On tensor products of complete intersections (Q2865922)

Let \(A\) be a Noetherian ring and let \(B\) and \(C\) be two \(A\)-algebras such that the tensor product \(B \otimes_A C\) is Noetherian. In the present paper, the author studies two properties of \(B \otimes C\) by using André-Quillen homology. He shows that \(B \otimes C\) is regular (resp.\ complete intersection) if \(B\) and \(C\) are also and if one of \(B_{\mathfrak q \cap B}\), \(C_{\mathfrak q \cap C}\) and \((B \otimes C)/\mathfrak q\) (resp.\ \(B_{\mathfrak q \cap B}\) and \(C_{\mathfrak q \cap C}\)) is formally smooth (resp.\ flat) over \(A_{\mathfrak q \cap A}\) for any maximal ideal \(\mathfrak q\) of \(B \otimes C\).