Commutative Algebra Seminar

Let R be a Noetherian ring and I an ideal. It is well known that if the associated graded ring gr_I(R) is Cohen-Macaulay or Gorenstein, then so is R. However, the converse does not hold true in general. Therefore, it is interesting to investigate under which conditions good properties of R transfer to gr_I(R) or, more generally, to Rees algebras. In this talk, we show that for some classes of extended Rees algebras, the quasi-Gorensteinness implies Gorensteinness.

Hilbert-Kunz multiplicity is an invariant of a local ring in characteristic p > 0. I will talk about global properties of this invariant viewed as a function on the spectrum of a ring, and compare the results to the classical theory of multiplicity developed in connection with resolution of singularities.

We study free resolutions of R = Q/I modules using A-infinity structures on Q-free resolutions. These higher homotopies give a very general change of rings theorem for free resolutions, and a construction of Q-free resolutions of all R-syzygies of R-modules. We use these tools to study Golod modules. The higher homotopies shed light on some old results, and give two new results: if the usual power series bound defining a Golod module is an equality in the first dim Q degrees, it is an equality in all degrees; and a construction of the minimal free resolution of every module over a Golod ring.

There are several different flavors of quantum sl(2) based on a common algebraic presentation given by generators and uncommunicative relations. In this talk I will discuss an algebra we call the unrolled quantum group and its category of representations. In particular, I will discuss how to classify all indecomposable projective modules in this category and give a graded quiver which describes the maps between these indecomposable projective modules. This work is motived by topological applications. Moreover, conjecturally the category of representations considered in this talk is equivalent to the category of representations of the so called singlet vertex algebra W(2, 2p-1) studied in Conformal Field Theory. This is joint work with Francesco Costantino and Bertrand Patureau.

Balmer has shown us how to define a topological space (known as the spectrum) for any tensor triangulated category. The definition is very much in the spirit of the spectrum of a commutative ring and, indeed, you can recover the spectrum of a commutative ring using Balmer's setup. More generally, Balmer's spectrum allows us to bring geometry into various new settings, including many of importance in representation theory. However, the power and generality of Balmer's setup is offset by the challenge of providing an explicit description in any given setting. In our work we show that Balmer's spectrum has a remarkably down-to-earth description when applied to the representation theory of complex Lie superalgebras. In this talk we will give an overview of this work while keeping it accessible to a general mathematical audience. This work is joint with Brian Boe and Daniel Nakano.

In this talk we discuss recent generalizations to non m-primary ideals of the conditions of minimal multiplicity. We discuss the interplay between these conditions and the depths of blowup algebras. We will also show a bound on the reduction number of ideals having a Cohen-Macaulay associated graded ring.

F-modules were introduced by Lyubeznik in order to study properties of local cohomology modules in positive characteristic. Since then, F-modules have been found to have different applications in commutative algebra and algebraic geometry.  The goal of this talk is twofold. First, to show how (certain) F-modules can be used to recover invariants associated to the singularities of a hypersurface in positive characteristic. And, secondly, to produce two new families of ideals for measuring singularities, the so called F-jumping and F-Jacobian ideals.