CARS

Matthew Bachmann

Yoneda Extensions Part 2: Applications

April 24

Today's talk will be about Auslander's celebrated theorem: If R is a complete CM ring and has finite CM type, then R has an isolated singularity. A key part of Auslander's proof is to show Ext¹(M,N) has finite length for arbitrary MCM modules. Huneke and Leuschke proved that Ext¹(M,N) has finite length under weaker assumptions using ideas from the correspondence we looked at in Part 1. Our goal for today will be to understand the proof from Huneke and Leushke and see how it can be used to generalize Aulander's result.

Nicholas Packauskas

PERFECTOIDBOIS III: THE FINAL CHAPTER

April 17

We close our (limited) expedition into perfectoid notions by describing how the techniques have influenced commutative algebra. In particular, we will give a road map to how the direct summand conjecture was finally proved in the remaining case of mixed characteristic by Yves André in 2016.

Frank Zimmitti

Perfectoid Fields

April 10

We introduce some basic notions in nonarchimedean analysis in order to define a perfectoid field, a basic object in Peter Scholze's theory of perfectoid spaces. After giving a few examples, we will describe the "tilting" functor, which to any perfectoid field, $K$, associates a perfectoid field, K^#, of characteristic p>0. Time permitting, we will discuss some basic ideas from John Tate's theory of rigid analytic geometry, which are necessary to discuss perfectoid K-algebras and higher dimensional perfectoid spaces.

Michael DeBellevue

Perfectoids I: Tasting the Ingredients

April 3

We begin the first of three talks on perfectoids and their role in proving the direct summand conjecture in mixed characteristic. In this talk I will begin with some remarks on the differences between mixed and equicharacteristic. I will then give a broad overview of the three major ingredients involved: Field theory, topology, and "almost mathematics". More specifically, I will prove the direct summand conjecture in equicharacteristic zero, introduce étale morphisms, and the category of almost modules over a non-noetherian ring.

Matthew Bachmann

Yoneda Extensions

March 27

Elements of Ext¹(A,B) can be viewed as extensions of B by A in a rather mysterious way. In my talk, we will unpack this correspondence (following Rotman). Time permitting, we will look at how the correspondence gives {equivalence classes of extensions} a group structure that makes it isomorphic to Ext¹(A,B).

Su Ji Hong

An example of Kac-Moody algebra and root system

March 13

I am reading through a book about Kac-Moody algebra. I will give basic definitions to define Kac-Moody algebra and show an example to make sense of the definitions. I am still confused about lots of parts so this will be an adventure for all of us.

Taran Funk

Detecting Flatness in Characteristic p>0

March 6

We will be discussing how use a ring, viewed as module over itself via the Frobenius map, to test for flatness of a (not necessarily finitely generated) module. This is some recent work by myself and Tom Marley.

Eric Hopkins

Complete Intersections

February 27

Complete Intersections are a class of Cohen-Macaulay local rings with a definition that looks bizarre. I will present this bizarre definition, then characterize them using properties of the ring and Koszul complexes. I will remind you of the relevant information from my previous talk on Koszul complexes.

Erica Musgrave

Graded Koszul Rings

February 13

Let S be a polynomial ring over k and R = S/I for some graded ideal I. It is well known that the projective dimension of k over S is finite. In addition, the minimal free resolution of k over R is linear. One could ask are there any other rings R that the minimal free resolution of k has these nice properties? If R is not a polynomial ring then k always has infinite projective dimension. However, it is possible for the minimal free resolution of k to be linear and such rings are called graded Koszul rings. In this talk, I will go over the definition of Koszul rings along with some examples and results about Koszul rings.

Andrew Connor

Asymptotic Resurgence via Integral Closure

February 6

In 2009, our very own Brian Harbourne introduced an invariant of symbolic powers called the resurgence. Like most such invariants, it's extremely hard to compute in general. This talks relates some recent results from DiPasquale, Francisco, Mermin, and Schweig (arXiv: 1808.01547) that establish how to feasibly compute the asymptotic version of resurgence. In light of the computational nature, I'll be skipping all the proofs and getting right to the calculations.

Josh Pollitz

Homotopical Commutative Algebra

January 30

Historically, the representation theory of a commutative noetherian ring is the study of its category of finitely generated modules, indecomposable modules, MCM modules, etc. Gaining structural insight on these categories, in general, can prove to be a challenging/impossible endeavor. More recently, inspiration from homotopy theory has led some commutative algebraists to study, loosely speaking, "homotopical representation theory". In this talk, I will cover the fundamental result in this direction, namely, the following theorem of Hopkins and Neeman: "There is a lattice isomorphism between the specialization closed subsets of Spec(R) and the thick subcategories of perfect complexes."

Amadeus Martin

Buchweitz Theorem - La Segunda Parte

January 23

Today we finish proving a theorem of Buchweitz, establishing an equivalence of the stable MCM module category and the singularity category over a Gorenstein ring. Last week we showed that MCM(S) stable is equivalent to the category of unbounded acyclic complexes of projectives. We will utilize this result to create the necessary functors to settle the claim of the theorem.

Amadeus Martin

Buchweitz Theorem - Part I

January 16

We will be proving a theorem of Buchweitz, establishing an equivalence of the stable MCM module category and the singularity category over a Gorenstein ring. This will be done in two parts. This week we introduce a third category of unbounded acyclic complexes of projectives, and show its equivalence to MCM stable.