring. (K2) For any G-torsor S over X, we have a well-defined map K(Rep(G))→K(X) sending a representation V of G to the vector bundle S×GV, and this map is a λ-morphism. (The V in S×GV is to be interpreted as a trivial vector bundle over X.) (K3) For any pair of vector bundles E and F over X, there exists a linear algebraic group scheme G and a G-torsor S such that both E and F lie in the image of the morphism K(Rep(G))→K(X) defined by S. ( G can be chosen to be a product of general linear groups.) From these three points, the fact that K(X) is a λ-ring follows via the detection criterion (Section 2.1). The same argument will clearly work for GW(X) provided the following three analogous statements hold. We now assume that chark≠2. (GW1) The symmetric representation ring GW(Rep(G)) is a λ-ring. (GW2) For any G-torsor S over X, we have a well-defined map GW(Rep(G))→GW(X) sending a symmetric representation (V,ν) of G to a symmetric vector bundle S×G(V,ν), and this map is a λ-morphism. (GW3) Any pair of symmetric vector bundles lies in the image of some common morphism GW(Rep(G))→GW(X) defined by a G-torsor as above. ( G can be chosen to be a product of split orthogonal groups.) Statement (GW1) is the main result of [49]. (This is where the assumption on the characteristic of k enters.) The remaining points (GW2) and (GW3) are discussed below: see Propositions 3.13 and 3.15. Any reader to whom (K2) and (K3) are obvious will most likely consider (GW2) and (GW3) equally obvious. The only point of note is that for (GW3) we have to allow étale G-torsors rather than just Zariski G-torsors. This modification does not create any problems as the notions of vector bundles for these two topologies are known to coincide: Proposition 3.6 For any scheme (X,O), the restriction functor from the étale site to the Zariski site induces an equivalence of categories Vec(Xet)⟶≃Vec(X)between the category of finite locally free O-modules in the étale topology and the category of finite locally free O-modules in the Zariski topology. This equivalence is compatible with the exact structure of the two categories and the usual duality functor HomO(−,O) on either side. Thus, K(X) and GW(X) may equivalently be defined in terms of étale vector bundles and symmetric étale vector bundles, respectively. References for Proposition 3.6 This is a classical and well-known result. The key ingredient, the Zariski local triviality of étale GLn-torsors, is due to Serre [39]. For a streamlined exposition, see for example the stacks project [41]: by Tag [03DX], we have an equivalence of the respective categories of quasi-coherent O-modules, and by [05VG/05B2] this equivalence restricts to an equivalence of the respective categories of finite locally free O-modules. In [03FH], the functor inducing the equivalence is identified with the left adjoint of the restriction functor, so the restriction itself must be an equivalence.□ Twisting by torsors Let (X,O) be a scheme. Fix a topology τ∈{Zariski,étale} on X and let Xτ denote the associated small site. Up to Proposition 3.13, this choice will be irrelevant, but ultimately, it will be the étale topology that we are interested in. We will refer to members of τ-coverings {Ui→X}i as τ-opens. Given a sheaf of not necessarily abelian groups G on Xτ, a (right) G-torsor on Xτ is a sheaf of sets S on Xτ with a right G-action such that: There exists a τ-covering {Ui→X}i such that S(Ui)≠∅ for all Ui. (Any such covering is said to split S.) For all τ-opens (U→X), and for one (hence for all) s∈S(U), the map G∣U→S∣Ug↦s.g is an isomorphism. Definition 3.7 Let S be a G-torsor as above. For any presheaf E of O-modules on Xτ with an O-linear left G-action, we define a new presheaf of O-modules on Xτ by S×ˆGE≔coeq((S×G)⊗E⇉S⊗E)=coker(⨁S,GE⟶⨁SE), where on any τ-open U, the morphism in the second line has the form s,g[v]↦s.g[v]−s[g.v] for s∈S(U), g∈G(U) and v∈E(U). (We use square brackets with a subscript on the left for an element of a direct sum that is concentrated in a single summand. A general element of ⨁s∈SXs is a finite sum of the form ∑s∈Ss[xs] in this notation.) If E is a sheaf on Xτ, we define S×GE as the τ-sheafification of S×ˆGE. Equivalently, we may define S×GE by the same formula as S×ˆGE provided we interpret the direct sum and cokernel as direct sum and cokernel in the category of sheaves of O-modules on Xτ. Remark 3.8 The sheaf of O-modules S×GE can alternatively be described as follows. Fix a τ-covering {Ui→X}i which splits S, and fix an element si∈S(Ui) for each Ui. Let gij∈G(Ui×XUj) be the unique element satisfying sj=si.gij. Then S×GE is isomorphic to the sheaf given on any τ-open (V→X) by {(vi)i∈∏iE(Vi)∣vi=gij.vjonVij}, where Vi≔V×XUi, Vij≔Vi×XVj. (We will not use this description in the following.) The above construction is known as twisting by a torsor or contracted product with a torsor. Many variants of the construction itself and much of the material that follows are well-known and may be found at various levels of generality in the literature, see for example [21, Chapter III] or [14]. However, having been unable to locate a reference that perfectly suits our needs, we instead include enough details for all claims to be easily checked. We begin by recalling some of the basic properties of S×GE. We call two presheaves of O-modules on Xτ τ-locally isomorphic if X has a τ-covering such that the restrictions to each open of the covering are isomorphic. A morphism