Some homological criteria for regular, complete intersection and Gorenstein rings (Q2806129)

Let \((A,\mathfrak m,k)\) be a local Noetherian ring. Theorems of Serre, Auslander-Bridger, and Avramov-Gasharov-Peeva tell us that \(A\) is regular (respectively, Gorenstein, respectively a complete intersection) if and only if the projective dimension (respectively, Gorenstein dimension, respectively Complete Intersection dimension) of \(k\) as an \(A\)-module is finite. If \(A\) contains a field of positive characteristic, then similar theorems have been established when the homomorphism \(A\to k\) is replaced by the Frobenius homomorphism \(A\to A\) or some other contracting homomorphism. In the author's words, ``We introduce here a class of local homomorphisms, in some sense larger than all above, for which these characterizations still hold, providing an unified treatment for this class of homomorphisms.''NEWLINENEWLINERecall that if \(f:A\to B\) is a homomorphism of commutative rings and \(M\) is a \(B\)-module, then the corresponding André-Quillen homology \(B\)-modules are denoted by \(\text{H}_n(A,B,M)\). If \(f:(A,\mathfrak m,k)\to (B,\mathfrak n,\ell)\) is a local homomorphism of Noetherian local rings, then the author says that \(f\) has the \(h_2\)-vanishing property if the homomorphism induced by \(f\), NEWLINE\[NEWLINE\text{H}_2(A,\ell,\ell)\to \text{H}_2(B,\ell,\ell)NEWLINE\]NEWLINEvanishes. The author proves a version of the three theorems mentioned above for homomorphisms which have the \(h_2\)-vanishing property. It is noteworthy that any local homomorphism that factors through a regular local ring has the \(h_2\)-vanishing property. Also, if \(A\) has a contracting endomorphism, then \(A\) necessarily contains a field, while this is not needed for \(h_2\)-vanishing homomorphisms.