Relative Hermitian Morita theory. I: Morita equivalences of algebras with involution
type: Item
instance of: scholarly article
title: Relative Hermitian Morita theory. I: Morita equivalences of algebras with involution
zbMATH Open document ID: 0806.16007
author: C. M. Marquez Hernandez
author: M. Victoria Reyes Sánchez
author: Alain Verschoren
published in: Journal of Algebra
publication date: 1994-06-07T00:00:00Z
review text: The aim is to develop a hermitian Morita theory for arbitrary involutive Grothendieck categories. By the Gabriel-Popescu Theorem this is equivalent to the relative Hermitian Morita theory for algebras with involution, where ``relative'' means ``with respect to a hereditary torsion theory, or equivalently, to an idempotent kernel functor''. More precisely, one considers the notion of a torsion triple \((A,\alpha,\sigma)\) where \(A\) is an algebra over a commutative ring with identity, \(\alpha\) is an involution of \(A\), and \(\sigma\) is an idempotent kernel functor in \(A\)-Mod. Then one introduces the notion of a relative hermitian Morita equivalence by means of so-called involutive relatively invertible bimodules, and finally one proves hermitian versions of Morita I and Morita II. For example, the hermitian Morita I shows that any relative hermitian Morita equivalence produces an equivalence between the corresponding categories of so-called \(\sigma\)-sesquilinear left \(A\)- modules. The paper contains no example, as well as no application of the theory developed here.
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 18E15
Mathematics Subject Classification ID: 16W10
Mathematics Subject Classification ID: 18E40
Mathematics Subject Classification ID: 16D20
Mathematics Subject Classification ID: 16S90
zbMATH DE Number: 554201
zbMATH Keywords: hermitian Morita theory
zbMATH Keywords: involutive Grothendieck categories
zbMATH Keywords: Gabriel- Popescu Theorem
zbMATH Keywords: relative Hermitian Morita theory
zbMATH Keywords: algebras with involution
zbMATH Keywords: hereditary torsion theory
zbMATH Keywords: idempotent kernel functor
zbMATH Keywords: torsion triple
zbMATH Keywords: involutive relatively invertible bimodules
zbMATH Keywords: \(\sigma\)-sesquilinear left \(A\)-modules
reviewed by: Toma Albu
MaRDI profile type: Publication
full work available at URL: jabr.1993.1246
OpenAlex ID: W1563048504
DOI: 10.1006/JABR.1993.1246
description: scientific article; zbMATH DE number 554201