# Commutative Algebra in the Southeast National Meeting Atlanta, GA September 24-26, 2010

The Commutative Algebra in the Southeast series of meetings represents a collaborative effort of commutative algebraists at Georgia State University, University of South Carolina and University of Central Florida to increase exposure of their research area in the Southeast through periodic meetings. Each academic year GSU, USC and UCF organize two regional meetings and one national. A link with past meetings can be found here. For some of the past meetings, including national meetings held in Atlanta, see the following links Spring 2010, Atlanta National Meeting-Spring 2009, Fall 2008, Atlanta National Meeting-Spring 2008, Spring 2007, Fall 2007 or Fall 2006 and earlier.

### Organizers

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

Invited speakers are:

 Hailong Dao, University of Kansas Sankar Dutta, University of Illinois at Urbana-Campaign Andy Kustin, University of South Carolina Alina Iacob, Georgia Southern University Paul C. Roberts, University of Utah Karl Schwede, University of Michigan Anurag K. Singh, University of Utah Adela Vraciu, University of South Carolina Wenliang Zhang, University of Michigan

Updated schedule:
 FRIDAY 3:00-4:00pm: Paul C. Roberts (colloquium at GSU) The Arithmetic Case in Commutative Algebra 4:30-5:30pm: Hailong Dao On the structure of Hom(M,N) SATURDAY 9:00-10:00am: Adela Vraciu The resolution of the Frobenius powers of the maximal ideal in a diagonal surface ring 10:15-11:15am: Anurag K. Singh Local cohomology supported at determinantal ideals 11:30-12:30pm: Alina Iacob Gorenstein flat complexes over noetherian rings Lunch Break 2:30-3:30pm: Sankar Dutta A connection between two sets of Conjectures 4:00-5:00pm: Karl Schwede Surjectivity of the trace map and F-singularities SUNDAY 9:00-10:00am: Paul C. Roberts Fontaine Rings and Local Cohomology 10:15-11:15am: Wenliang Zhang Quasilength, content of local cohomology, and closure operations on ideals 11:30-12:30pm: Andy Kustin Strata of Rational Plane Curves

Abstracts:
Hailong Dao, University of Kansas
On the structure of Hom(M,N): Let R be a regular local ring and M,N be reflexive modules over R. In this talk we will describe restrictions on the direct summands of Hom(M,N). Our results recover classical ones by Auslander, Auslander-Goldman, and a result by Griffith on simple vector bundles of small rank on the punctured spectrum of R.

Sankar Dutta, University of Illinois at Urbana-Campaign
A connection between two sets of Conjectures

Andy Kustin, University of South Carolina
Strata of Rational Plane Curves: Each rational plane curve is parameterized by an ordered triple of homogeneous forms in two variables. We relate the Zariski topology in the parameter space to the singularity configuration on the curves. Our tool in this investigation is the Hilbert-Burch matrix for the ordered triple of homogeneous forms. This is joint work with David Cox, Claudia Polini, and Bernd Ulrich.

Alina Iacob, Georgia Southern University
Gorenstein flat complexes over noetherian rings:We show that over a two sided noetherian ring the Gorenstein flat complexes are exactly the complexes of Gorenstein flat modules. We use this to show that over such rings the class of Gorenstein flat complexes is covering. This is joint work with Edgar Enochs and Sergio Estrada.

Paul C. Roberts, University of Utah
The Arithmetic Case in Commutative Algebra: One of the aims of the field of Commutative Algebra is to give a unified theory of the geometric properties of rings that are used in Algebraic Geometry and Number Theory. Nevertheless, there are many instances where the arithmetic case, where the ring does not contain a field, requires different methods than the geometric case, where the ring is defined over a field such as the field of complex numbers. In this talk I will survey some of the special aspects of the arithmetic case and describe how it has come up in recent work on several conjectures in the field.

Fontaine Rings and Local Cohomology: The Fontaine ring is a construction which allows one to apply methods of positive characteristic, and in particular the Frobenius map, to rings of mixed characteristic. In this talk I will describe the construction of the Fontaine ring and discuss recent progress in applying it to questions of local cohomology and the properties of the absolute integral closure of a domain of mixed characteristic.

