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# Vanishing of Ext and Tor over complete intersections

#### Abstract

Let (*R,* m ) be a local complete intersection, that is, a local ring whose m -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 Tor^{R}i (*M*, *N*) and Ext^{i}R (*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

*n*th 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 Tor

^{R}i (

*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 Tor

^{R}i (

*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*⊗

*Hom*

_{R}*(*

_{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 Ext

^{i}R (

*M, N*) has finite length for all

*i*» 0, then the Herbrand difference [18] is defined as length( Ext

^{2}R (

*M, N*)) – length( Ext

^{2n-1}R (

*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 Ext

^{i}R (

*M, N*) needed to ensure that Ext

^{i}R (

*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.^

#### Subject Area

Mathematics

#### Recommended Citation

Celikbas, Olgur, "Vanishing of Ext and Tor over complete intersections" (2010). *ETD collection for University of Nebraska - Lincoln*. AAI3411994.

http://digitalcommons.unl.edu/dissertations/AAI3411994