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A GerstenWitt spectral sequence for regular schemes
 Ann. Sci. ENS
"... Abstract. A spectral sequence is constructed whose nonzero E1terms are the Witt groups of the residue fields of a regular scheme X, arranged in GerstenWitt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of ..."
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Cited by 41 (7 self)
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Abstract. A spectral sequence is constructed whose nonzero E1terms are the Witt groups of the residue fields of a regular scheme X, arranged in GerstenWitt complexes, and whose limit is the four global Witt groups of X. There are several immediate consequences concerning purity for Witt groups of lowdimensional schemes. The Witt groups of punctured spectra of regular local rings are also computed. Let X be a regular integral separated noetherian scheme in which 2 is everywhere invertible. (We will maintain these hypotheses throughout the introduction.) It is now known to the experts that the Witt groups of the residue fields of X form a nonexact cochain complex
Triangular Witt Groups  Part I: The 12Term Localization Exact Sequence.
, 1999
"... . To a short exact sequence of triangulated categories with duality, we associate a long exact sequence of Witt groups. For this, we introduce higher Witt groups in a very algebraic and explicit way. Since those Witt groups are 4periodic, this long exact sequence reduces to a cyclic 12term one. Of ..."
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Cited by 17 (9 self)
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. To a short exact sequence of triangulated categories with duality, we associate a long exact sequence of Witt groups. For this, we introduce higher Witt groups in a very algebraic and explicit way. Since those Witt groups are 4periodic, this long exact sequence reduces to a cyclic 12term one. Of course, in addition to higher Witt groups, we need to construct connecting homomorphisms, hereafter called residue homomorphisms. Introduction. The reader is expected to have some interest in the usual Witt group, as defined for schemes by Knebusch in [9, definition p. 133]. This Witt group is obtained by considering symmetric vector bundles (up to isometry) modulo the bundles possessing a lagrangian  that is a maximal totally isotropic subbundle. This being said, the present article is maybe more about triangulated categories than about symmetric forms and that might possibly make it generalizable to invariants other than the Witt group. For the time being, it is not my goal to compute an...
cohomology, MayerVietoris, homotopy invariance and
 the Gersten Conjecture, Ktheory
"... Abstract. We establish a MayerVietoris long exact sequence for Witt groups of regular schemes. We also establish homotopy invariance for Witt groups of regular schemes. For this, we introduce Witt groups with supports using triangulated categories. Subsequently we use these results to prove the Ger ..."
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Abstract. We establish a MayerVietoris long exact sequence for Witt groups of regular schemes. We also establish homotopy invariance for Witt groups of regular schemes. For this, we introduce Witt groups with supports using triangulated categories. Subsequently we use these results to prove the GerstenWitt Conjecture for semilocal regular rings of geometric type over infinite fields of characteristic different from two. Dedicated to Professor Manuel Ojanguren on his sixtieth birthday. The Witt group W(X) of a scheme X was defined in Knebusch’s 1977 paper [11]. In this generality, that is when X is not assumed to be affine, very little is known about the contravariant functor W (−). Motivated by Ktheory analogies, we may first restrict our attention to regular schemes. Even then, such elementary questions as the existence of a
Koszul complexes and symmetric forms over the punctured affine space, preprint at http://www.math.uiuc.edu/Ktheory/0
, 2004
"... Abstract. Let X be a scheme which is not of equicharacteristic 2 and let UnX ⊂ AnX be the punctured affine nspace over X. If n ≡ ±1 modulo 4, we show that there exists a ±1symmetric bilinear space (E(n)X, ϕ(n)X) over UnX which can not be extended to the whole affine space AnX and which is locally ..."
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Abstract. Let X be a scheme which is not of equicharacteristic 2 and let UnX ⊂ AnX be the punctured affine nspace over X. If n ≡ ±1 modulo 4, we show that there exists a ±1symmetric bilinear space (E(n)X, ϕ(n)X) over UnX which can not be extended to the whole affine space AnX and which is locally metabolic for n ≥ 2. IfX is regular, contains 1 2 and is of finite Krull dimension, we show that the total Witt ring Wtot(UnX) of UnX is a free Wtot(X)module with two generators: the Witt classes of < 1> and of the above (E(n)X, ϕ(n)X). We describe Wtot(UnX) similarly when n is even.
