CARS

Peder Thompson

Cosupport: a Homological Dual to Support.

April 5

You probably know what the support of a finitely generated module is (i.e., the set of primes such that localization at those primes doesn't cause the module to vanish). But did you know you can also use the minimal injective resolution to detect the support of finitely generated modules by just looking at which primes "show up" in it? Ok, maybe you did, maybe you didn't - but either way, my goal is to carefully define a homological dual notion to this version of support: namely, cosupport. We will show that another type of resolution can be used to identify primes in the cosupport. Finally, we'll find a class of rings where the support coincides with the cosupport (and show that this doesn't always happen).

Seth Lindokken

hat Can Finite CM Type Do For You?

March 29

When studying the properties of a Cohen-Macaulay local ring, one can learn a lot from understanding Maximal Cohen-Macaulay (MCM) modules over the ring. In particular, if a ring has only finitely many MCM modules (finite CM type), then there are strong restrictions on its structure. In this talk we will explore two results in this direction. The first, from Auslander, says that Cohen-Macaulay rings of finite CM type are isolated singularities. The second, from Herzog, says that Gorenstein rings of finite CM type are hypersurface rings. We will close with a brief description of the rings that arise in the context of Herzog's result.

Chris Evans

Non-Noetherian Cohen-Macaulay Rings

March 15

Have you ever wondered what Cohen-Macaulay should mean for non-Noetherian rings? Apparently Graz did, and Hamilton/Marley decided to answer the question. More specifically, they gave a definition of Cohen-Macaulay that ensures that all (not necessarily Noetherian) regular rings are Cohen-Macaulay. There are some obvious difficulties to deal with, and the Cohen-Macaulay property isn't quite so well-behaved in the non-Noetherian setting. We shall delve into some of these issues.

Josh Pollitz

Support Varieties and Symmetric Complexity

March 1

We will define the complexity of a pair of finite modules over a noetherian local ring. We will see that complete intersections have "symmetric complexity" which will give us some interesting homological information and vanishing results for Ext (over complete intersections). To do this, we associate to each pair of finite R-modules its support variety which is a cone in affine c-space, where R is a complete intersection of codimension c. The dimension of this cone is exactly the complexity of the pair of modules and we will see that the cone has the symmetry property that we are after. My main reference is Support Varieties and Cohomology over Complete Intersections-Avramov and Buchweitz (2000).

Matthew Mills

An Introduction to Quiver Representations

February 23

A quiver is simply a directed graph. A representation of a quiver associates a vector space to each vertex and a linear map to each arrow. The study of quiver representations gives a nice way to visualize the modules over the path algebra associated to the quiver. The power of this approach is that quiver representations can be used to study any finite-dimensional algebra. In this talk we discuss the basic properties of quiver representations and the construction of the Auslander-Reiten quiver. We will then show how to use the Auslander-Reiten quiver to easily compute short exact sequences and the dimensions of the Hom and Ext spaces.

Neil Steinburg

The Auslander Reiten Condition

February 16

You've heard of Nakayama's lemma. You know it. You love it. So, why not learn more from the great Nakayama himself! Well, now you can with Nakayama's conjecture! Think of all the fun and excitement of Nakayama's lemma and triple it! With Nakayama's conjecture, you are on your way to proving some of the greatest conjectures in algebra, including the powerful Auslander Reiten conjecture. But, who wants to just focus on a bunch of conjectures that may or may not be true? Well, you're in luck, because we will not only look at this A-R conjecture, but also at the conditions of it (the Auslander-Reiten Condition) that can be proven to hold for a wide class of commutative rings outside the scope of the original conjecture. Want to know what happens when Ext_Rⁱ(M,M ⊕ R) = 0 for all positive i? Come to the talk and find out!

Andrew Windle

An Introduction to A-infinity Algebras

January 26

A-infinity algebras were invented in the sixties as a way to solve problems in topology, including descriptions of singular chain complexes of loop spaces. Since then A-infinity structures have been used to solve problems in commutative algebra, homological algebra, mirror symmetry, and mathematical physics. In this talk, we will spend time describing the motivation behind originally studying A-infinity algebras, provide a definition and a few examples, and describe some of the benefits of working with AM-infinity algebras compared to associative algebra and DG algebra structures.

Jason Lutz

Fractals & Mysterious Triangles

Jason Lutz

Using playing cards, we'll construct mysterious triangles and explore the hidden patterns in the apparent randomness of these triangles. (This is a practice talk for an on-campus interview. If you are able, please bring a deck of playing cards.)