Tor and Torsion on a Complete Intersection

✍ Scribed by David A Jorgensen

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 231 KB
• Volume: 195
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7064

✦ Synopsis

Let R, ᒊ be a complete intersection, that is, a ring whose ᒊ-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 R Ž . Tor M, N for all i 4 0 in terms of the intersection of certain affine algebraic i sets associated to M and N. We apply this condition to the study of torsion in tensor products. For example, we show 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.

R Ž . We also give a sufficient condition for the vanishing Tor M, N for all i 4 0 in i 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.