Multiplicities and Local Cohomology in Commutative Algebra

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This proposal studies two questions in the HomologicalConjectures in Commutative Algebra. The first is on intersection multiplicities defined by modules of finiteprojective dimension. This part of the proposal studiesthe action on thAbstracte 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 complicatedsituation in higher dimension. The second part concernsthe question of whether there are small annihilators of elementsof local cohomology groups in finite extensions. Thisquestion was inspired by a recent result of Ray Heitmann,who proved that small powers of the prime p achieved thisin rings of mixed characteristic in dimension three andused this fact to prove the Direct Summand Conjecture in thatcase. In this proposal we consider alternative approachesin mixed characteristic and whether similar results holdin characteristic zero.One of the problems that often arises in mathematicsis how to measure the order of tangency, or multiplicityof intersection, of subspaces of a space. These numbers areused, 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 canbe quite complicated. This proposal deals with questions that arisefrom attempting to measure these algebraically. More specifically,the methods use Homological Algebra, and they havegiven rise to a set of questions known as the "Homological Conjectures."The particular questions in this proposal concern determiningwhether these invariants are zero and studying their consequences forthe structure of rings and modules.