This Summer CARS is organized by Shalom Echalaz Alvarez and Kesavan Mohana Sundaram
Andrew Soto Levins
Gorenstein Dimension, Auslander Bounds, and Vanishing of Tor
June 10, 12 and 14
The Auslander bound of a module can be thought of as a generalization of projective dimension. For a finitely generated module M with finite projective dimension over a Noetherian local ring R, $\ext_{R}^{n}(M,N)=0$ for $n>\pdim_{R}M$ and $\ext_{R}^{\pdim_{R}M}(M,N)\neq 0$. We say that the Auslander bound of M is finite if for all finitely generated modules N such that whenever $\ext_{R}^{n}(M,N)$ for $n\gg 0$, there exists an integer b that only depends on M so that $\ext_{R}^{n}(M,N)=0$ for $n>b$. The Auslander bound is the least such b. The goal of these talks is to give an introduction to Gorenstein dimension and Auslander bounds. In the first talk I will define Gorenstein dimension and prove R is Gorenstein if and only if every finitely generated module has finite Gorenstein dimension if and only if the residue field has Gorenstein dimension.
In the second talk I will show that, under certain assumptions, M has finite Gorenstein dimension if it has a finite Auslander bound. In the third talk I will show how having a finite Auslander bound is related to the vanishing of Tor. My main references are Gorenstein Dimensions by Lars Christensen, my paper A Study on Auslander Bounds, and William Sanders thesis Categorical and homological aspects of module theory over commutative rings."
Nawaj K C
Modules of finite projective dimension
June 17, 19 and 21
I will talk about the objects in the title. Specifically, I want to exposit several proofs of the intersection theorem of Peskine-Szpiro, whatever that is. I know very little about this, but my plan is to learn it with you all during the lectures. Also, I will mostly focus on the insights and ideas (not that I can handle the details) so my talks will be very informal, somewhat careless, but quite easy.
Shalom Echalaz Alvarez
F-singularities
June 24
During this talk the main goal is get to Fedder's criterion. We will cover the basics of prime characteristic, review a proof of Kunz's theorem, and if we have time, go over some singularities.
Kesavan Mohana Sundaram
Symbolic Powers
June 26
Recently I started learning about Symbolic Powers and I think it is interesting. In this talk I will define symbolic powers and go over the examples and the first few facts that I learned.
Ryan Watson
Support Varieties (In Commutative Algebra)
July 15, 17 and 19
In the early 1970s Quillen developed geometric techniques to study the cohomology of finite groups. Following this work, in 1989 Avramov introduced a theory of support varieties for modules over complete intersections which assigns a projective variety to every pair of finitely generated modules M and N. Together with Buchweitz in 2000, they used these support varieties to prove various theorems about the vanishing of Ext among other things. Thirty years later in 2019, Josh Pollitz expanded this idea and defined support varieties over any local ring, not just complete intersections. In doing so, he was able to give a classification of complete intersection rings by looking at the support variety of the ring and more generally in terms of the derived category of the ring. In this series of talks I will first go over the definition of support varieties over complete intersection rings and look at some of the results one can prove using them. Then I'll go over the necessary background on DG algebras and DG modules and define what a support variety is over any local ring. Lastly, I'll go over the classification of complete intersections in terms of the support variety of the ring, and time permitting, the classification in terms of the derived category.
Kesavan Mohana Sundaram
A Zariski-Nagata theorem for smooth Z-Algebras
July 22 and 24
In a Polynomial ring over a perfect field, the symbolic powers of a prime ideal can be described via differential operators: a classical result by Zariski and Nagata says that the n-th symbolic power of a given prime ideal consists of the elements that vanish upto order n on the corresponding variety. However, this description fails in mixed characteristic. In these talks, I use p-derivations, a notion due to Buium and Joyal, to define a new kind of differential powers in mixed characteristic and prove that this new object does coincide with the symbolic powers of prime ideals. Throughout the talk, I will be following the paper, "A Zariski-Nagata theorem for Smooth Z-Algebras" by Alessandro De Stefani, Eloísa Grifo and Jack Jeffries. One can also refer to the Chapter 4 in the Symbolic Power lecture notes by Eloísa Grifo.
Zach Nason
Constructing the Derived Category
July 29, 31 and August 2
In homological algebra, the category of chain complexes of modules over a ring (or of abelian categories in general) is a (relatively) intuitive place to work in. However, true isomorphisms between complexes are rare in the category of chain complexes - we generally have to settle for homotopy equivalences and quasi-isomorphisms between complexes. In my talks, I'll go over how to force homotopy equivalences and quasi-isomorphisms to become true isomorphisms by first constructing the homotopy category and then the derived category. Since these two categories are generally not abelian, we'll go over the definition of a triangulated category as a replacement and will show that the homotopy category is triangulated (and will sketch out how the derived category is triangulated). If I have time, I'll go over the construction of derived functors in the derived category and will talk about forming the derived category of differential graded modules over a differential graded algebra.
Ana Podariu
Lefschetz Properties Part 2, Electric Boogaloo
August 7
We'll start with a review of the definitions of the Weak Lefschetz Property (WLP), Strong Lefschertz Property (SLP), and Strong Lefschetz property in the narrow sense (SLPn). We'll introduce representations of sl_2 in order to exposit a bit on interactions between tensor products and the Lefschetz properties.
Stephen Stern
What is a Scheme?
August 9
In the 1960’s Grothendieck published a treatise that reworked the theory of varieties in an encompassing theory of schemes. Algebraically speaking this replaces "finitely generated k-algebra” with "commutative ring”. In the same decade he defined local cohomology on schemes and the derived category. A necessary step to defining a scheme is defining a sheaf, which appeared in the 50’s. I will define sheaf and the stalk of a sheaf and construct an example where the notion of “geometry” is familiar, and hopefully define a scheme.
Danny Anderson
Gröbner Bases & Implicitization
August 14
Hate (t², t³)? Love x³ - y² = 0? Today, I'll provide a quick crash course into some Algebraic Geometry, where we'll throw those ugly parametrizations in the trash! We'll learn about the Multivariable Division Algorithm, Gröbner bases, varieties, and parametrizations! Disclaimer: We cannot solve parametrizations that have square roots, e^x, or trig functions. Also, parametrizations are not trash. They deserve love and kindness, just like the rest of us. The above was a joke and does not reflect the author's internalized trauma from Algebra II.
