In the introduction of the paper under review, the authors express that their present work is inspired by the earlier paper by S. Nasseh and S. Sather-Wagstaff [Proc. Am. Math. Soc. 145, No. 11, 4661–4674 (2017; Zbl 1394.13018)], and that it should be regarded as an addendum or an advertisement for the utility and the nice results established in [loc. cit.] concerning the vanishing of Ext and Tors fiber products.
Fix two local rings \((R,\mathfrak{m},k)\) and \((T,\mathfrak{n},k)\) with the same residue field \(k\) such that neither \(S\) nor \(T\) is a field. The fiber product, \(S\times_kT\), of \(S\) and \(T\) over \(k\) is the subset of the Cartesian product \(S\times T\) consisting of element \((s,t)\) such that the residue classes of \(s\) and \(t\) in \(k\) coincide. The fiber product \(S\times_kT\) then is a local ring with decomposable maximal ideal \(\mathfrak{m}\times \mathfrak{n}\) and will be denoted by \(R\) throughout the review. As a folklore result, \(R\) has depth at most \(1\) and so \(\text{pd}_R(M)\le 1\) for any \(R\)-module of finite projective dimension.
The present paper contains several results concerning the effect of Tor vanishings on the intrinsic properties of the \(R\)-modules involved in the Tor. An excerpt of the results of the paper is as follows (in each of the results below, \(M\) and \(N\) are assumed to be finitely generated modules over the fiber product \(R\), although some results may not need this finiteness assumption):
Theorem 1.16. Assume that neither \(S\) nor \(T\) is a discrete valuation domain. If \(\text{Tor}^R_m(M,N)=0\) for some \(m\ge 6\), then \(\text{pd}_R(M)\le 1\) or \(\text{pd}_R(N)\le 1\).
Theorem 1.18. If \(\text{Tor}^R_{2i+1}(M,N)=0=\text{Tor}^R_{2j}(M,N)\) for some \(i\ge 2\) and \(j\ge 3\), then \(\text{pd}_R(M)\le 1\) or \(\text{pd}_R(N)\le 1\).
Corollary 1.19. Assume that neither \(S\) nor \(T\) is a discrete valuation domain and \(M\) and \(N\) are torsionless \(R\)-modules. If \(\text{Tor}^R_4(M,N)=0\), then at least one of \(M\) and \(N\) has projective dimension at most \(1\).
Example 1.14. If \(S\) and \(T\) are discrete valuation domains, then \(\text{Tor}^R_m(S,S)=\text{Tor}^R_m(T,T)=0\) for every positive even integer \(m\) and \(\text{Tor}^R_m(S,T)=0\) for every odd positive integer.
In particular Example 1.14, above, answers in negative a question asked by Nasseh and Sather-Wagstaff in [loc. cit.] about the finiteness of the projective dimension of \(M\) or \(N\) provided \(\text{Tor}^R_4(M,N)=0\) (the \(R\)-modules \(S\) and \(T\) here both have infinite projective dimension). However, Corollary 1.19 (mentioned above) shows that under some mild conditions the Nasseh and Sather-Wagstaff question has an affirmative answer. Moreover, Theorem 1.18 above is a slight generalization of Theorem 1.1(b) of [loc. cit.].
One ingredient of the proofs of the aforementioned results of the paper under review is the following result of A. Dress and H. Krämer [Math. Ann. 215, 79–82 (1975; Zbl 0287.13005)]:
Let \(M\) be an \(R\)-module. Then the second syzygy, \(\Omega^2_R(M)\), of \(M\) decomposes as a direct sum: \(\Omega^2_R(M)\cong X\oplus Z\), where \(X\) is an \(S\)-module and \(Z\) is an \(T\)-module.