of presheaves of O-modules is said to be τ-locally an isomorphism if X has a τ-covering such that the restriction of the morphism to each open of the covering is an isomorphism. Lemma 3.9 For any presheaf Eas above, S×ˆGEis τ-locally isomorphic to E. For any sheaf Eas above, S×GEis τ-locally isomorphic to E. The canonical morphism S×ˆGE→S×GEis τ-locally an isomorphism for any sheaf as above.More precisely, the presheaves in (i) & (ii) are isomorphic over any τ-open (U→X)such that S(U)≠∅, and likewise the morphism in (iii) is an isomorphism over any such U. Proof For (i) of the lemma, let (U→X) be a τ-open such that S(U)≠∅. Fix any s∈S(U). For each (V→U) and each t∈S(V), there exists a unique element gt∈G(V) such that t=s.gt. Therefore, the morphism ⨁S∣UE∣U→E∣U sending t[v] to gt.v describes the cokernel defining S×ˆGE over U. Statements (ii) and (iii) of the lemma follow from (i).□ Lemma 3.10 The functor S×G−is exact, that is, it takes exact sequences of sheaves of O-modules with O-linear G-action on Xτto exact sequences of sheaves of O-modules on Xτ. In particular, the functor is additive. If Eis a sheaf of O-modules on Xτwith trivial G-action, then S×GE≅E. Given arbitrary sheaves of O-modules Eand Fwith O-linear G-actions on Xτ, consider E⊗F, ΛiEand E∨with the induced G-actions. We have the following isomorphisms of O-modules on Xτ, natural in Eand F: θ:S×G(E⊗F)≅(S×GE)⊗(S×GF).λ:S×G(ΛkE)≅Λk(S×GE).η:S×G(E∨)≅(S×GE)∨. Proof (i) If we fix s and U as in the proof of Lemma 3.9, then the induced isomorphism S×GE∣U→E∣U is functorial for morphisms of O-modules with O-linear G-action. The claim follows as exactness of a sequence of τ-sheaves can be checked τ-locally. (ii) When G acts trivially on E, the τ-local isomorphisms of Lemma 3.9 do not depend on choices and glue to a global isomorphism. (iii) It is immediate from Lemma 3.9 that in each case the two sides are τ-locally isomorphic, but we still need to construct global morphisms between them. For ⊗ and Λk, we first note that all constructions involved are compatible with τ-sheafification, in the following sense: let ⊗ˆ and Λˆ denote the presheaf tensor product and the presheaf exterior power, and let (−)+ denote τ-sheafification. Then, for arbitrary presheaves of O-modules E and F on Xτ, the canonical morphisms (E⊗ˆF)+→(E+)⊗(F+)(ΛˆkE)+→Λk(E+)(S×ˆGE)+→S×G(E+) are isomorphisms. (In the third case, this follows from Lemma 3.9.) The arguments for ⊗ and Λk are very similar, so we only discuss the latter functor. Let ⨁ˆ denote the (infinite) direct sum in the category of presheaves. We first check that the morphism ⨁ˆS(ΛˆkE)→Λˆk(⨁ˆSE) which identifies the summand s[ΛˆkE] on the left with Λˆk(s[E]) on the right induces a well-defined morphism S×ˆG(ΛˆkE)⟶λˆΛˆk(S×ˆGE). Secondly, we claim that λˆ is τ-locally an isomorphism. For this, we only need to observe that over any U such that S(U)≠∅, we have a commutative triangle where the diagonal arrows are induced by the isomorphisms of Lemma 3.9. For dualization, one of the τ-sheafification morphisms goes in the wrong direction, so the argument is slightly different. Again, we first construct a morphism of presheaves ηˆ:S×ˆGE∨→(S×ˆGE)∨. Over τ-opens U such that S(U)=∅, the left-hand side is zero, so we take the zero morphism. Over τ-opens U with S(U)≠∅, we define (S×ˆGE∨)(U)→ηˆ(S×ˆGE)∨(U)s[ϕ]↦(s.g[v]↦ϕ(g.v)). Over these U with S(U)≠∅, the morphism is in fact an isomorphism. To define a local inverse, pick an arbitrary s∈S(U), and send ψ on the right-hand side to s[v↦ψ(s[v])] on the left. Finally, given the morphism ηˆ, we consider the following square in which α and β are τ-sheafification morphisms: By Lemma 3.9, both α and β are τ-locally isomorphisms, and it follows that β∨ is likewise τ-locally an isomorphism. The diagonal morphism is defined as follows: over any U with S(U)=∅, it is the zero morphism, and over all other U, it is the composition of ηˆ with the (local) inverse of β∨. Thus, the dotted diagonal is a factorization of ηˆ over (S×GE)∨. The latter being a sheaf, this factorization must further factor through S×GE∨. We thus obtain the horizontal morphism of sheaves η which, being a τ-locally an isomorphism, must be an isomorphism.□ Twisting symmetric bundles Recall that a duality functor is a functor between categories with dualities F:(A,∨,ω)→(B,∨,ω) together with a natural isomorphism η:F(−∨)⟶≅F(−)∨ such that commutes. In the context of exact categories with duality, an exact duality functor is simply an exact functor which is moreover a duality functor in this sense [5, 1.1.15]. Any such exact duality functor induces a functor Fsym between the associated categories of symmetric spaces, sending a symmetric space (A,α) over A to the symmetric space (FA,ηAFα) over B, and descends to a morphism of Grothendieck–Witt groups GW(A,∨,ω)→GW(B,∨,ω) [37, Section 3.1]. Now consider the category Vec(Xτ) of τ-vector bundles over X, that is, the category of finite locally free O-modules over Xτ. Let GVec(Xτ) denote the category of τ-vector bundles over X equipped with a G-action and G-equivariant morphisms. Equip both categories with their usual dualities and exact sequences. By Lemma 3.9 and Lemma 3.10 (i), S×G− restricts to an exact functor GVec(Xτ)→Vec(Xτ). By Lemma 3.10 (iii), we moreover have a natural isomorphism η:S×G(−∨)≅(S×G(−))∨. The commutativity of the triangle in the definition of a duality functor is easily checked, so we deduce: Lemma 3.11 (S×G−,η)is an exact duality