CARS
This semester, CARS was organized by Ben Drabkin and Andrew Connor
Eric Hopkins
Koszul Complex
November 28
In honor of the 35,758th day since the birth of Jean-Louis Koszul, I'm going to
talk about what the Koszul Complex is and why we might care.
Michael DeBellevue
Alexander Duality and Linear Resolutions
November 14
This talk continues the exploration of the Stanley-Reisner correspondence.
We will prove some results concerning the Alexander dual of an ideal, and then go
through an overview of the proof of a theorem of Eagon and Reiner concerning linear
resolutions and cohen-macaulayness. Then, time permitting, we introduce polarization
as a technique to extend results concerning square-free monomial ideals to other
monomial ideals.
Michael DeBellevue
What are Stanley Reisner Rings? + Alexander Duality
November 7
In this talk I will introduce the basics of Stanley-Reisner theory, which is one
of the first subjects of algebraic combinatorics. We will discuss the combinatorics of
simplicial complexes, and establish the Stanley-Reisner correspondence between
simplicial complexes and monomial ideals. As an illustration of the Stanley-Reisner
correspondence in practice, we will cover the Alexander Duality, which allows one to
relate the associated primes of a squarefree monomial ideal with the complement of the
simplicial complex in the sphere.
Ben Drabkin
Symbolic Defect and Cover Ideals
October 31
Let R be a commutative Noetherian ring, and let I be an ideal in R. The
symbolic defect is a numerical measurement of the difference between the symbolic
and ordinary powers of I. In the case that I has sufficiently well-behaved symbolic
powers (i.e. its symbolic Rees algebra is finitely generated) we prove that the symbolic
defect of I grows eventually quasi-polynomially in m. Furthermore we describe more
specifically the growth of the symbolic defect in certain classes of ideals arising from
combinatorial structures.
Nick Packauskas
Asymptotic Behavior of Betti Sequences
October 24
We have seen so far that regular rings and hypersurface rings can be
classified by the behavior of the Betti sequences of their finitely generated modules. I
will further develop this idea to classify complete intersection rings using homological
criteria. This talk will provide (most of) the necessary background for my upcoming talk
in Commutative Algebra Seminar.
Josh Pollitz
The Derived Category of a Commutative Noetherian Ring
October 17
Today I will speak on well-known/classical constructions and results involving
the derived category of a commutative noetherian ring. The main goal is to give some of
the background necessary for tomorrow's talk in the algebra seminar.
Josh Pollitz
The Homotopy Category of Matrix Factorizations and the Category of Stable MCMs
October 10
Today's talk is the last in a series of four talks exploring the relationship
between matrix factorizations and MCM modules over a hypersurface ring. We will
describe precisely their relationship by discussing two equivalences of categories. In
particular, we will define the categories from the title and show that over a hypersurface
ring these categories are equivalent.
Amadeus Martin
Categories of Matrix Factorizations and Maximal Cohen Macaulay Module
October 3
Erica proved last week why minimal free resolutions over hypersurfaces are
eventually 2-periodic. The main tool to acquire this result was matrix factorizations. This
week we discuss the category of matrix factorizations mf(Q,f), where Q is a regular local
ring and f a non-zero divisor, and its connection to the category of MCM modules over
the hypersurface Q/f. We explore their relation and attempt to construct an equivalence
of categories.
Erica Hopkins
Free Resolutions and Maximal Cohen-Macaulay Modules
September 26
Today I will be continuing where Michael left off in our series of talks given by the MCMSG (Maximal Cohen-Macaulay Super-Group). In particular, my goal is to prove
Eisenbud's result that a module over a hypersurface ring has a periodic 2 free resolution
if and only if it is a maximal Cohen-Macaulay module with no free summand. In order to prove this result, I will be using matrix factorizations that Michael discussed last time. In
addition, I will define and give examples of regular sequences, depth, and maximal
Cohen-Macaulay modules. These definitions and results will be important for the next
two talks in this series.
Michael DeBellevue
An Introduction to Regular Rings, HyperSurfaces, and Matrix Factorizations
September 19
In this talk we will discuss some results about projective resolutions of
modules over regular rings and hypersurfaces. We will be discussing in particular a
result by Eisenbud that projective resolutions over hypersurfaces are eventually periodic
of period 2. We will introduce matrix factorizations, a tool which will be used in future
talks to prove this result. This talk is part of a four part series on maximal Cohen-
Macaulay modules, matrix factorizations, and (time permitting) some generalizations to
gorenstein rings.
Ben Drabkin
Hyperplane Arrangements
September 12
Hyperplane arrangements are arrangements of n−1 dimensional subspaces in
n-space, which have numerous interesting algebraic properties. This talk aims to give a
brief overview on the algebraic objects associated to hyperplane arrangements and why
they are cool. The talk will include an overview of all aspects of algebraic geometry
needed, so no geometric background is required.
Taran Funk
Local Cohomology, A Love Story
September 5
Today we will use some of the stuff and junk we discussed last week to
explore (and in some cases prove) some results in local cohomology. I plan to have the
majority of the talk accessible to everyone, so please stop me at any point for
clarification!
Taran Funk
(Co)Homological Stuff and Junk
August 29
I will talk about basic (co)homological algebra facts and other commutative
algebra tools. Everything mentioned today is likely to be used in nearly every talk you
will see in CARS/ CAS. Getting a first exposure now will likely help make future talks
less mysterious, but still just as magical. Be sure to tune in next week for some more
stuff (and junk).
