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 dimR≤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 language | English |
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Pages (from-to) | 146-178 |
Number of pages | 33 |
Journal | Journal of Algebra |
Volume | 605 |
DOIs | |
State | Published - 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