CARS
Moshen Gheibi
Introduction to Theory of Linkage
April 28
Linkage of algebraic varieties first appeared in the work of Noether, Halperin and Severi in late 19th century and early 20th century. They used linkage to study the curves in ℙ3. The idea of linkage is to pass from a given curve to another one which is simpler, without losing certain properties of the original curve. Iterating the procedure gives a series of curves which are in the same "linkage class". Peskine and Szpiro in 1974 reduced general linkage to certain questions on ideals over commutative algebras and after then, many works have been done to develop this theory in commutative algebra and algebraic geometry. In this talk I will try to address some important results in linkage of ideals and generalization of them for modules.
John Myers
You could have discovered Golod rings
April 21
A Golod ring is a type of commutative noetherian local ring that is encountered in the theory of infinite free resolutions. While the precise definition is not difficult to state (they are defined by an extremal growth condition on the Poincaré series of the residue field), the motivation leading up to the definition can be difficult to understand. The goal of this talk is to convince the audience that the definition is actually quite natural -- so natural, in fact, that they could have discovered the class of Golod rings for themselves. We will start from the basics, assuming no background in the theory of infinite free resolutions, and then progress through the main definitions and a few examples to a description of Golod’s main contribution to the theory, which is a characterization of the class of Golod rings in terms of “higher homology operations.” We will end with a discussion of how the class of Golod rings relates (or doesn’t) to the more familiar classes of commutative noetherian local rings, e.g., Cohen-Macaulay rings and complete intersections.
Michael Brown
Clifford Algebras
April 16
I will talk about a 1987 theorem of Buchweitz-Eisenbud-Herzog that relates matrix factorizations of quadratic polynomials to modules over Clifford algebras. I will carefully define what a Clifford algebra is along the way. My goal is to make the talk understandable to anyone who has been attending Mark Walker's hypersurfaces course.
Eric Canton
Uniform Convergence of F-signature of Pairs
April 7
The F-signature of a local ring R (of characteristic p>0) is a number between 0 and 1 which gives an asymptotic measure of the number of free summands of R considered as a module over itself via restriction of scalars along Frobenius. This information can be twisted by some element f in R and a real parameter t≥ 0, where now we require the projection mappings to be pre-multiplied "by" ft. Whereas before the F-signature of R gave a measure of the singularities of R, the F-signature of the pair (R, ft) measures both the singularities of R and of f. As it turns out, t → s(R, f^t) gives an almost-everywhere differentiable function from [0, 1] to [0, 1]; assuming f is an integer polynomial, we can consider the mod-p reduction fp in the power series ring Fp[[x1, ..., xd]], where Fp = ℤ/pℤ. When f is a product of r ≥2 linear polynomials, we will show that the sequence of functions {s(Rp, fpt)} converges uniformly on [0, 2/r] to the quadratic polynomial (rt/2 - 1)2.
Time permitting, I'll relate this circle of ideas to log canonical singularities arising in the characteristic 0 classification of algebraic varieties up to birational equivalence.
Josh Pollitz
Gorenstein Dimension
March 31
In 1969, Auslander and Bridger introduced a homological dimension for finitely generated modules called Gorenstein dimension (or G-dimension). In today's talk, we will discuss G-dimension and see how this is a refinement of projective dimension. In particular, we will sketch a proof of the Auslander-Bridger formula, which is an analogue of the Auslander-Buchsbaum formula, and see how G-dimension gives a characterization of a Gorenstein local ring.
Brittney Falahola
Applications of Dualizing Complexes
March 10
A well-known result of Bass states that Cohen-Macaulay local rings of type 1 are Gorenstein. Vasconcelos conjectured that one only needs the ring to be of type 1 to be Gorenstein. In 1977, Foxby proved this was indeed the case for rings which contain a field using a version of the Intersection Theorem. In 1980, Paul Roberts proved the more general statement that type 1 local rings are Gorenstein using dualizing complexes. In this talk, I will develop some of the background on dualizing complexes that is necessary in understanding the proof that local domains of type 1 are Gorenstein.
Jason Lutz
An Introduction to Quasi-Complete Intersection Ideals
March 3
Quasi-complete intersection ideals were first introduced by Rodicio in 1996, and are defined as ideals with "free exterior Koszul homology". We'll discuss this class of ideals and its relationship to the (two-step) Tate complex.
Seth Lindoken
On the Support of Local Cohomology
February 24
This talk will center on the following question: If R is a commutative Noetherian ring, I an n-generated ideal, and M a finitely generated R-module, is the support of the nth local cohomology module of M with respect to I a closed subset of Spec(R)? In particular, we will give an affirmative answer for the case n=2, which comes to us from the 2007 paper On the Support of Local Cohomology by Huneke, Katz, and Marley.
Andrew Windle
Hyman Bass and Ubiquity: Gorenstein Rings
February 17
In this talk, we will discuss the history and the ubiquity of Gorenstein rings, as presented by Craig Huneke in his paper "Hyman Bass and Ubiquity: Gorenstein Rings". In particular, we will outline the history of the study of Gorenstein rings, starting with Gorenstein, and culminating in Bass's famous paper On the Ubiquity of Gorenstein Rings. Along the way we will survey why Gorenstein Rings are ubiquitous and their basic properties, which appear throughout commutative algebra.
Doug Dailey
On the Decomposition of Flat, Cotorsion Modules
February 10
The decomposition of injective modules over Notherian rings is well-known. In this talk, I plan to cover a "dual" notion of this phenomenon, namely the decomposition of flat, cotorsion modules. The topic offers many simple, but interesting, results, and I hope to provide a solid introduction to the topic. I will be citing and following some of the results in Edgar Enochs's paper "Flat Covers and Cotorsion Modules" during this talk. Anyone who took Tom's class should be very comfortable with the material.
Peder Thompson
Complete Intersection Dimension
January 27
We will highlight portions of the paper
Complete Intersection Dimension by Avramov, Gasharov, and Peeva from 1997. This paper introduced a new homological dimension that has since gained attention. We'll take a look at the basic definitions, some motivation, and examples, which all work toward producing a class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. Along the way we'll see how CI-dimension (complete intersection dimension) interacts with G-dimension and projective dimension. The paper can be found at:
http://www.math.cornell.edu/~irena/papers/ci.pdf
