Schedule and Abstracts

All talks will take place in Avery Hall, room 19.

 Saturday, April 21, 2013 10:30 – 10:50 Registration 10:50 – 11:00 Opening Remarks 11:00 – 11:40 Kosmas Diveris Vanishing of self-extensions over some self-injective rings. 11:50 – 12:30 Grant Serio Calculating Buchsbaum-Rim Multiplicities 12:30 – 2:00 Lunch 2:00 – 2:20 Kat Shultis Free Resolutions and Sparse Determinantal Ideals Part I 2:30 – 3:10 Olgur Celikbas On Auslander's depth formula 3:20 – 3:40 Haydee Lindo Free Resolutions and Sparse Determinantal Ideals Part II 3:50 – 4:10 20 Minute Break 4:20 – 5:00 Richard Wicklein Codualizing Modules And Complexes 5:10 – 5:30 Jack Jeffries Splittings for Rings of Modular Invariants
 Sunday, April 22, 2013 9:30 – 10:10 Jonathan Montaño j-multiplicity and Monomial Ideals 10:20 – 10:40 Silvia Saccon Direct-sum behavior of maximal Cohen-Macaulay modules 10:50 – 11:30 Jason Hardin Generalizing a Bound of Avramov and Buchweitz 11:40 – 12:00 Alessio Sammartano Associated graded rings of numerical semigroup rings

## Abstracts

Speaker: Kosmas Diveris
Title: Vanishing of self-extensions over some self-injective rings.
Abstract: The Auslander-Reiten (AR) quiver of an Artin algebra is a combinatorial device for organizing the indecomposable modules over the algebra. For commutative self-injective rings (and even a bit more generally), the combinatorial structure of this quiver is well suited for investigating modules with eventually vanishing self-extensions. In fact, one can determine when the vanishing of self-extensions must begin for any such module based on its position in the AR quiver. In this talk, we will explain how one can use the combinatorial data of the AR quiver to prove this and discuss connections with conjectures of Tachikawa, Auslander and Reiten. This is based on joint work with Marju Purin of St. Olaf College.

Speaker: Grant Serio
Title: Calculating Buchsbaum-Rim Multiplicities
Abstract: The Buchsbaum-Rim multiplicity of a module is a generalization of the Hilbert-Samuel multiplicity of ideals. Many results which are true for H-S multiplicity have parallel statements in terms of B-R multiplicities, but whether they are true is still unknown. I explore some differences which make modules more difficult to work with than ideals, and calculate some examples of B-R multiplicities of some simple modules. All this is then related to the question of the relationship between B-R multiplicity of a module and the B-R multiplicity of powers of that same module.

Speaker: Kat Shultis
Title: Free Resolutions and Sparse Determinantal Ideals Part I
Abstract: A sparse generic matrix is a matrix whose entries are either zero or distinct variables. In this talk, we will explore recent work by Adam Boocher who extends results of Giusti and Merle by computing a minimal free resolution for the ideals of such matrices. We will start by discussing relevant background material, and then explain Boocher’s method of pruning. We will end with some of Boocher’s results and how he arrives at them using his pruning method.

Speaker: Olgur Celikbas
Title: On Auslander's depth formula
Abstract: In 1961 Auslander proved that finitely generated Tor-independent modules over regular local rings satisfy a remarkable depth equality, referred to as Auslander's depth formula. This formula is an important tool to compute the depth of tensor product of modules over certain commutative Noetherian local rings. The aim of my talk is to partially survey the literature on Auslander's depth formula with an emphasis on some related recent results.

Speaker: Haydee Lindo
Title: Free Resolutions and Sparse Determinantal Ideals Part II
Abstract: A sparse generic matrix is a matrix whose entries are either zero or distinct variables. In this talk, we will explore recent work by Adam Boocher who extends results of Giusti and Merle by computing a minimal free resolution for the ideals of such matrices. We will start by discussing relevant background material, and then explain Boocher’s method of pruning. We will end with some of Boocher’s results and how he arrives at them using his pruning method.

