A Hermitian analog of a quadratic form theorem of Springer
type: Item
instance of: scholarly article
title: A Hermitian analog of a quadratic form theorem of Springer
zbMATH Open document ID: 1452.11039
author: Stefan G. Gille
published in: Manuscripta Mathematica
publication date: 2020-08-12T00:00:00Z
review text: Let \(F\) be a complete discretely valued field of characteristic not 2 with residue field \(\overline{F}\). \textit{T. A. Springer} [Nederl. Akad. Wet., Proc., Ser. A 58, 352--362 (1956; Zbl 0067.27605)] showed that the Witt group of symmetric bilinear spaces over \(F\) decomposes as \(W(F) \cong W(\overline{F}) \oplus W(\overline{F})\). A corollary of this computation is the existence of second residue maps \(W(K) \rightarrow W(k)\), where \(K\) is the fraction field of a discrete valuation ring \(R\) with residue field \(k\), whose characteristic is not two. This map, denoted \(\partial_\pi\) here, depends on the choice of an uniformizer \(\pi\), i.e., generator of the maximal ideal \(\mathfrak{m}\) of \(R\), and fits into an exact sequence as the third map \[0 \rightarrow W(R) \rightarrow W(K)\rightarrow W(k) \rightarrow 0.\] The goal of this paper is to prove the analogous results for Witt groups of algebras with involution: Theorem B. Let \(R\) be a complete discrete valuation ring with fraction field \(K\) and residue field \(k\), and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of the first kind. Then \(W_\varepsilon(A_K , \tau_K ) \cong W_\varepsilon(A_k, \tau_K ) \oplus W_\varepsilon(A_k, \tau_k )\) for any \(\varepsilon \in \{\pm 1\}\). Theorem A. Let \(R\) be a semilocal Dedekind domain containing \(\frac{1}{2}\) with fraction field \(K\) and \((A, \tau)\) an \(R\)-Azumaya algebra with involution of first or second kind. We assume that \(R\) is Galois of degree 2 over the fix ring of \(\tau|_R\) if \(\tau\) is not of the first kind. Let \(\max R \subset \operatorname{Spec} R\) be the set of maximal ideals of \(R\) and \(X_\tau^{(1)} := \{\mathfrak{m} \in \max R \ | \ \tau(\mathfrak{m}) = \mathfrak{m}\} = \{\mathfrak{m}_1,\dots ,\mathfrak{m}_\ell\}\). (\(\ell = 0\) is possible). We set \(k_i := R/\mathfrak{m}_i\) for \(i = 1,\dots, \ell\), and denote by \(\tau_K\) (respectively, \(\tau_{k_i}\)) the induced involutions on \(A_K:= K\otimes_R A\) (…
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 11E70
Mathematics Subject Classification ID: 11E81
zbMATH DE Number: 7233346
zbMATH Keywords: Hermitian forms
zbMATH Keywords: quadratic forms
zbMATH Keywords: Witt group
zbMATH Keywords: algebras with involution
MaRDI profile type: Publication
full work available at URL: s00229-019-01155-4
OpenAlex ID: W2982288135
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Wikidata QID: Q126861831
DOI: 10.1007/S00229-019-01155-4
description: scientific article; zbMATH DE number 7233346