Asymptotic Behavior of Homology and Intersection Multiplicity

Abstract

In this thesis, we are primarily concerned with problems related to intersection multiplicity originally introduced by Serre. The characteristic p techniques, especially, the use of the Frobenius functor due to Kunz, Peskine and Szpiro are the major tools in our study. We will present four results. The first one, also the most important one, deals with the asymptotic length of homology for complexes of finitely generated free modules, under the iteration of the Frobenius functor. We first obtain asymptotic bounds on such length functions in lower dimensional cases, which generalizes a result of Dutta. Then, we demonstrate a surprising example to show that this upper bound does not hold in general. As an application, this study leads to a counterexample to a sufficient condition for an interesting case of nonnegativity of intersection multiplicity over complete intersections. In the third result, we use the Frobenius endomorphism and the vanishing of Ext to characterize finitely generated modules of finite projective dimension over complete intersections. This result completes the picture as analogous results had been proved for Tor by Avramov and Miller. Finally, we provide a slightly different proof of a result originally proved by Foxby on positivity of intersection multiplicity in the case when one of the intersecting modules has dimension one.