CARS
Commutative Algebra Reading Seminar
This semester, CARS was organized by Jake Kettinger and David Lieberman
Nawaj KC
The Total Rank Conjecture
December 8Suppose (R,m,k) is a local ring and M is a finite length R-module of finite projective dimension. Set b_i(M) = dim_k(Tor_i(M,k)). It was conjectured several decades ago that b_i(M) is bounded below by d choose i where d = dim (R). This easily stated conjecture is wide open even for regular local rings! A slightly weaker form of this conjecture, referred as the Total Rank Conjecture, was settled in 2017 by Walker employing K-theoretic methods. In this talk, I will sketch this proof in broad strokes. This talk is somewhat technical and I will assume that we know what Tor is, for example.
Taylor Murray
An Introduction to DG Algebras and DG Modules
December 1In this talk, we will introduce DG Algebras and DG Modules. In some sense a DG Algebra is a chain complex that has a ring structure, where multiplication "plays nicely" with the differential of the complex and the homological degrees of elements. We will define what a DG Algebra is by carefully describing what it means for multiplication to "play nicely". Next, we will discuss DG Modules. After looking at some examples of DG Algebras and DG Modules, we will state a nice lifting property for a particular class of DG Modules. If time permits, we will also discuss a few corollaries of the lifting property; one of which is the famous Künneth Theorem from Algebraic Topology. Familiarity with chain complexes, chain maps, homotopy, and homology is helpful but certainly not required to gain something from the talk.
Jake Kettinger
Geometry of Elliptic Fibrations Part 2
November 17First we will look at the group law on elliptic curves! Then we will see something I like to call the Fundamental Theorem of Elliptic Curves. These tools will help us study elliptic fibrations of higher index. We will go on a grand tour of examples of different elliptic surfaces!
The above is an example of the Bertini involution of the plane, which uses the group law of elliptic curves. Under the involution, the point P maps to the point Q. It begins by taking an elliptic fibration with nine base points and choosing one of the base points as the identity element under the group law for all of the curves in the fibration. This point is marked as O.
Then when P is not one of the nine base points, there is a unique elliptic curve that contains all nine base points and P. Then P is mapped to -P under a group homomorphism on the curve (which is why we have an involution: if P goes to -P, then -P goes to -(-P)=P).
To show P+Q=O, we can connect P and Q with the green line, and label the third point where the green line meets the curve as R. Then to find P+Q, we connect R to the identity O with the orange line. This new line is tangent to the curve at O, and so the "third" point where the orange line meets the purple curve is O itself. This shows that P+Q=O, and so we truly have Q=-P.
When P is O, the image of P is P. When P is one of the eight base points that are not O, the image of P is not defined. The Bertini involution is a rational transformation of the plane, and is just defined on a dense open subset of the plane. To have it be defined everywhere, you would have to blow up the eight points where it not defined.
David Lieberman
Local Cohomology Go Vroooooom!
November 10In this talk we will investigate the basics of local cohomology. Local cohomology is a powerful tool that is useful for studying the structure of rings and modules. After setting the scene and going over the basic definitions, we will discuss some examples, theorems, and properties of local cohomology, and identify some applications of the subject. In particular vanishing properties and some applications to Cohen-Macaulay and Gorenstein rings.
David Lieberman
The Ring of Differential Operators: An Amuse-Bouche
November 3In this talk, we give an introduction to differential operators. We will discuss what they are, how to find them, and what they can be used for. The rings of differential operators have wide applications in commutative algebra via D-module theory. Some applications include local cohomology, algebraic geometry, and invariant theory. The goal of this talk is that the audience gains an understanding what the ring of differential operators is, and walks away with just a taste of this (honestly pretty wacky) algebraic setting.
Michael DeBellevue and Dakota White
Hopf Algebra Structure of Tor and Ext Part II
October 27We'll continue discussing the Hopf algebra structure of Tor(k,k) and Ext(k,k). In this talk, we'll introduce Tor(k,k), define the operations on it necessary for it to have a bialgebra structure, discuss why its dual is Ext(k,k), and that the comultiplication on Tor(k,k) dualizes to the Yoneda product on Ext(k,k).
Michael DeBellevue and Dakota White
The Hopf Algebra Structure of Tor and Ext, Part I
October 20Ext_R(k,k) and Tor_R(k,k) enjoy a great deal of structure which controls their behavior and makes them easier to understand. In particular, they are dual to each other, in a manner which extends classical hom-tensor adjointness. In this talk we'll introduce tor and ext, and the notion of a Hopf algebra, which is the structure which precisely captures the duality of Tor and Ext. Hopf-ully we'll see you all there!