The Gersten Conjecture for Witt Groups in the Equicharacteristic Case
 DOCUMENTA MATH.
, 2002
"... We prove the Gersten conjecture for Witt groups in the equicharacteristic case, that is for regular local rings containing a field of characteristic not 2. ..."
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Cited by 7 (4 self)
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We prove the Gersten conjecture for Witt groups in the equicharacteristic case, that is for regular local rings containing a field of characteristic not 2.
An introduction to triangular Witt groups and a survey of applications, Algebraic and arithmetic theory of quadratic forms
 Contemp. Math
, 2004
"... Abstract. These are extended notes from a survey talk on Witt groups of triangulated categories, given at the Talca–Pucon Conference, December 2002. 1. ..."
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Cited by 6 (0 self)
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Abstract. These are extended notes from a survey talk on Witt groups of triangulated categories, given at the Talca–Pucon Conference, December 2002. 1.
A ZERO THEOREM FOR THE TRANSFER OF COHERENT WITT GROUPS
"... Abstract. Let R be a Gorenstein ring of finite Krull dimension and t ∈ R a regular element. We show that if the quotient map R → R/Rt has a flat splitting then the transfer morphism of coherent Witt groups Tr(R/Rt)/R: ˜ W i (R/Rt) → ˜ W i+1 (R) is zero for all i ∈ Z. As an application we give ano ..."
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Cited by 4 (0 self)
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Abstract. Let R be a Gorenstein ring of finite Krull dimension and t ∈ R a regular element. We show that if the quotient map R → R/Rt has a flat splitting then the transfer morphism of coherent Witt groups Tr(R/Rt)/R: ˜ W i (R/Rt) → ˜ W i+1 (R) is zero for all i ∈ Z. As an application we give another proof of the Gersten conjecture for Witt groups in the case of regular semilocal rings essentially of finite type over a field of characteristic not 2. Let ˜ W i (R) be the ith coherent Witt group of the Gorenstein ring of finite Krull dimension R as defined in [11], and let t ∈ R be a regular element, i.e. t is not a zero divisor in R. Then R/Rt is a Gorenstein ring, too (see e.g. [8], Proposition 3.1.19), and we have a transfer morphism (cf. [11] and [12]):
Projective pushforwards in the Witt theory of algebraic varieties
 Adv. Math
"... Abstract. We define pushforwards along projective morphisms in the Witt theory of smooth quasiprojective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pullbacks and projection formulas. 1. ..."
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Cited by 4 (0 self)
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Abstract. We define pushforwards along projective morphisms in the Witt theory of smooth quasiprojective varieties over a field. We prove that they have standard properties such as functoriality, compatibility with pullbacks and projection formulas. 1.
Products of degenerate quadratic forms
 Compos. Math
"... Abstract. We challenge the classical belief that products of degenerate quadratic forms must remain degenerate and we show that this fails in general, e.g. over tensor triangulated categories with duality. This opens new ways of constructing nondegenerate quadratic forms and hence classes in Witt ..."
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Abstract. We challenge the classical belief that products of degenerate quadratic forms must remain degenerate and we show that this fails in general, e.g. over tensor triangulated categories with duality. This opens new ways of constructing nondegenerate quadratic forms and hence classes in Witt groups. In addition, we encapsulate in a Leibniztype formula the behaviour of the product with respect to the symmetric cone construction. We illustrate these ideas by computing the total Witt group of regular projective spaces. Contents
Shifted Witt groups of semilocal rings
"... Abstract. We show that the oddindexed derived Witt groups of a semilocal ring with trivial involution vanish. We show that this is wrong when the involution is not trivial and we provide examples. ..."
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Abstract. We show that the oddindexed derived Witt groups of a semilocal ring with trivial involution vanish. We show that this is wrong when the involution is not trivial and we provide examples.