Homological properties of modules over complete intersections.

Abstract

Let <italic>R</italic> be a commutative, Notherian local hypersurface and <italic>M</italic>, <italic>N</italic> be finitely generated modules over <italic>R</italic>. Two properties of the pair (<italic>M</italic>, <italic> N</italic>) are studied. We say that (<italic>M</italic>, <italic>N</italic>) is <italic>rigid</italic> if for any integer <italic>i</italic> &ge; 0, Tor<italic>_i<super>R</super></italic>(<italic>M</italic>, <italic>N</italic>) = 0 implies Tor<italic>_j<super>R</super></italic>(<italic>M</italic>, <italic> N</italic>) = 0 for all <italic>j</italic> &ge; <italic>i</italic>. If <italic> l</italic>(<italic>M</italic> &otimes;<italic>_R</italic> <italic> N</italic>) < infinity, then (<italic>M</italic>, <italic>N</italic>) is said to <italic>intersect decently</italic> if dim <italic>M</italic> + dim <italic> N</italic> &le; dim <italic>R</italic>. The main goal of this thesis is to understand these two properties and their implications. One of the main technical tools is theta<italic><super>R</super></italic>(<italic>M</italic>, <italic> N</italic>), a function previously defined by Hochster. It is known that theta<italic><super> R</super></italic>(<italic>M</italic>, <italic>N</italic>) = 0 implies <italic> M</italic> and <italic>N</italic> intersect decently. It is shown in this thesis that the vanishing of theta<italic><super>R</super></italic>(<italic> M</italic>, <italic>N</italic>) = 0 also implies the rigidity of (<italic> M</italic>, <italic>N</italic>). Many techniques from Intersection Theory are exploited to study vanishing of the theta function. It is shown, for example, that if <italic>R</italic> contains a field and has isolated singularity and if dim <italic>M</italic> + dim <italic>N</italic> &le; dim <italic>R</italic>, then theta<italic><super>R</super></italic>(<italic>M</italic>, <italic> N</italic>) = 0. As an application, it is proved that the high depth of the tensor product of two modules forces all the higher Tor modules to vanish. When <italic>R</italic> is a local complete intersection, a new function eta<italic><super> R</super></italic>(<italic>M</italic>, <italic>N</italic>) is introduced using an asymptotic definition, which can be viewed as a generalization of both theta<italic><super> R</super></italic>(<italic>M</italic>, <italic>N</italic>) and the Serre intersection multiplicity chi<italic><super>R</super></italic>(<italic>M</italic>, <italic> N</italic>). We show that eta is well defined through a delicate study of the complexity (polynomial growth rate) of <italic>l</italic>(Tor<italic>_i<super>R</super></italic>(<italic>M</italic>, <italic>N</italic>)), which we call tcx(<italic>M</italic>, <italic>N</italic>). In particular, when <italic>l</italic>(Tor<italic>_i<super>R</super></italic>(<italic> M</italic>, <italic>N</italic>)) < infinity for <italic>i</italic> >> 0, we have tcx(<italic>M</italic>, <italic>N</italic>) = cx(<italic>M</italic>, <italic> N</italic>). Finally, applications of eta<italic><super>R</super></italic>(<italic> M</italic>, <italic>N</italic>) are discussed, including new results about vanishing of Tor and some dimension inequalities over complete intersections which extend results by Hochster and Roberts.