Relationships Among Hilbert-Samuel Multiplicities, Koszul Cohomology, and Local Cohomology

Abstract

We consider relationships among Hilbert-Samuel multiplicities, Koszul cohomology, and local cohomology. In particular, we investigate upper and lower bounds on the ratio e(I,M)/l(M/IM) for m-primary ideals I of the local ring (R,m) and finitely generated quasi-unmixed R-modules M and, in joint work with Linquan Ma, Pham Hung Quy, Ilya Smirnov, and Yongwei Yao, give a precise formula for the upper bound for all finitely-generated R-modules M and show that the ratio is bounded away from 0 whenever M is quasi-unmixed. We also, as independent work, give a characterization of quasi-unmixed R-modules M whose local cohomology is finite length up to some index in terms of asymptotic vanishing of Koszul cohomology on parameter ideals up to the same index. We show that if M is an equidimensional module over a complete local ring, then M is asymptotically Cohen-Macaulay if and only if the supremum of the set of lengths of lower Koszul cohomology modules on systems of parameters if finite if and only if M is Cohen-Macaulay on the punctured spectrum.