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Register a new userOn motivic obstructions to Witt cancellation for quadratic forms over schemes
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Authors
Matthias Wendt
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The paper provides computations of the first non-vanishing $mathbb{A}^1$-homotopy sheaves of the orthogonal Stiefel varieties which are relevant for the unstable isometry classification of quadratic forms over smooth affine schemes over perfect fields of characteristic $ eq 2$. Together with the $mathbb{A}^1$-representability for quadratic forms, this provides the first obstructions for rationally trivial quadratic forms to split off a hyperbolic plane. For even-rank quadratic forms, this first obstruction is a refinement of the Euler class of Edidin and Graham. A couple of consequences are discussed, such as improved splitting results over algebraically closed base fields as well as examples where the obstructions are nontrivial.
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We introduce a Bredon motivic cohomology theory for smooth schemes defined over a field and equipped with an action by a finite group. These cohomology groups are defined for finite dimensional representations as the hypercohomology of complexes of equivariant correspondences in the equivariant Nisnevich topology. We generalize the theory of presheaves with transfers to the equivariant setting and prove a Cancellation Theorem.
We define push-forwards for Witt groups of schemes along proper morphisms, using Grothendieck duality theory. This article is an application of results of the authors on tensor-triangulated closed categories to such structures on some derived categories of schemes together with classical derived functors.
In this note, we provide an axiomatic framework that characterizes the stable $infty$-categories that are module categories over a motivic spectrum. This is done by invoking Luries $infty$-categorical version of the Barr--Beck theorem. As an application, this gives an alternative approach to Rondigs and O stvae rs theorem relating Voevodskys motives with modules over motivic cohomology, and to Garkushas extension of Rondigs and O stvae rs result to general correspondence categories, including the category of Milnor-Witt correspondences in the sense of Calm`es and Fasel. We also extend these comparison results to regular Noetherian schemes over a field (after inverting the residue characteristic), following the methods of Cisinski and Deglise.
We show that the category of motivic spaces with transfers along finite flat morphisms, over a perfect field, satisfies all the properties we have come to expect of good categories of motives. In particular we establish the analog of Voevodskys cancellation theorem.
In this paper, we compute the (total) Milnor-Witt motivic cohomology of the complement of a hyperplane arrangement in an affine space.
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