Tor and Torsion on a Complete Intersection

AbstractLet (R,m) be a complete intersection, that is, a ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. SupposeMandNare finitely generatedR-modules. We give a necessary condition for the vanishing of TorRi(M,N) for alli⪢0 in terms of the intersection of certain affine algebraic sets associated toMandN. We apply this condition to the study of torsion in tensor products. For example, we show that ifRis a domain andMis anR-module of infinite projective dimension then there exist infinitely manynfor which the tensor product ofMwith one of itsnth syzygy modules has torsion.We also give a sufficient condition for the vanishing TorRi(M,N) for alli⪢0 in terms of the ability to leftMandNto “disjoint” complete intersections of smaller codimension. We use this condition to construct tensor products of non-free modules which are maximal Cohen–Macaulay