This semester, CARS was organized by Nick Packauskas and Josh Pollitz
Josh Pollitz
Koszul Cohomological Operators and Varieties
April 18
Using the work of Avramov-Gasharov-Peeva, Eisenbud, and Gulliksen, if R is
a complete intersection then ExtR*(M,N) is a finitely generated graded module over
the polynomial ring S=R[X1,...,Xn] where the Xi have cohomological degree 2 for all
finitely generated R-modules M and N. This structure on the Ext-modules have been
exploited to prove interesting homological properties over a complete intersection. The
goal of today's talk is to show how this graded module structure over the polynomial ring
is obtained via a certain Koszul complex. To do this, I will first give a gentle introduction
to DG homological algebra. Next, I will talk about cohomological operators that arise
from a certain Koszul complex and how this gives ExtR*(M,N) the structure of a
finitely generated module over S. Time permitting, I will talk about some of the things I
have been looking at recently.
Mohsen Gheibi
Absolute, Relative, and Tate Cohomology of Modules of Finite G-Dim
April 11
Auslander and Bridger (in 1969) introduced Gorenstein dimension of finitely
generated modules as a generalization of projective dimension. Over Gorenstein rings,
every finitely generated module has finite G-dimension, similar as the fact that over
regular rings any module has finite projective dimension. In this talk, we introduce
relative cohomology of a finitely generated module of finite G-dimension and show that
the relative cohomology functors treat modules of Gorenstein dimension zero as
projective modules.
Andrew Windle
Cohomological Operators on Quotients by Exact Zero Divisors
April 4
Let S be a commutative ring, I an ideal generated by a regular sequence.
Eisenbud (and many others) provide a construction of a family of cohomological
operators on the quotient ring S/I that played an important role in understanding the
homological properties of modules over complete intersections. In this talk, we provide a
similar construction of cohomological operators for the case that I is generated by an
exact zero divisor instead of a regular sequence.
Ben Drabkin
Symbolic Powers and the Containment Problem
March 28
Given a radical ideal, I, in a commutative Noetherian ring, the nth symbolic
power of I is given by the intersection of the saturations of In at each associated prime
of I. The symbolic powers of an ideal are interesting both as algebraic objects, and for
their geometric significance -- in particular, the nth symbolic power of I is the set of
polynomials vanishing to order at least n on the variety corresponding to I. Containment
relationships between ordinary and symbolic ideals are a source of great interest. This
talk will give an introduction to symbolic powers and cover some of the theorems,
conjectures, and counterexamples of the containment problem.
Eric Canton
Group (Co)Homology, Tate Cohomology, and the Extension/Ramification Theory of Valuation Spaces, Part 2
March 7
I will introduce the group (co)homology functors, explain them as some
favorite derived functors of many audience members, and prove basic functorial
properties of them. For this, I follow (for example) Serre's beautiful treatment in "Local
Fields". Several important morphisms will be introduced via the the abstract language of
(co)effacable cohomological delta functors. Grothendieck introduced these ideas in his
"Tohoku paper" [written before derived categories were invented], which give slick
(though highly abstract) morphisms amongst cohomological delta functors. I will
introduce this language and exhibit the stylistic power and beauty of this approach with
several examples.
Nick Packauskas
Netflix And(re) Quill(en Homology)
February 28
We have been developing the Andre-Quillen (co)homology functors for
several weeks. Today we will study ring homomorphisms by examining the vanishing of
AQ homology for certain types of maps. In particular, we will get a characterization of
regular and complete intersection rings in terms of their AQ homology.
Josh Pollitz
The Cotangent Complex and Andre-Quillen Homology
February 21
Today, we will construct the cotangent complex of a map of commutative
rings. We will see that the cotangent complex is a simplicial module that is used to
compute Andre-Quillen (co)homology. Next, we will touch on some basic properties of
Andre-Quillen (co)homology. Finally, we will finish with a specific example where we
explicitly compute Andre-Quillen (co)homology.
Seth Lindokken
Simplicial Algebras and Simplicial Resolutions: Part II
February 14
A continuation of last week's talk.
Seth Lindokken
Simplicial Algebras and Simplicial Resolutions
February 7
Last week we saw the Jacobi-Zariski sequence that related the modules of
Kahler differentials associated to various ring homomorphisms. Andre-Quillen homology
can be thought of as the objects that extend this sequence into a long exact sequence.
To make this precise, we need to familiarize ourselves with some new homological
tools. In this talk we will recall the notion of a simplicial algebra and set the stage for
simplicial resolutions, which will turn out to be the "right" gadgets to consider.
Andrew Conner
Kahler Differentials: Some Basic Properties and Exact Sequences
January 31
This talk is the first in a series reading through the paper "Andre-Quillen
homology of commutative algebras" by Srikanth Iyengar. This week, we'll discuss the
first section of the paper, which covers Kahler differentials. We'll define Kahler
differentials, discuss some of their properties, look at an example or two, and find some
exact sequences of Kahler modules. This will be useful to us later, since Andre-Quillen
homology is built as the derived functor of these sequences later in the paper.
Luigi Ferraro
Depth of Ext Algebras
January 24
We'll start by talking about depth of graded algebras (not necessarily
commutative). Then we are going to use this theory to understand the depth of the ext
algebra of a local ring. The reason why we are interested in this invariant is because if
this depth is at least 2 (and the ring is Gorenstein) we know the structure of stable
cohomology. We will find formulae for the depth of ExtR(k,k) in the following cases: R
regular, R complete intersection, R Gorenstein of codimension 1, 2, 3 and R Golod.
