Commutative Algebra Seminar

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Abstract goes here.

The notion of totally reflexive module has been introduced by Auslander and Bridger. A finitely generated module M over a noetherian ring R is called "totally reflexive" if the following two conditions are satisfied: (1) Ext^i_R(M,R)=0 for all i>0, (2) Ext^i_R(Tr(M),R)=0 for all i>0. Here, Tr(M) is the Auslander transpose of M. There is a famous question that "Does the condition (1) imply the condition (2)?". In 2004, D. Jorgensen and L. Sega gave a counter example. However, there are many cases where the question is hold. In this talk, we will consider when the question is hold.

Let R be a commutative Noetherian ring. In this talk, we introduce the local homology functor lambda^W with cosupport in a subset W of Spec R. If W is equal to a closed subset V(I) for an ideal I of R, then lambda^W coincides with an ordinary local homology functor, which is isomorphic to the left derived functor of I-adic completion functor. We show several results about lambda^W, where W is not necessarily closed or specialization-closed. As an application, we can give a simple proof of a classical theorem by Raynaud and Gruson, which states that the projective dimension of a flat R-module is less than or equal to the Krull dimension of R. This talk is based on a joint work with Yuji Yoshino.

A classical theorem of Auslander-Buchsbaum asserts that the divisor class group of a regular local ring A is trivial. Several years later, Claborn and Fossum conjectured that the same result ought to hold for all cycles of positive codimension on Spec A. In this talk, we shall discuss the storied history of this problem, its connection with higher algebraic K-theory, and some recent progress.

In my talk I will report on joint work with Manuel Blickle. I will explain how one can generalize the definition of test ideals tau to so-called Cartier modules in a functorial way. We obtain several transformation rules with respect to f^! and f_* for various classes of morphisms f: X -> Y, e.g. for f smooth one has an isomorphism f^! tau = tau f^!. Part of the reason for working in this generality is that one has an equivalence with constructible etale p-torsion sheaves up to nilpotence of Cartier modules and these results further support the idea that the test module construction relates to etale nearby cycles similarly to the complex situation where multiplier ideals relate to complex nearby cycles.

One of the central problem in linkage theory (or liaison) is to understand the behavior of singularities. Given a variety X, the general equations of X will define a complete intersection V containing X as a component. The closure of V-X gives a variety Y as the residual part of X in V . Then Y is called a general link of X via V . Even for X nonsingular, Y is usually not nonsingular. But sometimes Y could have better singularities than X. In this talk, we will discuss how techniques from birational geometry can used in the study of singularities under linkage. We will show that MJ-singularties can be preserved under the process of linkage. This also leadsus to show the minimal log discrepancies are increased under linkage.

Given a radical ideal I in a regular ring R, the containment problem of symbolic and ordinary powers of I consists of determining which symbolic powers of I are contained in each power of I. By work of Ein-Lazersfeld-Smith and Hochster-Huneke, there is a uniform answer to this question, but the containments it provides are not necessarily best possible.

In this talk, we will discuss the containment problem and present new results for the case when R/I is F-pure or when R/I is strongly F-regular (each); in particular, that a conjecture of Harbourne holds in the F-pure case. This is joint work with Craig Huneke.

Abstract goes here.

The title refers to a conjecture on the sum of the ranks of all the free modules occurring in a free resolution of a module of finite projective dimension over a local ring. It is a weakened form of the Buchsbaum-Eisenbud-Horrocks Conjecture, which concerns a lower bound on the ranks of the individual free modules in such a resolution. I’ll sketch the proof of the total rank conjecture in certain cases and discuss possible generalizations.