Decent intersection and or-rigidity for modules over local hypersurfaces (Q2838105)

Let \(A\) be a noetherian local ring which is a hypersurface, say \(A=R/(f)\) where \(R\) is a regular local ring, and \(M\) and \(N\) two finitely generated \(A\)-modules. The usual change of rings spectral sequence from \(R\) to \(A\) shows that the Tor modules are eventually periodic (\(\mathrm{Tor}^A_{i}(M,N)=\mathrm{Tor}^A_{i+2}(M,N)\) for \(i\gg 0\)). When the length \(l(\mathrm{Tor}^A_i(M,N))<\infty\) for all \(i>>0\) the function NEWLINE\[NEWLINE \theta^A(M,N) = l(\mathrm{Tor}^A_{2n}(M,N))- l(\mathrm{Tor}^A_{2n+1}(M,N))NEWLINE\]NEWLINE (for \(n\) sufficiently large) was introduced in [\textit{M. Hochster}, Algebraic geometry, Proc. Conf., Chicago Circle 1980, Lect. Notes Math. 862, 93--106 (1981; Zbl 0472.13005)]. The vanishing of \(\theta^A(M,N)\) is known (when \(A\) is an admissible hypersurface) to imply decent intersection [loc.cit.]. In the paper under review it is also shown to imply rigidity of Tor under the same hypotheses, and moreover, several results are obtained around these three topics: decent intersection, vanishing of \(\theta\), and rigidity of Tor.NEWLINENEWLINEIn order to give an idea of the results, we reproduce here just one of them: let \(A\) be a local hypersurface ring with an isolated singularity (in this case \(\theta^A(M,N)\) is defined). If \(A\) is excellent and contains a field, then dim \(M\) + dim \(N\) \(\leq\) dim \(A\) implies \(\theta^A(M,N)=0\).NEWLINENEWLINESince the paper was written, some other papers appeared dealing with similar questions, some of them are listed in the bibliography.