A generalization of the Auslander-Buchsbaum formula (Q1964128)

Let \(R\) be a local ring and let \(M\) and \(N\) be finitely generated non-zero \(R\)-modules. The paper is concerned with formulas of the invariant \({\mathfrak q}^R(M,N)= \sup \{ i \mid \text{Tor}^R_i(M,N)\not= 0 \}\). The map \(R'\leftarrow Q\) is called a deformation if it is a surjective local homomorphism of local rings whose kernel is generated by a \(Q\)-regular sequence contained in the maximal ideal of \(Q\) and a diagram \(R\to R'\leftarrow Q\) is a quasi-deformation if \(R'\) is a flat \(R\)-module and \(R'\leftarrow Q\) is a deformation. If \(M\) is a finite complete intersection module, i.e. there exists a quasi-deformation \(R\to R'\leftarrow Q\) with \(\text{pd}_QM\otimes_R R'<\infty\), and \({\mathfrak q}^R(M,N)<\infty\), then \[ {\mathfrak q}^R(M,N)= \sup \{ \text{depth}_{R_p}R_p -\text{depth}_{R_p}M_p -\text{depth}_{R_p}N_p\mid p\in \text{spec}(R)\}. \] This is a generalization of the classical Auslander-Buchsbaum formula. It is also shown that when \(R\) is a Golod ring the condition \({\mathfrak q}^R(M,N)<\infty\) implies that either \(M\) or \(N\) has finite projective dimension, then in this case the preceding formula is true.