Speaker: Richard Wicklein
Title: Codualizing Modules And Complexes
Abstract: Let $R$ be a commutative, noetherian ring. A finitely generated $R$-module $C$ is said to be semdualizing if $\operatorname{Ext}_R^i (C,C)=0$ for all $i>0$ and $R \xrightarrow\cong \operatorname{Hom}_R(C,C)$. When $R$ is local, an artinian $R$-module $T$ is said to be quasidualizing if $\operatorname{Ext}_R^i (T,T)=0$ for all $i>0$ and $\widehat{R} \xrightarrow\cong \operatorname{Hom}_R(T,T)$. Using the notion of $I$-cofiniteness, we introduce a unifying notion that recovers each of the above notions as special cases. This is based on joint work with Sean Sather-Wagstaff.

Speaker: Jack Jeffries
Title: Splittings for Rings of Modular Invariants
Abstract: Rings of polynomial invariants of finite group actions are among the most classical objects in commutative algebra. There are many beautiful theorems ensuring that the invariant ring has good properties when the order of the group is invertible. However, if the order of the group is not a unit (i.e., is divisible by the characteristic of the ground field), many of these properties become more subtle. In this talk, I aim to illustrate some of the differences in invariant theory in this setting, and to describe some of my work in progress in this area.

Speaker: Jonathan Montaño
Title: j-multiplicity and Monomial Ideals
Abstract: The j-multiplicity was introduced by Achiles and Manaresi in 1993 as a generalization of the Hilbert-Samuel multiplicity for arbitrary ideals in a Noetherian local ring. In this talk, I will review some of its properties and applications. I will also report joint work with Jack Jeffries, where we show that the j-multiplicity of a monomial ideal is equal to the normalized volume of a region. This result is an extension of Teissier's volume interpretation of the Hilbert-Samuel multiplicity for m-primary monomial ideals.

Speaker: Silvia Saccon
Title: Direct-sum behavior of maximal Cohen-Macaulay modules
Abstract: In the last half century, there has been a lot of interest in the study of direct-sum behavior of finitely generated modules over commutative Noetherian local rings. While every module is a direct sum of indecomposable modules, these decompositions need not be unique. We restrict our attention to one-dimensional Noetherian local rings $(R, \mathfrak{m})$ whose $\mathfrak{m}$-adic completion $\widehat{R}$ is reduced and to the class of maximal Cohen-Macaulay $R$-modules. The direct-sum behavior of this class of modules is captured by the monoid of isomorphism classes of maximal Cohen-Macaulay modules (together with $[0_R]$) with operation induced by the direct sum. The key to understanding this monoid is determining which ranks occur for indecomposable maximal Cohen-Macaulay modules. While it is known how this class of modules behaves when the ring has finite Cohen-Macaulay type, the question is still open when the ring has infinite Cohen-Macaulay type. In this talk, I will discuss progress made in this direction, and describe the direct-sum behavior of this class of modules when the ring has infinite Cohen-Macaulay type.

Speaker: Jason Hardin
Title: Generalizing a Bound of Avramov and Buchweitz
Abstract: Let M be a finitely generated module over a complete intersection of codimension two. In 2000, Avramov and Buchweitz provided a bound on the degrees of the generators of the graded module Ext$^*$(M,k) in terms of the Betti numbers of M. The proof relies on properties specific to the codimension two case and is not generalizable to higher codimension. Using recent work of Burke and Walker, we'll discuss a result which provides a bound in any codimension in terms of the regularity of coherent sheaves. We'll see that in the codimension two case, this bound implies the Avramov-Buchweitz bound.

Speaker: Alessio Sammartano
Title: Associated graded rings of numerical semigroup rings
Abstract: The goal is to give numerical characterizations of semigroup rings such that the tangent cone is a complete intersection. A main tool in this study is the so called Apery Set of a numerical semigroup. If time allows, other properties such as the Cohen-Macaulayness and the Gorensteinness will be explored. This is a joint work with D'Anna and Micale, published on J. Commutative Algebra (2011) and J. Pure Applied Algebra (2013).