functor GVec(Xτ)→Vec(Xτ). A symmetric space (E,ε) over GVec(Xτ) is a symmetric τ-vector bundle on which G acts through isometries. For any such symmetric space, we now define S×G(E,ε)≔(S×G−)sym(E,ε). The τ-local isomorphisms of Lemma 3.9 are isometries in this case, that is, S×G(E,ε) is τ-locally isometric to (E,ε). Lemma 3.12 For symmetric spaces (E,ε)and (F,ϕ)over GVec(Xτ), the natural isomorphisms θand λof Lemma3.10(iii) respect the symmetric structures. Proof Temporarily writing F for the functor S×G−, checking the claim amounts to the following. For symmetric bundles (E,ε) and (F,ϕ), the isomorphism θE,F is an isometry from F(E⊗F) to FE⊗FF with respect to the induced symmetries if and only if the outer square of the diagram on the left commutes: Similarly, for a symmetric vector bundle (E,ε), the isomorphism λE is an isometry from F(ΛkE) to Λk(FE) if and only if the outer square of the diagram on the right above commutes. In both cases, we already know that the upper square commutes for all E and F, by naturality of θ and λ. So it suffices to verify that the lower square commutes. This can be checked τ-locally, and follows easily from the descriptions of η, θ and λ given in the proof of Lemma 3.9.□ Proofs of the statements We are now ready to prove the statements (K2), (K3) and (GW2), (GW3) made above. Recall from Proposition 3.6 that K(X) and GW(X) may equivalently be defined in terms of étale vector bundles over X. Proposition 3.13 (K2 & GW2) Let π:X→Spec(k)be a scheme over some field k. For any algebraic group scheme Gover k, and for any étale G-torsor S, the maps K(RepkG)→K(X)GW(RepkG)→GW(X)V↦S×Gπ*V(V,ν)↦S×Gπ*(V,ν)are well-defined λ-morphisms. Proof The homomorphism between the K-groups is induced by the composition of exact functors RepkG⟶π*GVec(Xet)⟶S×G−Vec(Xet). It follows from Lemma 3.11 that this composition is an exact duality functor. So the composition also induces a well-defined homomorphism between the respective Grothendieck–Witt groups. Lemma 3.10(iii) and Lemma 3.12 imply that both homomorphisms are λ-morphisms.□ Proposition 3.14 (K3) Let Xbe as in Proposition3.13, and let Vnbe the standard representation of GLn. Any vector bundle Eis isomorphic to S×GLnπ*Vnfor some Zariski GLn-torsor S. For any two vector bundles Eand F, there exists a Zariski GLn×GLm-torsor Ssuch that E≅S×GLn×GLmπ*VnF≅S×GLn×GLmπ*Vm.Here GLmis supposed to act trivially on Vn, and GLnis supposed to act trivially on Vm. Proposition 3.15 (GW3) Let Xbe a scheme over a field k of characteristic not two. Let (Vn,qn)be the standard representation of Onover k, equipped with its standard symmetric form. Any symmetric étale vector bundle (E,ε)is isometric to S×Onπ*(Vn,qn)for some étale On-torsor S. For any two symmetric étale vector bundles (E,ε)and (F,ϕ), there exists an étale On×Om-torsor Ssuch that (E,ε)≅S×On×Omπ*(Vn,qn)(F,ϕ)≅S×On×Omπ*(Vm,qm).Here, Omis supposed to act trivially on Vn, and Onis supposed to act trivially on Vm. Proof of Proposition 3.14 Identify π*Vn with O⊕n, and let S be the Zariski sheaf of isomorphisms S≔Iso(O⊕n,E) with GLn=Aut(O⊕n) acting by precomposition. This is a Zariski GLn-torsor as E is Zariski locally isomorphic to O⊕n. Moreover, evaluation f[v]↦f(v) defines an isomorphism ev:S×ˆGO⊕n⟶≅E. Indeed, this can be checked (Zariski) locally—for any s∈Iso(O⊕n,E)(V), the restriction of ev to V factors as (S×ˆGO⊕n)∣V⟶≅O∣V⊕n⟶s≅E∣V, where the first arrow is the isomorphism f[v]↦gf(v) of Lemma 3.9 determined by s. (Compare [14, Theorem 3.1(4)].) Suppose S is a Zariski G-torsor, S′ is a Zariski G′-torsor, and E is a Zariski sheaf of O-modules with O-linear actions by both G and G′. Then if G′ acts trivially, (S×S′)×G×G′E≅S×GE. We can, therefore, take S≔Iso(O⊕n,E)×Iso(O⊕m,F). □ Proof of Proposition 3.15 We have On=Aut(O⊕n,qn). So let S be the étale sheaf of isometries S≔Iso((O⊕n,qn),(E,ε)) with On acting by precomposition. This is an étale On-torsor since any symmetric étale vector bundle (E,ε) is étale locally isometric to (O⊕n,qn) (cf. [24, 3.6]). The rest of the proof works exactly as in the non-symmetric case.□ 4. (No) splitting principle The splitting principle in K-theory asserts that any vector bundle behaves like a sum of line bundles. There are two incarnations: The algebraic splitting principle: For any positive element e of a λ-ring with positive structure A, there exists an extension of λ-rings with positive structure A↪Ae such that e splits as a sum of line elements in Ae. The geometric splitting principle: For any vector bundle E over a scheme X, there exists an X-scheme π:XE→X such that the induced morphism π*:K(X)↪K(XE) is an extension of λ-rings with positive structure, and such that π*E splits as a sum of line bundles in K(XE). Both incarnations are discussed in [18, I]. An extension of a λ-ring with positive structure A is simply an injective λ-morphism to another λ-ring with positive structure A↪A′, compatible with the augmentation and such that A≥0 maps to A≥0′. 