Jake Kettinger
The Geometry of Elliptic Fibrations
October 13In algebraic geometry, we study surfaces over algebraically closed fields, such as the complex numbers. When looked at as manifolds over the real numbers, these are 4-manifolds! We take a tour through many foundational results of projective curves, such as Bézout's Lemma and the Cayley-Bacharach Theorem. We use these in conjunction with the tool of blowing up a space to develop the study of families of elliptic curves called Halphen surfaces and the classification of elliptic fibrations associated to these surfaces.
The above is an example of an elliptic fibration with the nine base points (-1,1), (0,1), (1,1), (-1,0), (0,0), (1,0), (-1,-1), (0,-1), and (1,-1). You can click and drag the orange point to show the unique cubic curve that contains all nine base point and the orange point.
Juliann Geraci
Construction of Free Resolutions Through Simplicial Complexes
October 6From a simplicial complex Δ we can build a chain complex C(Δ) which gives an algebraic encoding of information about Δ. We will recognize a fundamental connection between simplicial complexes and commutative algebra through C(Δ), which enables us to understand a result of Bayer, Peeva, and Sturmfels that gives an effective way to describe some resolutions in terms of labeled simplicial complexes.
Shahriyar Roshan Zamir
Geometric Invariant Theory, Part 2
September 22In the second installment of GIT we will enjoy the fruit of our labor from last week. Turns out the algebra Pol(Mat_{rxm})^{SL_r(k)}, which was generated by bracket functions, is the coordinate ring of the Grassmannian variety; a fact I intend to prove today. However, we need to do some work to understand and be able to think about the Grassmannian variety in the projective space. Doing so involves an important idea: given a set of objects how can we, in some form, think of them as points in a desired space. I'd like to point out last week's talk is not a prerequisite for this talk and we will only use the main result from last week.
Shahriyar Roshan Zamir
Geometric Invariant Theory, Part 1
September 15Geometric Invariant Theory (GIT) grew out of the following question: suppose S represents the set of regular functions between two varieties X and Y, each of which endowed with an action of a linear algebraic group G. This induces a G action on S, and we are interested in what can be said about the invariant elements of S i.e. is it a finitely generated k-Algebra. Like almost everything else in modern algebra, a partial solution was given by Hilbert. In this talk, I hope to give you an idea of classical (19th Century) GIT by proving what's called the first fundamental theorem of invariant theory. I will give, with proof, an explicit description of the algebra Pol(Mat_{r \times m})^SL_r(k) (whatever that means.) In my subsequent talk next week, I will show you why we care about this invariant algebra. I will prove, next week, that the obscure looking set above is the coordinate ring of the Grassmannian Variety, which we will construct from scratch. Finally, I'd like to point out that GIT also stands for the video game Grand Inner-city Theft, which is GTA's (Grand Theft Auto) less commercially successful cousin. (You can look it up.)
Matthew Bachmann
Nice Properties, Nice Modules, and other Nice Things
September 8For a commutative Noetherian ring R, the Auslander-Reiten Conjecture (ARC) asks if the vanishing of Ext(M,M) and Ext(M,R) in positive degrees implies M is a free module. Though simple in appearance, not much progress has been made on ARC - it is not even known in the Gorenstein local case. However, the conjecture has inspired a large body of work that can be summarized as follows,
if Hom(M,N) is nice and Ext(M,N)=0 in certain degrees, M and N are also nice. In this talk I will try to convince you that vanishing of certain Ext modules - a homological computation - can provide us with concrete geometric information about when a module is nice. If you participated in the summer course over Bruns and Herzog, this discussion may look pretty familiar. I will also give a brief description of where the Auslander-Reiten Conjecture stands currently and what tools are being used to approach the problem.
Nawaj KC
A Sales Pitch for CARS
September 1This talk is an invitation to the uninitiated to our favorite and most beautiful field. Our aim is to give some flavor of pre-modern commutative algebra. We pursue the following theme: algebra forces geometry to happen the way it does. The talk is mostly examples and therefore only a rudimentary familiarity with linear algebra is assumed. For the experts, we will study the solution set of two varieties Z(x+y+z) and Z(xyz) in affine 3-space with some of our deep, remarkable results: Krull's Height Theorem, the Jacobian criterion, and the equidimensionality of Cohen Macaulay rings.