Vanishing of Ext and Tor over complete intersections

Abstract

Let (R, [special characters omitted]) be a local complete intersection, that is, a local ring whose [special characters omitted]-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of [special characters omitted](M, N) and [special characters omitted](M, N).

In this context, M satisfies Serre's condition ( S_n) if and only if M is an n th syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r − 1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of [special characters omitted](M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and M ⊗_R N force the vanishing of [special characters omitted] (M, N) for all i ≥ 1. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and M ⊗_R Hom_R( M, R) are torsion-free, then M is free if and only if M has complexity at most one.

If R is a hypersurface and [special characters omitted](M, N) has finite length for all i » 0, then the Herbrand difference [18] is defined as length([special characters omitted](M, N)) – length([special characters omitted](M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of [special characters omitted](M, N) needed to ensure that [special characters omitted](M, N) = 0 for all i » 0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.