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Vanishing of Tor on a complete intersection
Abstract
Let (R, m) be a complete intersection, that is, a local ring whose m-adic completion is the quotient of a regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation deals mostly with the vanishing of Tor$\sbsp{i}{R}(M,\ N)$ and various consequences. The even and odd Betti sequences of an R-module M are eventually given by polynomials of the same degree d. The non-negative integer d + 1 is called the complexity of M. We use this notion of complexity to study the vanishing of Tor$\sbsp{i}{R}(M,\ N).$ In particular, we prove several theorems on the rigidity of Tor which are generalizations and, in some cases, 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 on the minimum of the complexities of M and N. We give examples showing this dependence is sharp. We also show that if $M\otimes\sb{R}\ N$ has finite length, two consecutive vanishing Tors of sufficiently high degree force the vanishing of all higher Tors. Associated to each R-module M is a certain affine algebraic set. We give a necessary condition for the vanishing of Tor$\sbsp{i}{R}(M,\ N)$ for all $i\gg0$ in terms of the intersection of these sets. We then apply this condition to the study of torsion in tensor products of modules. We also give a sufficient condition for the vanishing of Tor$\sbsp{i}{R}(M,\ N)$ for all $i\gg0$ in terms of lifting M and N to "disjoint" complete intersections of lower codimension. We use this condition to construct tensor products of non-free modules which are maximal Cohen-Macaulay. Finally, we prove that if M has complexity greater than 1, the Betti sequence of M is eventually strictly increasing. This was proved independently by Avramov, Gasharov and Peeva in (AGP) using somewhat different techniques.
Subject Area
Mathematics
Recommended Citation
Jorgensen, David Allen, "Vanishing of Tor on a complete intersection" (1996). ETD collection for University of Nebraska-Lincoln. AAI9628236.
https://digitalcommons.unl.edu/dissertations/AAI9628236