4.1. No splitting principle for GW For GW(X), the analogue of the geometric splitting principle fails: Over any field of characteristic not two, there exists a (smooth, projective) scheme X and a symmetric vector bundle (E,ε) over X such that there exists no X-scheme π:X(E,ε)→X for which the class of π*(E,ε) in GW(X(E,ε)) splits into a sum of symmetric line bundles. The natural analogue of the algebraic splitting principle could be formulated using the notion of a real λ-ring: Definition 4.1 A real λ-ring is a λ-ring with positive structure A in which any line element squares to one. This property is clearly satisfied by the Grothendieck–Witt ring GW(X) of any scheme X. However, an algebraic splitting principle for real λ-rings fails likewise: There exist a real λ-ring A and a positive element e∈A that does not split into a sum of line elements in any extension of real λ-rings A↪Ae. The failure of both splitting principles is clear from the following simple counterexample: Lemma 4.2 Let P2be the projective plane over some field kof characteristic not two. Consider the element e≔H(O(1))∈GW(P2). There exists no extension of λ-rings GW(P2)↪Aesuch that Aeis real and such that esplits as a sum of line elements in Ae. Proof For any element a in a real λ-ring that can be written as a sum of line elements, the Adams operations ψn are given by ψn(a)={rank(a)ifniseven,aifnisodd. However, for e≔H(O(1))∈GW(P2), we have ψ2(e)≠2, so ψ2(e) cannot be a sum of line bundles, neither in GW(P2) itself nor in any real extension. Explicitly, GW(P2)=GW(k)⊕Za with a2=0 and ϕ·a=rank(ϕ)·a for any ϕ∈GW(k), and in this notation e=⟨1,−1⟩+a (see Example 6.3 below). We deduce from this that e2=2⟨1,−1⟩+4a. The second exterior power of e can be computed directly as λ2(e)=det(H(O(1)))=(O,−id)=⟨−1⟩. So altogether we find that ψ2(e)=e2−2λ2(e)=2⟨1,−1⟩+4a−2⟨−1⟩=2+4a, which differs from 2, as claimed.□ We note in passing the following general property of real λ-rings, from which we recover a classical result on torsion in Witt rings [8, Theorem V.6.3]: Proposition 4.3 A real λ-ring in which every positive element can be written as a sum of line elements contains no p-torsion elements for any odd prime p. Corollary 4.4 Let Rbe a connected semi-local ring in which 2 is invertible. Then GW(R)and W(R)contain no p-torsion elements for odd primes p. Proof of Proposition 4.3 On an arbitrary λ-ring A, we can consider the Witt operations θk:A→A. For primes p, they are related to the Adams operations by ψp(a)=ap+pθp(a) [10, 4.5]. More importantly for us, they are also related by ψp(a)=pp−1ap+θp(pa) (4.1) for all a∈A: see [12, Proposition 2.4] or [13, Proposition 1], but note that Clauwens writes −θp for what we call θp. It is clear from (4.1) that θp(0)=0. Now assume that A is a real λ-ring in which all positive elements decompose into sums of line elements. Then as in the proof of Lemma 4.2, we find that ψn=id for all odd n. In particular, for all odd primes p, Equation (4.1) simplifies to a=pp−1ap+θp(pa). Thus, if pa=0 in A, we find that a=0+θp(0)=0.□ Proof of Corollary 4.4 For GW(R), the claim is immediate from Proposition 4.3 and the proof of Proposition 3.4. For W(R), note that any p-torsion element for an odd prime p is necessarily contained in the fundamental ideal I(R), and that the fundamental ideal GI(R) of GW(R) projects isomorphically onto I(R) (c.f. Remark 5.2).□ 4.2. A splitting principle for étale cohomology Despite the negative result above, we do have a splitting principle for Stiefel–Whitney classes of symmetric bundles. Let X be any scheme over Z[12]. Proposition 4.5 For any symmetric bundle (E,ε)over Xthere exists a morphism π:X(E,ε)→Xsuch that π*(E,ε)splits as an orthogonal sum of symmetric line bundles over X(E,ε)and such that π*is injective on étale cohomology with Z/2-coefficients. Proof Recall the geometric construction of higher Stiefel–Whitney classes of Delzant and Laborde, as explained for example in [15, Section 5]: given a symmetric vector bundle (E,ε) as above, the key idea is to consider the scheme of non-degenerate one-dimensional subspaces π:Pnd(E,ε)→X, that is, the complement of the quadric in P(E) defined by ε. (This is an algebraic version of the projective bundle associated with a real vector bundle in topology; cf. [47, Lemma 1.7].) Let O(−1) denote the restriction of the universal line bundle over P(E) to Pnd(E,ε). This is a subbundle of π*E, and by construction the restriction of π*ε to O(−1) is non-degenerate. Let w be the first Stiefel–Whitney class of this symmetric line bundle O(−1). The étale cohomology of Pnd(E,ε) decomposes as Het*(Pnd(E,ε),Z/2)=⨁i=0r−1π*Het*(X,Z/2)·wi, and the higher Stiefel–Whitney classes of (E,ε) can be defined as the coefficients of the equation expressing wr as a linear combination of the smaller powers wi in Het*(Pnd(E,ε),Z/2). We only need to note two facts from this construction. First, over Pnd(E,ε) we have an orthogonal decomposition π*(E,ε)≅(O(−1),ϵ′)⊥(E″,ε″), where E″=O(−1)⊥ and ε′ and ε″ are the restrictions of π*ε. Secondly, π induces a monomorphism from the étale cohomology of X to the étale cohomology of Pnd(E,ε). So the proposition is proved by iterating this construction.□ 5. The γ-filtration on the Grothendieck–Witt ring From now on, we assume that X is connected. As we have seen, GW(X) is a pre- λ-ring with positive structure, and we can consider the associated γ-filtration FγiGW(X) of GW(X). The image of this filtration under the canonical epimorphism GW(X)↠W(X) will be denoted FγiW(X). In particular, by definition, Fγ1GW(X)=Fclas1GW(X)≔ker(rank:GW(X)→Z),Fγ1W(X)=Fclas1W(X)≔ker(rank¯:W(X)→Z/2). For a field, or more generally for a connected semi-local ring R, we also write GI(R) and I(R) instead of Fclas1GW(R) and Fclas1W(R), respectively. 