Homological properties of modules over complete intersections.

Let R be a commutative, Notherian local hypersurface and M, N be finitely generated modules over R. Two properties of the pair (M, N) are studied. We say that (M, N) is rigid if for any integer i ≥ 0, Tor iR(M, N) = 0 implies TorjR(M, N) = 0 for all j ≥ i. If l(M ⊗R N) M, N) is said to intersect decently if dim M + dim N ≤ dim R. The main goal of this thesis is to understand these two properties and their implications. One of the main technical tools is thetaR(M, N), a function previously defined by Hochster. It is known that theta R(M, N) = 0 implies M and N intersect decently. It is shown in this thesis that the vanishing of thetaR( M, N) = 0 also implies the rigidity of ( M, N). Many techniques from Intersection Theory are exploited to study vanishing of the theta function. It is shown, for example, that if R contains a field and has isolated singularity and if dim M + dim N ≤ dim R, then thetaR(M, N) = 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 R is a local complete intersection, a new function eta R(M, N) is introduced using an asymptotic definition, which can be viewed as a generalization of both theta R(M, N) and the Serre intersection multiplicity chiR(M, N). We show that eta is well defined through a delicate study of the complexity (polynomial growth rate) of l(Tor iR(M, N)), which we call tcx(M, N). In particular, when l(ToriR( M, N)) i >> 0, we have tcx(M, N) = cx(M, N). Finally, applications of etaR( M, N) are discussed, including new results about vanishing of Tor and some dimension inequalities over complete intersections which extend results by Hochster and Roberts.Ph.D.MathematicsPure SciencesUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/126005/2/3224859.pd