On class groups of lattices
type: Item
instance of: scholarly article
title: On class groups of lattices
zbMATH Open document ID: 0617.16002
DOI: 10.1016/0022-4049(87)90064-8
author: Yanyan Li
published in: Journal of Pure and Applied Algebra
publication date: 1987-01-01T00:00:00Z
review text: The genus of an order over a Dedekind domain in a central simple algebra can be divided in so-called spinor genera by a local conjugation condition on the completions of the order at the prime ideals of the centre. In this paper the author aims to derive general formulae for SG(\(\Lambda)\), the group of spinor genera, similar to the formulae for the locally free class groups, using a formalism introduced by A. Fröhlich, also M. Kneser, in the computation of class groups of algebras, resp. spinor class groups of lattices on quadratic spaces. In Section 3 of the paper the author extends the Fröhlich-Jacobinski formula for the locally free class group. In Section 4 the author obtains a Fröhlich-Jacobinski formula for the group of spinor genera in the genus of a lattice on a quadratic space. In the particular case where L is a maximal R-lattice on a regular quadratic space A the spinor genera group may be expressed in terms of the ray class group \(Cl_ q(R)\) of R relative to q, in fact \(SG(L)=Cl_ q(R)/G(A)Gl_ q(R)^ 2\) where G(A) is generated by the classes of prime ideals \(\rho\) of R dividing the discriminant of A; this corresponds to R. Swan's formula for class groups of maximal orders.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16L30
Mathematics Subject Classification ID: 16P10
zbMATH DE Number: 4000143
zbMATH Keywords: genus
zbMATH Keywords: order
zbMATH Keywords: Dedekind domain
zbMATH Keywords: central simple algebra
zbMATH Keywords: group of spinor genera
zbMATH Keywords: class groups of algebras
zbMATH Keywords: lattices on quadratic spaces
zbMATH Keywords: locally free class group
zbMATH Keywords: ray class group
zbMATH Keywords: maximal orders
MaRDI profile type: Publication
full work available at URL: 0022-4049(87)90064-8
OpenAlex ID: W2049251792
cites work: Spinor class groups of orders
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cites work: Klassenzahlen indefiniter quadratischer Formen in drei oder mehr Veränderlichen
cites work: Q5675787
cites work: Q4061092
cites work: Q3934492
cites work: Projective modules over binary polyhedral groups.
description: scientific article; zbMATH DE number 4000143