Cohomological invariants for quadratic forms over local rings
type: Item
instance of: scholarly article
title: Cohomological invariants for quadratic forms over local rings
zbMATH Open document ID: 1441.11072
author: Jeremy Allen Jacobson
published in: Mathematische Annalen
publication date: 2018-02-12T00:00:00Z
review text: The main result of the paper under discussion (Theorem 5.5) is the following variant of the Milnor conjecture for local rings: if \(A\) is a local ring in which \(1+1\) is invertible, then there is a well-defined group homomorphism \[e_n : I^n(A) \rightarrow H_{\text{et}}^n(A,\mathbb{Z}/2), \ \text{given by}\] \[\langle \! \langle a_1,\dots,a_n \rangle \! \rangle \mapsto (a_1) \cup \dots \cup (a_n),\] whose kernel is exactly \(I^{n+1}(A)\). A key tool is the proof of the Gersten conjecture for Witt groups (Theorem 3.8) for such \(A\) under the additional assumption that \(A\) be regular and unramified. In the last section (no. 6), the author provides sufficient conditions for the triviality of the intersection \(\bigcap I^n(A)\) (which in the case of fields follows the Arason-Pfister Hauptsatz), such as \(A\) being a regular ring for which the Gersten conjecture holds.
reviewed by: Adam Chapman
Mathematics Subject Classification ID: 11E08
Mathematics Subject Classification ID: 14F20
zbMATH DE Number: 6837430
zbMATH Keywords: cohomological invariants
zbMATH Keywords: quadratic forms
zbMATH Keywords: local fields
zbMATH Keywords: Etale cohomology
zbMATH Keywords: Witt groups
MaRDI profile type: Publication
full work available at URL: s00208-017-1561-z
OpenAlex ID: W2731725630
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DOI: 10.1007/S00208-017-1561-Z
description: scientific article; zbMATH DE number 6837430