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. Each term
GSU and USC organize a regional seminar and a national one. This Spring, GSU
will organize the national seminar. There
was a national meeting in Columbia, SC in November 2008, see the following
link. For some of the
past seminars, see the following links Fall 2008,
Spring 2008,
Spring 2007, Fall 2007 or Fall 2006 and earlier.
Florian
Enescu (Georgia State University) fenescu@gsu.edu
Yongwei Yao (Georgia State University) yyao@gsu.edu
The
main speaker for this meeting is the Henry J. Bischoff Professor Craig Huneke from the University
of Kansas. He will deliver a colloquium on Friday, Februrary 27 at
Georgia State University and a research talk later in the meeting.
Invited
speakers are:
Joe Brennan, University of Central Florida
Hailong Dao, University of Kansas
Craig Huneke, University of Kansas
Lee Klingler, Florida Atlantic University
Andrew Kustin, University of South Carolina
Jinjia Li, Middle Tennessee State University
David Jorgensen, University of Texas at Arlington
Adela Vraciu, University of South Carolina
Schedule (all talks will take place in 796 COE, the 7th floor of the College of Education Building - Map , address: 30 Pryor St SW, Atlanta GA):
Friday
Colloquium Talk (refreshments served at 2:00pm)
2:30-3:30pm Craig Huneke, How many times does a polynomial vanish along an algebraic subset of points?
Abstract: This talk will discuss the title question. It is not too hard
to figure out what it should mean for a polynomial f(X_1,...,X_n) to
have an k-fold zero at a point (a_1,a_2,...,a_n) in complex n-space.
One demands that f and all its partial derivatives up to k-1 st order vanish at that
point. But there are several possibilities if one asks that f vanish
along an algebraic subset X of complex n-space. (An algebraic set is
the set of zeroes of some set of polynomials.) The differences between
possible definitions leads to what are called symbolic powers of ideals.
Many open questions pertain to symbolic powers. We will discuss some
elementary ones.
3:45-4:45pm Joseph Brennan, Cut ideals and wheels
Saturday
9:00am Refreshments
9:30-10:30am Lee Klingler, Finitely generated modules over commutative Noetherian rings
11:00-12:00pm David Jorgensen, On the vanishing of Ext_R(M,M)
Lunch break
2:00-3:00pm Adela Vraciu, Canonical modules of rings which are almost Gorenstein
3:15-4:15pm Jinjia Li, Some observations on rigidity of Frobenius endomorphism
4:30-5:30pm Hailong Dao, On complexities of modules
Sunday
9:00am Refreshments
9:30-10:30pm Andrew Kustin, Generic Gaussian Ideals
11:00-12:00pm Craig Huneke, Bounding multiplicities in graded rings
Abstracts
Joe Brennan, University of Central Florida
Abstract: We will survey the progress in investigating on cut ideals and report on the (very very latest) progress in the area.
Hailong Dao, University of Kansas
Abstract: Let R be a local ring. The complexity of a pair of R-modules
(M,N) is defined by Avramov-Buchweitz to be the polynomial growth rate
of the sequence of number of generators of the modules Ext_i(M,N). In
this talk I will describe some recent results on this interesting
invariant especially when R is Cohen-Macaulay (part of this is joint
work with Oana Veliche).
Craig Huneke, University of Kansas
Abstract: This talk is based on preliminary work with M. Mustata, S.
Takagi, and K. Watanabe. We study a conjectured inequality relating
the multiplicities of two homogeneous systems of parameters in a
non-negatively graded Noetherian ring over a field. We sketch a proof
which uses reduction to characteristic $p$ and the fact (proved by
Hochster and myself) that a certain graded plus closure is
Cohen-Macaulay.
Lee Klingler, Florida Atlantic University
Abstract: A survey of results from two joint projects.
First (joint with L. Levy): For which rings is a ``reasonable''
description of all indecomposable finitely generated modules possible?
We show that the answer is an extension of the notion of Dedekind-like
rings introduced by Levy twenty-five years ago.
Second (joint with W. Hassler, R. Karr, and R. Wiegand): It is known
that indecomposable finitely generated modules over Dedekind-like rings
have bounded torsion-free rank (in fact less than or equal to 2 at each
minimal prime). Are there any other rings for which the indecomposable
finitely generated modules have bounded torsion-free rank? We show
that the answer is ``no'' by constructing, for each (local)
non-Dedekind-like ring and each possible rank, infinitely many pairwise
non-isomorphic indecomposable finitely generated modules.
