Some remarks on relative Tor and regular homomorphisms
The author proves a base change for relative \(\text{ Tor}\): given an \(S\)-module \(M\) and an \(S \otimes_R T\)-module \(N\), we have \(\text{ Tor}_n^{S \mid R}(M,N) = \text{ Tor}_n^{S \otimes_R T \mid T}(M \otimes_R T, N)\). Some application of this formula is given, e.g.\ for every homomorphism \(f: A \to B\) of Noetherian commutative rings, if \(\text{Tor}_n^{B \otimes_A B \mid B} (B,-)\) is zero for all \(n \geq m\) then the fibres of \(f\) are geometrically regular of dimension at most \(m\).
scientific article; zbMATH DE number 2067067
scientific article