Tor and Torsion on a Complete Intersection Original Research Article

Let (R, image) be a complete intersection, that is, a ring whose image-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 much greater-than 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 much greater-than 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.