Andrew Kustin, University of South Carolina
Abstract: The content of a polynomial is the ideal generated by its
coefficients. This notion was introduced by Gauss who observed that the
content of the product of two polynomials is the product of contents,
for polynomials in one variable with integer coefficients. Now a days,
the content of the product of two polynomials f and g is called the
Gaussian ideal, G(f,g), of the two polynomials.
Consider two generic polynomials f=x_0+...+x_nt^n and g=y_0+...+y_mt^m,
where the coefficients of f and g are indeterminates over the base
field k. Let R be the polynomial ring
k[{x_i,y_j}]. Corso, Vasconcelos, and Villarreal found that the primary
decomposition of the generic Gaussian ideal G(f,g) consists of three
components: the content of f, the content of g, and one other,
interesting, component that they call L(f,g). Furthermore, they proved
that R/L(f,g) is a Gorenstein ring. Their proof that R/L(f,g) is
Gorenstein amounted to observing that R/L(f,g) is the trivial extension
of a certain residual intersection ring by its canonical module. The
residual intersection and its canonical module had been studied by
Huneke and Ulrich. Corso, Vasconcelos, and Villarreal then examined the
primary decomposition of the content of the product of three generic
polynomials: f, g, and h. It is not hard to guess what they found: the
content of f, the content of g, the content of h, L(f,g), L(f,h),
L(g,h), and one other, interesting, component that they call L(f,g,h).
Corso, Vasconcelos, and Villarreal conjecture that R/L(f,g,h) is a
Gorenstein ring. We discuss a new proof that R/L(f,g) is a Gorenstein
ring with an eye toward proving the [CVV] conjecture. Our proof
involves resolving the coordinate ring of a variety of commutative
squares.
Jinjia Li, Middle Tennessee State University
Abstract: Avramov and Miller proved that over complete intersections, the Frobenius endomorphism (regarded as a module) is
always rigid. In general, this cannot be generalized to the Gorenstein case. Explicit examples of nonrigid Frobenius
endomorphism will be constructed. On the other hand, it is not clear to which extent one can generalize Avramov
and Miller's result. Some partial results in this regard will also be discussed. Part of this work is joint with Claudia
Miller.
David Jorgensen, University of Texas at Arlington
Abstract: The vanishing of Ext_R^i(M,M) for all i>0
happens quite easily for finitely generated modules
M of either finite projective or finite injective
dimension. But is this the only way the vanishing
can occur? We will see that although in some cases
the answer to this question is yes, it is not always
so, even over commutative local rings R. We will
also give a characterization of such modules in
terms of presentations of R by Gorenstein rings.
This involves joint work with G. Leuschke and
S. Sather-Wagstaff.
Adela Vraciu, University of South Carolina
Abstract: We
study the minimal free resolution of the canonical module of a ring
which is almost Gorenstein. For certain rings of this type, which
includeTeter rings (Gorenstein modulo the socle), we prove that there
is a copy of the residue field that splits off the second syzygy of the
canonical module. As a consequence, we see that such rings do not admit
non-trivial totally reflexive modules.
This is joint work with Janet Striuli.
Other participants:
Muslim Baig, Georgia State University
Julian Chan, University of Utah
Annanth Hariharan, University of Kansas
Jong-Wook Kim, Georgia State University
Sara Malec, Georgia State University
Hannah Robbins, Wake Forest University
Bart Snapp, Coastal Carolina University
Sandra Spiroff, University of Mississippi
Harrison Stalvey, Georgia State University,
Janet Striuli, Fairfield University
Javid Validashti, University of Kansas
Two dinners are planned one on Friday at 6pm at Chateau Saigon and the other on Saturday at 6:30pm at
Mali Restaurant . Directions will be provided at the conference. Please let us know by Thursday February 26 if you plan to
attend any of the dinners.
For those interested in attending the conference, please complete our registration form. We have limited funds for
financial support and plan to answer any funding requests after January 28, 2009. So please send your requests by this date.
A few lodging recommendations:
Downtown Atlanta Wyndham.
Downtown Altanta Residence Inn.
North-East Atlanta Courtyard by Marriott.
North-East Atlanta Holiday Inn
Downtown hotels tend to be expensive, but they are within walking
distance to GSU. However, we will organize rides between GSU, hotels
and dining destinations. We would appreciate if you let us where you
plan to stay in case you need rides. There are many other hotels in the
downtown area with varying lodging rates. If you plan to stay there and
have questions regarding a hotel, do let us know.
The meeting is partially supported by the National Security Agency and
the Mathematics and Statistics Department at Georgia State University.