Commutative Algebra Seminar

Fall 2014, Friday 3:10−4:00, LCB 323

Date Speaker Title — click for abstract (if available)
August 29 Youngsu Kim
University of California, Riverside
Quasi-Gorensteinness of Extended Rees Algebras
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.
September 19 Ilya Smirnov
University of Virginia
Global properties of Hilbert-Kunz multiplicity
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.
September 26 Jesse Burke
UCLA
Higher homotopies and Golod rings
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.
September 26 (4:00pm) Bregje Pauwels
UCLA
Separable and Galois extensions in tensor triangulated categories
October 3 Nathan Geer
Utah State University
Some remarks on quantum sl(2)
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.
October 10 Zheng Yang
University of Nebraska-Lincoln
Cohomology over graded complete intersections of codimension two
October 22 Linquan Ma
Purdue University
The vanishing conjecture for maps of Tor and derived splinters
October 31 Billy Sanders
University of Kansas
Homological dimension, a derived equivalence, and the New Intersection Theorem
November 7 Jonathan Kujawa
University of Oklahoma
Tensor triangular geometry for Lie superalgebras
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.
November 14 Sean Sather-Wagstaff
North Dakota State University
Two questions about Ext
November 21 Jonathan Montaño
Purdue University
Minimal multiplicities and depth of blowup algebras
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.
December 5 Pedro Teixeira
Knox College
Syzygy gap fractals and F-threshold functions
December 12 Felipe Perez
University of Michigan
Using F-modules to measure singularities
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.

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