5.1. Comparison with the fundamental filtration Proposition 5.1 For any connected semi-local commutative ring Rin which 2 is invertible, the γ-filtration on GW(R)is the filtration by powers of the augmentation ideal GI(R), and the induced filtration on W(R)is the filtration by powers of the fundamental ideal I(R). Proof As we have already noted in the proof of Proposition 3.4, all positive elements of the Grothendieck–Witt ring GW(R) can be written as sums of line elements. Thus, the claim concerning GW(R) is immediate from Lemma 2.5. Moreover, the fundamental filtration on W(R) is the image of the fundamental filtration on GW(R).□ Remark 5.2 In the situation above, the projection GW(R)→W(R) restricts to an isomorphism of GW(R)-modules GIi(R)→Ii(R) for every i>0. So grγiGW(R)≅grγiW(R) in positive degrees i. This fails for general schemes in place of R (see Section 6). Remark 5.3 It may seem more natural to define a filtration on GW(X) starting with the kernel not of the rank morphism but of the rank reduced modulo two, as for example in [3]: GI′(X)≔ker(GW(X)→H0(X,Z/2)). For connected X, GI′(X) is isomorphic to a direct sum of GI(X) and a copy of Z generated by the hyperbolic plane H. In particular, GI(X) and GI′(X) have the same image in W(X). However, even over a field, the filtration by powers of GI′ does not yield the same graded ring as the filtration by powers of ( GI or) I. For example, for X=Spec(R), we find GIn(R)/GIn+1(R)≅Z/2(n>0),(GI′)n(R)/(GI′)n+1(R)≅Z/2⊕Z/2. It is the filtration by powers of GI that yields an associated graded ring isomorphic to Het*(R,Z/2) in positive degrees, not the filtration by powers of GI′. 5.2. Comparison with the classical filtration The classical filtration on the Witt ring of a scheme is given by the kernels of the first two étale Stiefel–Whitney classes w1 and w2 on the Grothendieck–Witt ring and of the induced classes w¯1 and w¯2 on the Witt ring Fclas2GW(X)≔ker(Fclas1GW(X)⟶w1Het1(X,Z/2)),Fclas2W(X)≔ker(Fclas1W(X)G⟶w¯1Het1(X,Z/2)),Fclas3GW(X)≔ker(Fclas2GW(X)⟶w2Het2(X,Z/2)),Fclas3W(X)≔ker(Fclas2W(X)G⟶w¯2Het2(X,Z/2)/Pic(X)). Proposition 5.4 Let Xbe any connected scheme over a field of characteristic not two (or, more generally, any scheme such that the canonical pre- λ-structure on GW(X)is a λ-structure). Then Fγ2GW(X)=Fclas2GW(X),Fγ2W(X)=Fclas2W(X). Proof The first identity is a consequence of Lemma 2.6. In our case, the group of line elements is the multiplicative group of isomorphism classes of symmetric line bundles over X, or, equivalently, of étale O1-torsors, and hence may be identified with Het1(X,O1)=Het1(X,Z/2). Under this identification, the determinant GW(X)→Het1(X,Z/2) is precisely the first Stiefel–Whitney class w1. In particular, the kernel of the restriction of w1 to Fclas1GW(X) is Fγ2GW(X), as claimed. For the second identity, it suffices to observe that Fclas2GW(X) maps surjectively onto Fclas2W(X).□ In order to analyse the relation of Fγ3GW(X) to Fclas3GW(X), we need a few lemmas concerning products of ‘reduced line elements’: Lemma 5.5 Let u1,…,ul,v1,…,vlbe line elements in a pre- λ-ring Awith positive structure. Then γk(∑i(ui−vi))can be written as a linear combination of products (ui1−1)⋯(uis−1)(vj1−1)⋯(vjt−1)with s+t=kfactors. Proof This is easily seen by induction over l. For l=1 and k=0, the statement is trivial, while for l=1 and k≥1, we have γk(u−v)=γk((u−1)+(1−v))=γ0(u−1)γk(1−v)+γ1(u−1)γk−1(1−v)=±(v−1)k∓(u−1)(v−1)k−1. For the induction step, we observe that every summand in γk(∑i=1l+1ui−vi)=∑i=0kγi(∑i=1lui−vi)γk−i(ul−vl) can be written as a linear combination of the required form.□ Lemma 5.6 Let Xbe a scheme over Z[12], and let u1,…,un∈GW(X)be classes of symmetric line bundles with Stiefel–Whitney classes w1(ui)=:u¯i. Let ρdenote the product ρ≔(u1−1)⋯(un−1).Then wi(ρ)=0for 0<i<2n−1, and w2n−1(ρ)=∏1≤i1<⋯<ik≤nwithkodd(u¯i1+⋯+u¯ik)=∑r1,…,rn:2r1+⋯+2rn=2n−1u¯12r1⋯u¯n2rn. Proof The lemma generalizes [32, Lemma 3.2/Corollary 3.3]. The first part of Milnor’s proof applies verbatim. Consider the evaluation map Z/2〚x1,…,xn〛⟶ev∏iHeti(X,Z/2) sending xi to u¯i. The total Stiefel–Whitney class w(ρ)=1+w1(ρ)+w2(ρ)+⋯ is the evaluation of the power series ω(x1,…,xn)≔(∏∣ϵ∣even(1+ϵx)∏∣ϵ∣odd(1+ϵx))(−1)n, where the products range over all ϵ=(ϵ1,…,ϵn)∈(Z/2)n with ∣ϵ∣≔ϵ1+⋯+ϵn even or odd, and where ϵx denotes the sum ∑iϵixi. As Milnor points out, all factors of ω cancel if we substitute xi=0 for some i. More generally, all factors cancel whenever we replace a given variable xi by the sum of an even number of variables xi1+⋯+xi2l all distinct from xi. Indeed, consider the substitution xn=αx with ∣α∣ even and αn=0. Write x=(x′,xn), ϵ=(ϵ′,ϵn) and α=(α′,0), so that the substitution may be rewritten as xn=α′x′. Then (ϵ′,ϵn)(x′,α′x′)=(ϵ′+α′,ϵn+1)(x′,α′x′), but the parities of ∣(ϵ′,ϵn)∣ and ∣(ϵ′+α′,ϵn+1)∣ are different. Thus, the corresponding factors of ω cancel. It follows that ω−1 is divisible by all sums of an odd number of distinct variables xi1+⋯+xik. Therefore, ω=1+(∏∣ϵ∣oddϵx)·f(x) (5.1) for some power series f. In particular, ω has no non-zero coefficients in positive total degrees below ∑kodd(nk)=2n−1, proving the first part of the lemma. For the second part, we need to show that the constant coefficient of f is 1. This can be seen by considering the substitution x1=x2=⋯=xn=x in (5.1): we obtain (1(1+x)K)±1=1+xKf(x,…,x), with K=∑kodd(nk)=2n−1, and as (1+xK)=1+xKmod2 for K a power of two, this equation can be rewritten as (1+xK)∓1=1+xKf(x,…,x). The claim follows. Finally, the identification of the product expression for w2n−1(ρ) with a sum is [22, Lemma 2.5]. It is verified by showing that all factors of the product divide the sum, using similar substitution arguments as above.