Chow-Witt groups and Grothendieck-Witt groups of regular schemes

By Jean Fasel, V. Srinivas. 2009

56 citations recorded by the source corpus.

Let A be a noetherian commutative Z[1/2]-algebra of Krull dimension d and let P be a projective A-module of rank d. We use derived Grothendieck-Witt groups and Euler classes to detect some obstructions for P to split off a free factor of rank one. If d <= 3, we show that the vanishing of its Euler class in the corresponding Grothendieck-Witt group is a necessary and sufficient condition for P to have a free factor of rank one. If d is odd, we also get some results in that direction. If A is regular, we show that the Chow-Witt groups defined by Morel and Barge appear naturally as some homology groups of a Gersten-type complex in Grothendieck-Witt theory. From this, we deduce that if d = 3 then the vanishing of the Euler class of P in the corresponding Chow-Witt group is a necessary and sufficient condition for P to have a free factor of rank one. Crown Copyright (C) 2008 Published by Elsevier Inc. All rights reserved.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/a286d9e67c045ed14b3a98e01e92700dae1d1773

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Citations in the corpus

The Mayer-Vietoris principle for Grothendieck-Witt groups of schemes (2008, background): “Grothendieck-Witt groups naturally occur in A1-homotopy theory [19] and are to oriented Chow groups what algebraic K-theory is to ordinary Chow groups; see [3, 6, 11].”
Aspects of Enumerative Geometry with Quadratic Forms (2017, methodology): “The foundations of this theory have been laid by work of Barge-Morel [7], Fasel [14], Fasel-Srinivas [15] and Morel [34, 35] (and many others), and first steps in this direction have been taken by Hoyois [20], Kass-Wickelgren [24, 25] and Pauli [42].”
Hermitian K-theory, derived equivalences and Karoubi's fundamental theorem (2012, methodology): “roups and our careful study of the multiplicative structure of higher Grothendieck-Witt groups in Sections 5 and 8 should be of independent interest. Results of this article have already been used in [FS09], [Zib11], [PW10], in the solution of a conjecture of Suslin on the structure of stably free p”
Algebraic vector bundles on spheres (2012, unknown): “Indeed, the Gersten–Grothendieck–Witt spectral sequence E(n− 1) defined in [15] shows that the edge homomorphism E(n− 1)n−1,0 2 = H n−1(An \ 0, I) → GWn−1(An \ 0) induces an isomorphism Hn−1(An \ 0, I) → GWn−1 red (A n \ 0).”