Tensor functors and finite representation type
type: Item
instance of: scholarly article
title: Tensor functors and finite representation type
zbMATH Open document ID: 0808.16017
DOI: 10.1016/0022-4049(94)90113-9
author: Maher Zayed
published in: Journal of Pure and Applied Algebra
publication date: 1995-03-16T00:00:00Z
review text: Let \(A\) be a finite-dimensional algebra over an infinite field \(K\). Given a field extension \(L\) of \(K\), one may consider the two algebras \(A\) over \(K\) and \(L \otimes_{K} A\) over \(L\). The tensor functor \(F_ L = L \otimes_ K - : \text{Mod} (A) \to \text{Mod} (L \otimes_ K A)\) from the category of all (left) \(A\)-modules to the category of all (left) \(L \otimes_ K A\)-modules, given by \(X \mapsto F_ L(X) = L \otimes_ K X\) is a faithful functor. The main result in the paper says that if for any extension \(L\) over \(K\) the functor \(F_ L\) is dense, then \(A\) is representation-finite. Moreover, the author shows that \(A\) is of finite- representation type if and only if for each separable extension \(L\) over \(K\), \(F_ L\) is dense.
Mathematics Subject Classification ID: 16G10
Mathematics Subject Classification ID: 16G60
Mathematics Subject Classification ID: 16D90
Mathematics Subject Classification ID: 12F10
Mathematics Subject Classification ID: 16P10
Mathematics Subject Classification ID: 18A22
Mathematics Subject Classification ID: 16E30
zbMATH DE Number: 579997
zbMATH Keywords: dense functors
zbMATH Keywords: finite-dimensional algebra
zbMATH Keywords: field extension
zbMATH Keywords: tensor functor
zbMATH Keywords: faithful functor
zbMATH Keywords: representation-finite
zbMATH Keywords: finite-representation type
zbMATH Keywords: separable extension
reviewed by: Changchang Xi
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90113-9
OpenAlex ID: W2037162425
cites work: Q4056148
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cites work: Model theory
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description: scientific article; zbMATH DE number 579997