Karl Schwede, University of Michigan
Surjectivity of the trace map and F-singularities: Suppose that R is a normal domain with fraction field K and that L is a finite separable extension of K. Set S to be the integral closure of R in L. The trace map Tr : L \to K induces a map Tr : S \to R. If R contains a field of charactersitic zero, then Tr is always surjective, but in characteristic p > 0, this is not the case. In this talk we will discuss conditions on the singularities of R and the branch locus which guarantee that Tr must be surjective (which generalizes a well known corollary of Abhyankar's lemma). This is joint work with Kevin Tucker.

Anurag K. Singh, University of Utah
Local cohomology supported at determinantal ideals:We will discuss a vanishing theorem for local cohomology supported at (not necessarily generic) determinantal ideals. This is based on work in progress with Uli Walther.

Adela Vraciu, University of South Carolina
The resolution of the Frobenius powers of the maximal ideal in a diagonal surface ring: We let $R=k[x, y, z]/(x^n+y^n+z^n)$, where $k$ is a field, and $I_N=(x^N, y^N, z^N)\subset R$. We study the $R$-free resolution of $I_N$. We write $N=an+r$, with $0\le r < n$. Once the characteristic of $k$ is fixed, the value of $a$ determines whether $I_N$ has finite or infinite projective dimension. When the projective dimension is infinite, the value of $r$, together with the parity of $a$ determine the periodic part of the resolution. This is joint work with Andy Kustin and Hamid Rahmati.

Wenliang Zhang, University of Michigan
Quasilength, content of local cohomology, and closure operations on ideals:After a brief discussion of notions of quasilength and content of local cohomology recently introduced by Hochster and Huneke, I will explain how to use these ideas to define a closure operation on ideals in commutative rings of all characteristic that agrees with tight closure in positive characteristic. This is a joint work with Mel Hochster.

Participants
 Brett Barwick, University of South Carolina Joe Brennan, University of Central Florida Luis Nunez-Betancourt, University of Michigan Ela Celikbas, University of Nebraska Olgur Celikbas, University of Kansas C-Y. Jean Chan, Central Michigan University Cătălin Ciupercă, North Dakota State University Hailong Dao, University of Kansas Sankar Dutta, University of Illinois at Urbana-Champaign Florian Enescu, Georgia State University Daniel Hernández, University of Michigan Jen-Chieh Hsiao, Purdue University Ryan Karr, University of Central Florida Andy Kustin, University of South Carolina Alina Iacob, Georgia Southern University Anton Leykin, Georgia Tech Jinjia Li, University of Louisville Laura Lynch, University of Nebraska Sara Malec, Georgia State University Chad Matthews, Georgia State University Lance Miller, University of Utah Anton Preslicka, Georgia State University Paul Roberts, University of Utah Karl Schwede, University of Michigan Soumya Deepta Sanyal, University of Missouri, Columbia Farbod Shokrieh, Georgia Tech Anurag K. Singh, University of Utah Bart Snapp, Ohio State University Sandra Spiroff, University of Mississippi Branden Stone, University of Kansas Adela Vraciu, University of South Carolina Roger Wiegand, University of Nebraska Sylvia Wiegand, University of Nebraska Emily Witt, University Of Michigan Yongwei Yao, Georgia State University Josephine Yu, Georgia Tech Wenliang Zhang, University of Michigan

Applications for funding:
For those interested in attending the conference and need financial support, please complete our funding application/registration form.. We have limited funds for financial support. We especially encourage graduate students and junior faculty to apply for support by indicating this on the form.

We ask everyone who intends to attend to still register for the conference by using the above form (just leave blank the appropriate items, if no financial support is requested).

Deadline for the applications for funding: August 1, 2010. There is no deadline to register for the conference.

All talks will be in Classroom South (CLSO) 408. A campus map can be found here: Map. The building is number 3 on the map.
On Saturday and Sunday there will be refreshments served at 8:30am.

Lodging: We have reserved a block of rooms at Holiday Inn Downtown Atlanta. This hotel is within short walking distance to the Department as well as the MARTA station. One can book a room either online or by phone by mentioning the Group Code: CAS. The rate is 94 dollars plus tax and fees. The deadline for reservations is September 3.

Airport: The Airport is served by MARTA and there is a MARTA station in the terminal (SOUTH TERMINAL). The stop for the hotel is Peachtree Center and the ride takes approximately 30-35 minutes.
The meeting is partially supported by the National Security Agency and Georgia State University Research Foundation.