Let (R, m) be a Noetherian local ring and I an m-primary ideal. In this paper, we study an inequality involving the number of generators, the Loewy length and the multiplicity of I. There is strong evidence that the inequality holds for all integrally closed ideals of finite colength if and only if Spec R has sufficiently nice singularities. We verify the inequality for regular local rings in all dimensions, for rational singularity in dimension 2, and cDV singularities in dimension 3. In addition, we can classify when the inequality always hold for a Cohen-Macaulay R of dimension at most two. We also discuss relations to various topics: classical results on rings with minimal multiplicity and rational singularities, the recent work on p(g) ideals by Okuma-Watanabe-Yoshida, a conjecture of Huneke, Mustata, Takagi, and Watanabe on F-threshold, multiplicity of the fiber cone, and the h-vector of the associated graded ring.

Let R be a polynomial ring over a field. We describe the extremal rays and the facets of the cone of local cohomology tables of finitely generated graded R-modules of dimension at most two. Moreover, we show that any point inside the cone can be written as a finite linear combination, with positive rational coefficients, of points belonging to the extremal rays of the cone. We also provide algorithms to obtain decompositions in terms of extremal points and facets.

Let R be a polynomial or power series ring over a field k. We study the length of local cohomology modules H-I(j) (R) in the category of D-modules and F-modules. We show that the D-module length of H-I(j) (R) is bounded by a polynomial in the degree of the generators of I. In characteristic p > 0 we obtain upper and lower bounds on the F-module length in terms of the dimensions of Frobenius stable parts and the number of special primes of local cohomology modules of R/I. The obtained upper bound is sharp if R/I is an isolated singularity, and the lower bound is sharp when R/I is Gorenstein and F-pure. We also give an example of a local cohomology module that has different D-module and F-module lengths.

Let (R, m) be a Noetherian local ring, and let M be a finitely generated R-module of dimension d. We prove that the set {l(M/IM)/e(I, M)}(root I=m) is bounded below by 1/d!e((R) over bar) where (R) over bar = R/ Ann(M). Moreover, when (M) over cap is equidimensional, this set is bounded above by a finite constant depending only on M. The lower bound extends a classical inequality of Lech, and the upper bound answers a question of Stuckrad-Vogel in the affirmative. As an application, we obtain results on uniform behavior of the lengths of Koszul homology modules.

In a formally unmixed Noetherian local ring, if the colength and multiplicity of an integrally closed ideal agree, then R is regular. We deduce this using the relationship between multiplicity and various ideal closure operations.

We establish the continuity of Hilbert–Kunz multiplicity and F-signature as functions from a Cohen–Macaulay local ring (R,m,k) of prime characteristic to the real numbers at reduced parameter elements with respect to the m-adic topology.

9.

Smirnov, Ilya

Stockholm University, Faculty of Science, Department of Mathematics.

We study Hilbert-Kunz multiplicity of the powers of an ideal and establish existence of the second coefficient at the full level of generality, thus extending a recent result of Trivedi. We describe the second coefficient as the limit of the Hilbert coefficients of Frobenius powers and show that it is additive in short exact sequences and satisfies a Northcott-type inequality.