On The Gersten-Witt Complex of an Azumaya Algebra with Involution

2021/02/11 by Uriya A. First, First, Uriya A.
Mathematics · #16H05 #19G38 #Advanced Topics in Algebra #Algebraic Geometry (math.AG) #Algebraic structures and combinatorial models #FOS: Mathematics #Finite Group Theory Research #K-Theory and Homology (math.KT) #Number Theory (math.NT) #Primary: 11E57 #Secondary: 11E39

paper · pdf · doi:10.48550/arxiv.2102.06264

openalex publication_date 2021/02/11 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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, ind A≤ 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.

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