Affine quadrics and the Picard group of the motivic category

Vishik, Alexander

Abstract

In this paper we study the subgroup of the Picard group of Voevodsky's category of geometric motives DMgm(k; Z/2) generated by the reduced motives of affine quadrics. Our main tools here are the functors of Bachmann [On the invertibility of motives of affine quadrics, Doc. Math. 22 (2017), 363-395], but we also provide an alternative method. We show that the group in question can be described in terms of indecomposable direct summands in the motives of projective quadrics over k. In particular, we describe all the relations among the reduced motives of affine quadrics. We also extend the criterion of motivic equivalence of projective quadrics.

Citation

Vishik, A. (2019). Affine quadrics and the Picard group of the motivic category. Compositio Mathematica, 155(8), 1500-1520. https://doi.org/10.1112/S0010437X19007401

• Journal Article Type: Article
• Acceptance Date: Apr 3, 2019
• Online Publication Date: Jul 4, 2019
• Publication Date: Aug 1, 2019
• Deposit Date: Nov 19, 2018
• Publicly Available Date: Nov 19, 2018
• Journal: Compositio Mathematica
• Print ISSN: 0010-437X
• Electronic ISSN: 1570-5846
• Publisher: Foundation Compositio Mathematica
• Peer Reviewed: Peer Reviewed
• Volume: 155
• Issue: 8
• Pages: 1500-1520
• DOI: https://doi.org/10.1112/S0010437X19007401
• Keywords: Algebra and Number Theory
• Public URL: https://nottingham-repository.worktribe.com/output/1174008
• Publisher URL: https://www.cambridge.org/core/journals/compositio-mathematica/article/affine-quadrics-and-the-picard-group-of-the-motivic-category/ADCA7F8CABFFCEB903DA970E72E0FB0D
• Contract Date: Nov 19, 2018