Complexity and Tor on a Complete Intersection

Let (R, ) be a complete intersection, that is, a local ring whose -adic completion is the quotient of a regular local ring by a regular sequence. SupposeMis a finitely generatedR-module. It is known that the even and odd Betti sequences ofMare eventually given by polynomials of the same degreen; the complexity ofMis the nonnegative integern + 1. We use this notion of complexity to study the vanishing of TorRi(M, N) for finitely generated modulesMandNover a complete intersectionR. We prove several theorems dealing with rigidity of Tor, which are generalizations and, in certain situations, improvements of known results. The main idea of these rigidity theorems is that the number of consecutive vanishing Tors required in the hypothesis of a rigidity theorem depends more on the minimum of the complexities ofMandNrather than on the codimension ofR. We give examples showing that this dependence is sharp. We also show that ifM RNhas finite length, then, for sufficiently high indices, two consecutive vanishing Tors force the vanishing of all higher Tors.