This proposal studies two questions in the Homological Conjectures in Commutative Algebra. The first is on intersection multiplicities defined by modules of finite projective dimension. This part of the proposal studies the action on thAbstract e Chow group defined by such a module. A theorem of the Principal Investigator and V. Srinivas answers this question in the case of dimension zero, and this research will investigate the more complicated situation in higher dimension. The second part concerns the question of whether there are small annihilators of elements of local cohomology groups in finite extensions. This question was inspired by a recent result of Ray Heitmann, who proved that small powers of the prime p achieved this in rings of mixed characteristic in dimension three and used this fact to prove the Direct Summand Conjecture in that case. In this proposal we consider alternative approaches in mixed characteristic and whether similar results hold in characteristic zero.
One of the problems that often arises in mathematics is how to measure the order of tangency, or multiplicity of intersection, of subspaces of a space. These numbers are used, for example, to count how many solutions a given equation has or how many geometric objects of a certain type there are. While computing the order of tangency may seem simple for curves intersecting in the plane, in higher dimension it can be quite complicated. This proposal deals with questions that arise from attempting to measure these algebraically. More specifically, the methods use Homological Algebra, and they have given rise to a set of questions known as the "Homological Conjectures." The particular questions in this proposal concern determining whether these invariants are zero and studying their consequences for the structure of rings and modules.
