On the decomposition of lattices over orders
type: Item
instance of: scholarly article
title: On the decomposition of lattices over orders
zbMATH Open document ID: 0907.16004
author: Hiroaki Hijikata
published in: Journal of the Mathematical Society of Japan
publication date: 1998-10-21T00:00:00Z
review text: Let \(R\) be a Dedekind domain with field of fractions \(K\), and let \(\Lambda\) be an order in a finite dimensional \(K\)-algebra \(A\) that need not be separable. For two \(\Lambda\) lattices \(L\) and \(L'\), write \(KL\gg KL'\) if each \(A\)-indecomposable direct summand of \(KL'\) occurs strictly more often in \(KL\) than in \(KL'\). Write \(M\sim L\) if the \(P\)-adic completions \(M_P\cong L_P\) for all maximal ideals \(P\). Two theorems are proved. The first is a form of the Roiter-Jacobinski Divisibility Theorem. It says that if \(M\sim L'\oplus L''\), then \(M\cong M'\oplus M''\), where \(M'\sim L'\) and \(M''\sim L''\). Further, if \(KM\gg KL'\), then \(M\cong L'\oplus M''\). The second theorem is a form of Jacobinski-Swan cancellation. Namely, let \(G(L)\) denote the set of lattices \(M\) with \(M\sim L\), and write \(nL\) for the direct sum of \(n\) copies of \(L\). Then the mapping \(G(L)\to G(nL)\) given by \(X\mapsto X\oplus(n-1)L\) is injective. -- The proofs avoid the strong approximation methods in the classical versions by applying methods of K-theory at the adelic level.
Mathematics Subject Classification ID: 16H05
Mathematics Subject Classification ID: 16D70
Mathematics Subject Classification ID: 16G30
zbMATH DE Number: 1172200
zbMATH Keywords: lattices over orders
zbMATH Keywords: orders in nonseparable algebras
zbMATH Keywords: Jacobinski-Swan cancellation
reviewed by: William H. Gustafson
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2969/jmsj/04930431
OpenAlex ID: W2009143227
DOI: 10.2969/JMSJ/04930431
description: scientific article; zbMATH DE number 1172200