Tor and torsion on a complete intersection

[摘要] Let (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. Suppose M and N are finitely generated R-modules. We give a necessary condition for the vanishing of Tor(i)(R)(M, N) for all i >> 0 in terms of the intersection of certain affine algebraic sets associated to M and N. We apply this condition to the study of torsion in tensor products. For example, we shaw that if R is a domain and M is an R-module of infinite projective dimension then there exist infinitely many n for which the tensor product of M with one of its nth syzygy modules has torsion. We also give a sufficient condition for the vanishing Tor(i)(R)(M, N) for all i >> 0 in terms of the ability to left M and N to ''disjoint'' complete intersections of smaller codimension. We use this condition to construct tensor products of non-free modules which are maximal Cohen-Macaulay. (C) 1997 Academic Press.

[发布日期] 1997-09-15