□ Remark 5.7 Milnor’s proof in the case when X is a field k uses the relation a∪2=[−1]∪a in H2(k,Z/2), which does not hold in general. Proposition 5.8 Let Xbe a connected scheme over Z[12]. Then wi(FγnGW(X))=0for0<i<2n−1. In particular Fγ2GW(X)⊂Fclas2GW(X),Fγ3GW(X)⊂Fclas3GW(X). Proof Let x≔γk1(x1)⋯γkl(xl) be an additive generator of GWn(X), that is, xi∈ker(rank) and ∑ki≥n. By writing each xi as [Ei,ϵi]−[Fi,ϕi] for certain symmetric vector bundles (Ei,ϵi) and (Fi,ϕi) and successively applying the splitting principal for étale cohomology (Proposition 4.5) to each of these, we can find a morphism Xx→X, which is injective on étale cohomology with Z/2-coefficients, and such that each π*xi is a sum of differences of line bundles. By Lemma 5.5, each γki(π*xi) can, therefore, be written as a linear combination of products (u1−1)⋯(um−1) with m=ki factors, where each ui is the class of some line bundle over Xx. Using the naturality of the γ-operations, it follows that π*x can be written as a linear combination of such products with m≥n factors. By Lemma 5.6, the classes wi vanish on every summand of this linear combination for 0<i<2n−1. So wi(π*x)=0 for all 0<i<2n−1, and by the naturality of Stiefel–Whitney classes and the injectivity of π* on cohomology, we may conclude that wi(x) vanishes in this range.□ 5.3. Comparison with the unramified filtration Here, we quickly summarize some observations on the relation of the γ-filtration with the ‘unramified filtration’. First, let X be an integral scheme with function field K, and let FK*GW(X) denote the unramified filtration of GW(X), given by the preimages of GIi(K) under the natural map GW(X)→GW(K). Said map is a morphism of augmented λ-rings, so FγiGW(X) maps to FγiGW(K)=GIi(K) and we obtain: Proposition 5.9 For any integral scheme X, the γ-filtration on GW(X)is finer than the unramified filtration, that is FγiGW(X)⊂FKiGW(X)for all i.□ Following [3, Section 2.2], we define the unramified Grothendieck–Witt group of X as GWur(X)≔⋂x∈X(1)im(GW(OX,x)→GW(K)), where X(1) denotes the set of codimension one points of X. Let us consider the functors GW and GWur as the presheaves on our given integral scheme X that send an open subset U⊂X to GW(U) or GWur(U), respectively. Then GWur is a sheaf, and we have a sequence of morphisms of presheaves GW→GW+→GWur↪GW(K), where (−)+ denotes sheafification and GW(K) is to be interpreted as the constant sheaf with value GW(K). The unramified filtration of GWur is obtained by intersecting the fundamental filtration on GW(K) with GWur: FKiGWur≔GWur∩GIi(K). This is a filtration by sheaves, and the unramified filtration FKiGW is given by the preimage of FKiGWur under the above morphisms. When X is regular integral of finite type over a field of characteristic not two, the purity results of Ojanguren and Panin [33; 34, Theorem A] imply that the morphism GW+→GWur is an isomorphism. If we further assume that the field is infinite, a result of Kerz and Müller–Stach yields the following: Proposition 5.10 For any regular integral scheme of finite type over an infinite field of characteristic not two, the γ-filtration and the unramified filtration have the same sheafifications: (FγiGW)+=(FKiGW)+=FKiGWur. Proof As already mentioned, the results of Ojanguren and Panin imply that GW+ injects into GW(K) in this situation, with image GWur. In particular, the stalks of GWur are those of GW: GWx=(GWur)x=GW(OX,x). Consequently, the unramified filtration has stalks (FKiGW)x=(FKiGWur)x=GW(OX,x)∩GIi(K). The γ-filtration FγiGW on the other hand, also viewed as a presheaf, has stalks FγiGW(OX,x). By Proposition 5.1 above and [26, Corollary 0.5], these stalks agree.□ Both propositions apply verbatim to the Witt ring W in place of GW. If, in addition to the assumptions of Proposition 5.10, our scheme is separated and of dimension at most three, then by [7] the Witt presheaf W is already a sheaf, and hence also FKiW is a filtration by sheaves. This justifies the claim made in the introduction that the ‘the unramified filtration of the Witt ring is the sheafification of the γ-filtration’ in this situation. 6. Examples All our examples will be smooth quasi-projective varieties over a field of characteristic different from two. The lower-degree pieces of the filtrations on the K-, Grothendieck–Witt and Witt rings will, therefore, always fit the following pattern: Fγ0K=Ftop0K=K,Fγ0GW=GW,Fγ0W=W,Fγ1K=Ftop1K=ker(rank),Fγ1GW=ker(rank),Fγ1W=ker(rank¯),Fγ2K=Ftop2K=ker(c1),Fγ2GW=ker(w1),Fγ2W=ker(w¯1),Fγ3K⊂Ftop3K=ker(c2),Fγ3GW⊂ker(w2),Fγ3W⊂ker(w¯2). (For the topological filtration Ftop* on the K-ring, see [17, Example 15.3.6]. The symbols ci denote the Chern classes with values in Chow groups.) Accordingly, the first Chern class c1 and the first Stiefel–Whitney classes w1 and w¯1 induce isomorphisms: grγ1K≅Picgrγ1GW≅Het1(−,Z/2)grγ1W≅Het1(−,Z/2). Some details concerning the computations for each of the following examples are provided at the end of this section. Example 6.1 (Curve). Let C be a smooth curve over a field of 2-cohomological dimension at most 1, for example over an algebraically closed field or over a finite field. Then grγ*GW(C)=grclas*GW(C)≅Z⊕Het1(C,Z/2)⊕Het2(C,Z/2),grγ*W(C)=grclas*W(C)≅Z/2⊕Het1(C,Z/2). Example 6.2 (Surface). Let X be a smooth surface over an algebraically closed field. Setting FclasiGW(X)=FclasiW(X)≔0 for i>3, we obtain