Title: Uniform Artin-Rees bounds for Syzygies
Abstract: Let (R,m) be a Noetherian local ring of dimension d, M a finitely generated R-module, and F• a finite free resolution of M. Let Zi be the ith syzygy module in the resolution. Eisenbud and Huneke raised the following uniform Artin-Rees style question: Given an ideal I, is there a uniform k such that for all i > 0 and for all n > 0 the containment Zi ∩ In Fi-1 ⊆ In-kZi? They were able to show that this is true if M has both constant rank and finite projective dimension on the punctured spectrum. We show, in fact, that not only is the general answer yes, but that the uniformity is true to a much greater extent. While the beginning of the resolution may be quite badly behaved, once we get to the dth syzygy, we get uniform behavior for all modules and all ideals in R. Specifically, we show that there exists k ≥ 0 such that for all finitely generated modules M, for all ideals I, and for i > d, Zi ∩ In Fi-1 ⊆ In-kZi. The proof relies on Huneke's uniform Artin-Rees theorem, and can also be viewed as an extension of that theorem. This work is joint with Aline Hosry and Janet Striuli.
Title: Rigidity of Ext and Tor with coefficients in residue fields
Abstract: This is joint work with Lars Christensen and Srikanth Iyengar. Let R be a commutative Noetherian ring and p a prime ideal. Let k(p) denote the residue field of Rp. We say that Ext*R(k(p), -) is rigid at n if whenever ExtnR(k(p),M)=0 for any R-module M (not necessarily finitely generated) then ExtiR(k(p), M)=0 for all i ≥ n. We define TorR*(k(p),-) to be rigid at n in an analogous manner. We show that Ext*R(k(p), -) is rigid at n = dim R + dim Rp + 1 and TorR*(k(p),-) is rigid at n = dim R_p +1.
Title: Local cohomology of powers of monomial ideals
Abstract: Let (R,m) be a commutative local ring of dimension d and I an R-ideal. Let L(M) be the length of the module M. The asymptotic behavior of the sequence L(H0m(R/In)) for n > 0 has been studied by several authors. For example, Dale Cutkosky proved that if R is analytically unramified and d > 0, the limit of the sequence L(H0m(R/In))/nd exists for any I. In this work, we focus on the sequence L(Him(R/In)) for i>0. We are able to show that for a monomial ideal I, if L(Him(R/In)) is finite for n >> 0, then this sequence coincides with a quasi-polynomial. Moreover, for a family of square-free quadratic monomial ideals we show that the limit of the sequence L(Him(R/In))/nd exists. We also prove the existence of a related limit that involves the minimal degree of the modules Him(R/In). This is joint work with Hailong Dao.
Title: Composition series of local cohomology modules
Abstract: A long standing problem in algebraic geometry and commutative algebra is to determine whether every irreducible curve in projective three-space is the intersection of two hypersurfaces. One way to approach this problem is via the study of local cohomology modules. As modules over the ring, local cohomology modules are huge (neither finitely generated nor Artinian), hence intractable. However, as modules over the Weil algebra D they have a finite composition series and become manageable. Hence an important task is to understand the D-module structure of local cohomology modules. In this talk we describe their composition series. This is joint work with Robin Hartshorne.
Title: Questions about Boij-Söderberg theory
Abstract: Boij-Söderberg theory is concerned with numerical invariants of graded modules and coherent sheaves up to scalar multiple. There is a rich theory behind the case of polynomial rings and projective spaces, respectively, and a number of generalizations have appeared. I'll highlight some questions and possible research directions. This is based on http://arxiv.org/abs/1606.01867
Title: Fundamental groups of F-regular singularities via F-signature
Abstract: Inspired by the analogy between F-regular singularities in characteristic p > 0 and KLT singularities in characteristic zero, it is natural to hope that the punctured spectrum of an F-regular singularity has finite etale fundamental group. I will talk about joint work with Carvajal and Tucker which proves exactly this. In fact, we prove more, we bound the size of the fundamental group by the reciprocal of the F-signature. As another consequence of this work, we obtain new purity of the branch locus results for rings with "mild" singularities.