Regularity and Related Properties in Tensor Products of Algebras Over a Field

It has been established, quite recently, that a tensor product of k-algebras, if Noetherian, it inherits the concepts of Cohen–Macaulay ring, Gorenstein ring, and complete intersection ring. However, it is well known that a tensor product of two extension fields is not necessarily regular. In 1965, Grothendieck showed that it is, in fact, regular if one of the two fields is separable and finitely generated. Since then, many articles appeared in the literature featuring partial results on this topic. The problem on the transfer or defect of regularity in more general settings remains elusively open. This survey paper tracks and studies some of these works which deal with this problem; precisely, we shed brighter light on the main results and examples published, chronologically, in [6, 7, 28]].