grclas*GW(X)≅Z⊕Het1(X,Z/2)⊕Het2(X,Z/2)⊕CH2(X),grγ*W(X)=grclas*W(X)≅Z/2⊕Het1(X,Z/2)⊕Het2(X,Z/2)/Pic(X). However, in general, Fγ3GW(X)⊊Fclas3GW(X)=CH2(X). For a concrete example, consider the product X=C×P1, where C is any smooth projective curve. In this case, Fclas3GW(X)≅Pic(C),Fγ3GW(X)≅Pic(C)[2](kernelofmultiplicationby2). Example 6.3 ( Pr). Let Pr be the r-dimensional projective space over a field k. We first describe its Grothendieck–Witt ring. Let a≔H0(O(1)−1) and ρ≔⌈r2⌉. Then GW(Pr)≅{GW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1⊕ZaρifrisevenGW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1⊕(Z/2)aρifr≡−1mod4GW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1ifr≡−1mod4. The multiplication is determined by the formula ϕ·ai=rank(ϕ)ai for ϕ∈GW(k) and i>0, and by the vanishing of all higher powers of a (that is, ai=0 for all i≥ρ when r≡−1mod4; ai=0 for all i>ρ in the other cases). (Over k=C, this agrees with the ring structure of KO(CPn) as computed by Sanderson [36, Theorem 3.9].) In this description, FγiGW(Pr) is the ideal generated by FγiGW(k) and a⌈i2⌉. In particular, Fγ3GW(X) is again strictly smaller than Fclas3GW(X): Fclas3GW(Pr)=Fγ3GW(k)+(a2,2a),Fγ3GW(Pr)=Fγ3GW(k)+(a2). The associated graded ring looks very similar to the ring itself: grγ*GW(Pr)≅{grγ*GW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1⊕Zaρifrisevengrγ*GW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1⊕(Z/2)aρifr≡−1mod4grγ*GW(k)⊕Za⊕Za2⊕⋯⊕Zaρ−1ifr≡−1mod4, with a of degree 2. In the Witt ring, all the hyperbolic elements ai vanish, so obviously grγ*W(Pr)≅grγ*W(k). Example 6.4 ( A1−0). For the punctured affine line over a field k, we have GW(A1−0)≅GW(k)⊕W(k)εˆFγiGW(A1−0)≅GIi(k)⊕Ii−1(k)εˆ, for some generator εˆ∈Fγ1GW(A1−0) satisfying εˆ2=2εˆ. In this example, Fγ3GW(A1−0)=ker(w2). Example 6.5 ( A4n+1−0). For punctured affine spaces of dimensions d≡1mod4 with d>1, there is a similar result for the Grothendieck–Witt group [6] GW(A4n+1−0)≅GW(k)⊕W(k)εˆ, for some εˆ∈Fγ1GW(A1−0). However, in this case εˆ2=0, and the γ-filtration is also different from the γ-filtration in the one-dimensional case. This is already apparent over the complex numbers, where we find FγiGW(AC5−0)≅FγiW(AC5−0)≅{Z/2εˆfori=1,20fori≥3. In particular, in this example, Fγ3W(X)≠Fclas3W(X), the latter being non-zero since since w2 and w¯2 are zero. Calculations for Example 6.1 (Curve) Consider the summary at the beginning of this section. In dimension 1, we have Ftop2K=0, so Fγ2K=ker(c1)=0. Moreover, by [48, proof of Corollary 3.7], w2 is surjective for the curves under consideration, with kernel isomorphic to the kernel of c1. So w2 is an isomorphism. It follows that Fγ3GW=Fγ3W=0, and hence that grγ*GW=grclas*GW and grγ*W=grclas*W. These graded groups are computed in [loc. cit., Theorem 3.1 and Corollary 3.7].□ Calculations for Example 6.2 (Surface) The classical filtration is computed in [48, Corollary 3.7/4.7]. In the case X=C×P1, Walter’s projective bundle formula [44, Theorem 1.5] and the results on GW*(C) of [48, Theorem 2.1/3.1] yield Here, π:X↠C is the projection and Ψ∈GW1(P1) is a generator. Writing Hi:K→GWi for the hyperbolic maps, we can describe the additive generators of GW(X) explicitly as follows: 1 (the trivial symmetric line bundle), aL≔π*L−1, for each symmetric line bundle L on C, that is for each L∈Pic(C)[2], b≔H0(π*L1−1), where L1 is a line bundle of degree 1 on C (hence a generator of the free summand of Pic(C)), c≔H−1(1)·Ψ=H0(FΨ); here FΨ=O(−1)−1 with O(−1) the pullback of the canonical line bundle on P1, dN≔H−1(π*N−1)·Ψ=H0((π*N−1)·FΨ), for each N∈Pic(C). In this list, the generators appear in the same order as the direct summands of GW(X) that they generate appear in the formula above. An alternative set of generators is obtained by replacing the generators dN by the following generators: dN′≔H0(π*N⊗O(−1)−1)={dN+cifNisofevendegreedN+b+cifNisofodddegree. The only non-trivial products of the alternative generators are aLc=aLdN′=dL′+c(=dL). Moreover, the effects of the operations γi on the alternative generators is immediate from Lemma 6.6 below. So Lemma 2.5 tells us that Fγ3GW has additive generators γ1(aL)·γ2(c)=aL·(−c)=dL with L∈Pic(C)[2]. Thus, Fγ3GW(X)≅Pic(C)[2], viewed as subgroup of the last summand in the formula above. We also find that Fγ4GW(X)=0.□ Calculation of the ring structure on GW(Pr) (Example 6.3) By [44, Theorems 1.1 and 1.5], the Grothendieck–Witt ring of projective space can be additively described as GW(Pr)={GW(k)⊕Za1⊕⋯⊕ZaρifrisevenGW(k)⊕Za1⊕⋯⊕Zaρ−1⊕(Z/2)H0(FΨ)ifr≡−1mod4GW(k)⊕Za1⊕⋯⊕Zaρ−1ifr≡−1mod4, where ai=H0(O(i)−1) and Ψ is a certain element in GWr(Pr). Moreover, by tracing through Walter’s computations, we find that H0(FΨ)=−∑j=1ρ(−1)j(r+1ρ−j)aj. (6.1) Indeed, we see from the proof of [44, Theorem 1.5] that FΨ=O⊕N−λρ(Ω)(ρ) in K(Pr), where Ω is the cotangent bundle of Pr and N is such that the virtual rank of this element is zero. The short exact sequence 0→Ω→O⊕(r+1)(−1)→O→0 over Pr implies that λρ(Ω)=λρ(O⊕(r+1)(−1)−1)inK(Pr), from which (6.1) follows by a short computation. An element Ψ∈GWr(Pr) also exists in the case r≡−1mod4, and (6.1) is likewise valid in this case. However, in this case, we see from Karoubi’s exact sequence GW−1(Pr)⟶FK(Pr)⟶H0GW0(Pr) that H0(FΨ)=0. We can thus rewrite the above result for the Grothendieck–Witt group as GW(Pr)={GW(k)⊕Za1⊕⋯⊕Zaρifriseven(GW(k)⊕Za1⊕⋯⊕Zaρ−1⊕Zaρ)/2hrifr≡−1mod4(GW(k)⊕Za1⊕⋯⊕Zaρ−1⊕Zaρ)/hrifr≡−1mod4, with hr≔∑j=1ρ(−1)j(r+1ρ−j)aj. To see that we can alternatively use powers of a≔a1 as generators, it suffices to observe that for all k≥1, ak=ak+(alinearcombinationofa,a2,…,ak−1), (6.2) which follows inductively from the recursive relation ak=(a+2)ak−1−ak−2+2a, (6.3) for all k≥2. ( a0≔0.) Next, we show that ak=0 for all k>ρ. Let x≔O(1), viewed as an element of K(Pr). The relation (x−1)r+1=0 in K(Pr) implies that (x−1)+(x−1−1)=∑i=2r(−1)i(x−1)i, so that we can compute ak=[H(x−1)]k=H([FH(x−1)]k−1(x−1))=H([(x−1)+(x−1−1)]k−1(x−1))=H((x−1)2k−1+higherordertermsin(x−1))=0for2k−1>r,or,equivalently,fork>ρ. Equation (6.2) also allows us to rewrite hr in terms of the powers of a. Inductively, we find that hr=(−a)ρ for all odd r, where ρ=⌈r2⌉.