Tor and torsion on a complete intersection
Let \((R, {\mathfrak m})\) be a complete intersection ring, i.e. \(\widehat R=Q/(x)\) with a regular local ring \((Q, {\mathfrak n},k)\) and a \(Q\)-regular sequence \(x=(x_1, \dots, x_r)\), and let \(M,N\) be two finitely generated \(R\)-modules. The author studies the vanishing of \(\text{Tor}^R_i (M,N)\) for \(i\gg 0\) and its relation to the existence of torsion elements in \(M \otimes_RN\). His first result says that \(\text{Tor}^R_i (M,N)=0\) for \(i\gg 0\) implies the triviality of the intersection \({\mathcal V} (Q,x,M) \cap {\mathcal V} (Q,x,N)= \{0\}\). Here \({\mathcal V} (Q,x,M)\) is the cohomological variety \[ {\mathcal V} (\text{Ann}_{\mathcal R} (\text{Ext}^*_R(M,k)))\subseteq \mathbb{A}^r_k \] of the module \(M\) defined by \textit{L. L. Avramov} [cf. Invent. Math. 96, No. 1, 71-101 (1989; Zbl 0677.13004)]. As an application, the author shows that if \(M\) is a generically free maximal Cohen-Macaulay \(R\)-module of complexity \(c=\dim {\mathcal V} (Q,x,M)\), and if \(S_i\) is an \(i\)-th syzygy module of \(M\) for \(i=1, \dots, c\in \{1,r\}\), then the tensor product \((M \oplus S_1 \oplus \cdots \oplus S_c) \otimes_R M^*\) has torsion elements. Finally, the author also shows a sufficient condition for the vanishing of \(\text{Tor}^R_i (M,N)\) for \(i\gg 0\). More precisely, he proves that this is the case, if \(M\) lifts to \(Q/(x_1, \dots, x_i)\) and \(N\) to \(Q/(x_j, \dots, x_r)\) for \(1\leq i<j\leq r\geq 2\). He applies this condition to provide a source of maximal Cohen-Macaulay tensor products \(M\otimes_RN\) in case \(r\geq 2\), where neither \(M\) nor \(N\) need to be free.
scientific article; zbMATH DE number 1088613