Regularity of tensor products of \(k\)-algebras
type: Item
instance of: scholarly article
zbMATH Open document ID: 1308.13035
author: Samir Bouchiba
author: S. Kabbaj
publication date: 2014-10-17T00:00:00Z
full work available at URL: https://arxiv.org/abs/1202.5615
Mathematics Subject Classification ID: 13H10
Mathematics Subject Classification ID: 13B02
zbMATH DE Number: 6357906
zbMATH Keywords: regular ring
zbMATH Keywords: complete intersection ring
zbMATH Keywords: Gorenstein ring
zbMATH Keywords: Cohen-Macaulay ring
zbMATH Keywords: separable extension
MaRDI profile type: Publication
OpenAlex ID: W2963900589
arXiv ID: 1202.5615
DOI: 10.7146/MATH.SCAND.A-18000
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title: Regularity of tensor products of \(k\)-algebras
published in: Mathematica Scandinavica
review text: Grothendieck raised the following question: Is \(A\otimes_{k} B\) regular if \(A\) and \(B\) are regular algebras over \(k\)? It is well known that in general \(A\otimes_{k} B\) is not regular if \(A\) and \(B\) are regular algebras over \(k\) (see Remark 7, \textit{M. Tousi} and \textit{S. Yassemi} [J. Algebra 268, No. 2, 672--676 (2003; Zbl 1087.13506)]).NEWLINENEWLINEIn 1965, \textit{A. Grothendieck} proved [Publ. Math., Inst. Hautes Étud. Sci. 24, 1--231 (1965; Zbl 0135.39701)] that \(K\otimes_{k} L\) is a regular ring provided \(K\) or \(L\) is a finitely generated separable field extension of \(k\).NEWLINENEWLINEIn this paper, the authors generalize the above mentioned result by establishing necessary and sufficient conditions for a noetherian tensor product of two extension fields of \(k\) to inherit regularity in various settings of separability (Theorem 2.4). The authors also consider the case when \(A\) and \(B\) are not necessarily field extensions of \(k\) and prove the following: Let \(A\) and \(B\) be two \(k\)-algebras such that \(A\otimes_{k} B\) is noetherian. Suppose that \(A\)(or \(B\)) is residually separable. Then \(A\otimes_{k} B\) is regular if and only if \(A\) and \(B\) are regular (Theorem 2.11).
description: scientific article; zbMATH DE number 6357906