□ Calculation of the γ-filtration on GW(Pr) (Example 6.3, continued) We claim above that FγiGW(Pr) is the ideal generated by FγiGW(k) and a⌈i2⌉. Equivalently, it is the subgroup generated by FγiGW(k) and by all powers aj with j≥i2. To verify the claim, we note that by Lemma 6.6 below, we have γi(aj)=±aj for i=1,2, while for all i>2 we have γi(aj)=0. In particular, a=a1∈Fγ2GW(Pr), and, therefore, aj∈Fγ2jGW(Pr). This shows that all the above named additive generators indeed lie in FγiGW(Pr). For the converse inclusion, we note that by Lemma 2.5, FγiGW(Pr) is additively generated by FγiGW(k) and by all finite products of the form ∏jγij(aαj), with ∑jij≥i. Such a product is non-zero only if ij∈{0,1,2} for all j, in which case it is of the form ±∏jaαj with at least i2 non-trivial factors. By (6.2), each non-trivial factor aαj can be expressed as a non-zero polynomial in a with no constant term. Thus, the product itself can be rewritten as a linear combination of powers aj with j≥i2.□ Calculations for Example 6.4 (A1−0) The Witt group of the punctured affine line has the form W(A1−0)≅W(k)⊕W(k)ε, where ε=(O,t), the trivial line bundle with the symmetric form given by multiplication with the standard coordinate (cf. [6]). It follows that GW(A1−0)≅GW(k)⊕W(k)εˆ, where εˆ≔ε−1. As for any symmetric line bundle, ε2=1 in the Grothendieck–Witt ring; equivalently, εˆ2=−2εˆ. To compute the γ-filtration, we need only observe that GW(A1−0) is generated by line elements. So FγiGW(A1−0)=(Fγ1GW(A1−0))i=(GI(k)⊕W(k)εˆ)i=GIi(k)⊕Ii−1(k)εˆ. The étale cohomology of A1−0 has the form Het*(A1−0,Z/2)≅Het*(k,Z/2)⊕Het*(k,Z/2)w1ε. Recall that when we write ker(w1) and ker(w2), we necessarily mean the kernels of the restrictions of w1 and w2 to ker(rank) and ker(w1), respectively. An arbitrary element of GW(A1−0) can be written as x+yεˆ with x,y∈GW(k). For such an element, we have w1(x+yεˆ)=w1x+rank(y)w1ε, so the general fact that ker(w1)=Fγ2GW is consistent with our computation. When rank(y)=0, we further find that w2(x+yεˆ)=w2x+w1y∪w1ε, proving the claim that ker(w2)=Fγ3GW in this example.□ Calculations for Example 6.5 (A4n+1−0) Balmer and Gille show in [6] that for d=4n+1 we have W(Ad−0)≅W(k)⊕W(k)ε for some symmetric space ε of even rank r such that ε2=0 in the Witt ring. Let εˆ≔ε−r2H. Then GW(Ad−0)≅GW(k)⊕W(k)εˆ with εˆ2=0. As the K-ring of Ad−0 is trivial, that is, isomorphic to Z via the rank homomorphism, FγiGW(Ad−0) maps isomorphically to FγiW(Ad−0) for all i>0. We now switch to the complex numbers. Equipped with the analytic topology, AC4n+1 is homotopy equivalent to the sphere S8n+1, so we have a comparison map GW(AC4n+1−0)→KO(S8n+1). As the λ-ring structures on both sides are defined via exterior powers, this is clearly a map of λ-rings. In fact, it is an isomorphism, as we see by comparing the localization sequences for ACd−0◦↪ACd|↩{0}, as in the proof of [47, Theorem 2.5]. The λ-ring structure on KO(S8n+1) can be deduced from [1, Theorem 7.4]—as a special case, the theorem asserts that the projection RP8n+1↠RP8n+1/RP8n≃S8n+1 induces the following map in KO-theory: Here, λ is the canonical line bundle over the real projective space, λˆ≔λ−1, and f is some integer. Thus, γt(2f−1λˆ)=(1+λˆt)2f−1 and we find that γi(εˆ)=ciεˆ for ci≔(2f−1i)2i−f. Note that ci is indeed an integer: by Kummerʼs theorem on binomial coefficients, we find that the highest power of two dividing (2f−1i) is at least f−1−k, where k is the highest power of two such that 2k≤i. In fact, modulo two we have c2≡1 and ci≡0 for all i>2. So the γ-filtration is as described.□ Finally, here is the lemma referred to multiple times above. Lemma 6.6 Let Lbe a line bundle over a scheme Xover Z[12]. Then γ2(H(L−1))=−H(L−1),and γi(H(L−1))=0in GW(X)for all i>2. Proof Let us write λt(x)=1+xt+λ2(x)t2+⋯ for the total λ-operation, and similarly for γt(x). Then λt(x+y)=λt(x)λt(y), γt(x+y)=γt(x)γt(y), and γt(x)=λt1−t(x). Let a≔H(L−1). From λt(a)=λt(HL)λt(H1)=1+(HL)t+det(HL)t21+(H1)t+det(H1)t2=1+(HL)t+⟨−1⟩t21+(H1)t+⟨−1⟩t2, we deduce that γt(a)=1+(HL−2)t+(1+⟨−1⟩−HL)t21+(H1−2)t+(1+⟨−1⟩−H1)t2=1+(HL−2)t−H(L−1)t21+(H1−2)t=[1+(HL−2)t−H(L−1)t2]·∑i≥0(2−H1)iti. Here, the penultimate step uses that H1≅1+⟨−1⟩ when two is invertible. In order to proceed, we observe that H1·Hx=H(FH1·x)=2Hx for any x∈GW(X). It follows that (2−H1)i=2i−1(2−H1), and hence that [1+(HL−2)t−H(L−1)t2]·(2−H1)iti=2i−1(2−H1)(1−2t)ti, for all i≥1. This implies that the above expression for γt(a) simplifies to 1+H(L−1)t−H(L−1)t2, as claimed.□ Acknowledgements I thank Pierre Guillot for getting me started on these questions. 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The Quarterly Journal of Mathematics – Oxford University Press

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