On the Gersten–Witt complex of an Azumaya algebra with involution

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Abstract

Let (A,σ) be an Azumaya algebra with involution over a regular ring R. We prove that the Gersten–Witt complex of (A,σ) defined by Gille is isomorphic to the Gersten–Witt complex of (A,σ) defined by Bayer-Fluckiger, Parimala and the author. Advantages of both constructions are used to show that the Gersten–Witt complex is exact when dim⁡R≤3, indA≤2 and σ is orthogonal or symplectic. This means that the Grothendieck–Serre conjecture holds for the group R-scheme of σ-unitary elements in A under the same hypotheses; R is not required to contain a field.

Original languageEnglish
Pages (from-to)146-178
Number of pages33
JournalJournal of Algebra
Volume605
DOIs
StatePublished - 1 Sep 2022

Bibliographical note

Publisher Copyright:
© 2022 Elsevier Inc.

Keywords

  • Azumaya algebra
  • Derived category
  • Gersten-Witt complex
  • Grothendieck-Serre conjecture
  • Hermitian category
  • Hermitian form
  • Involution
  • Regular local ring
  • Triangulated category

ASJC Scopus subject areas

  • Algebra and Number Theory

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