Regularity of tensor products of \(k\)-algebras (Q2925738)

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).