GSU-USC Commutative Algebra Seminar, March 2007
Atlanta, GA
March 17-18, 2007



The seminar represents a collaborative effort of commutative algebraists at Georgia State University and University of South Carolina to increase exposure of their research area in the South-East through periodic meetings. There will a meeting in Columbia, SC in April 2007, see the following webpage.

Organizers

Florian Enescu (Georgia State University) fenescu@gsu.edu
Yongwei Yao (Georgia State University) yyao@gsu.edu

Speakers and Titles

Cătălin Ciupercă, North Dakota State University, Projectively equivalent ideals
Abstract: Projective equivalence was introduced by Samuel and further developed by Nagata. As shown by McAdam, Ratliff and Sally, the set of integrally closed ideals projectively equivalent to a given ideal I is linearly ordered by inclusion and eventually periodic. In this talk we present the development of these concepts and give an introduction to projectively full ideals.

Neil Epstein, University of Michigan, Pieces of closure

Abstract: I will discuss recent work on breaking closure operations into specific, useful parts in various different ways. This applies to integral closure, tight closure, and other closures, and includes "special (parts of)" closures and "interiors" of closures. This is useful, for instance, in analyzing "spreads" (how many elements does it take to generate an ideal which gives the closure of a given ideal?) and extending the Brian\,con-Skoda Theorem. It also gives us a partial answer to a question of Holger Brenner on continuous closure. Portions of this work are joint with Mel Hochster.

Ananth Hariharan, University of Kansas, The Gorenstein Colength of an Artinian Local Ring
Abstract: The question we are interested in is the following: Given an Artinian local ring, how 'close' can we get to it by an Artinian Gorenstein local ring. In this talk I will make the notion of being 'close' precise. I will also mention some past results and discuss some current ones.

Andy Kustin, University of South Carolina, Divisors over determinantal rings defined by two by two minors
Abstract.

Sandra Spiroff, Seattle University, Commutativity of Intersection with Divisors.
Abstract: Let A be a Noetherian ring and f a non-zero non-unit of A. Intersection with the divisor (f) gives a map from the Chow group of A to the Chow group of the hypersurface determined by f. Using methods of algebraic geometry, it has been proven that intersection with divisors is commutative up to rational equivalence. In order to better understand the algebra behind this construction, we investigated a proof using purely algebraic means. The case where the divisors intersect in codimension 2 is straightforward, but when they intersect in codimension 1 the result is not so clear. I will present our new proof. This is joint work with Paul Roberts.

Adela Vraciu, University of South Carolina, Monotype closure operations
Abstract: We study closure operations from an axiomatic point of view. There are a number of standard properties that most closure operations satisfy. In addition to these standard properties, we introduce the monotype property, which seems to be specific to tight closure. We show that tight closure satisfies this property, while integral closure, and more generally, any closure that can be obtained as an intersection of other closures (for example the integral closure is obtained by intersecting expansion contractions from module-finite extensions of the ring) does not satisfy the monotype property. This is joint work with Mel Hochster and Anurag Singh.

Schedule:


Saturday

9:30-10:00am Coffee

10:00-10:50am Andy Kustin

11:15-12:05pm Neil Epstein

12:05-1:30pm Lunch Break

1:30-2:20pm Adela Vraciu

2:45-3:35pm Cătălin Ciupercă

4:30-8:00pm Party at Ioana's and Florian's place.

Sunday

9:30-10:00am Coffee

10:00-10:50am Sandra Spiroff

11:15-12:05 Ananth Hariharan

All talks will take place in 796 COE.
Return to the Commutative Algebra Meetings in the Southeast home page.