Talks

• 11-12-2020 - Kailey Perry (UArk) - Hensel's Lemma and Henselian Rings

Abstract: In this talk, we will introduce Henselian rings and Hensel's lemma. We will compute examples of Henselian rings and discuss completion as well as some important consequences of Hensel's lemma.

• 11-19-2020 - Nicholas Cox-Steib (MU) - A Noetherian Ring of Infinite Krull Dimension

Abstract: The purpose of this talk is to work through Nagata’s example of a Noetherian ring with Infinite Krull dimension. This talk is intended to be somewhat self-contained, so we begin by reviewing the Noetherian finiteness condition and the theory of Krull dimension.

• 12-03-2020 - Trevor Arrigoni (KU) - Regularity and Differential Operators

Abstract: In general, we can think about regular local rings as being smooth in some sense. In most introductory courses, this isn't fleshed out much. In this talk, we will flesh out this analogy in the case of rings coming from classical algebraic geometry, which will lead into a discussion of differential operators.

• 12-10-2020 - David Liberman (UNL) - Local Cohomology - The Basics

Abstract: In 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 properties and useful applications of local cohomology. In particular we will discuss some basic properties, vanishing properties, and some applications to Cohen-Macaulay and Gorenstein rings.

• 02-11-2021 - Alexander Duncan (UArk) - Hilbert Polynomials

Abstract: Studying the Hilbert polynomial for a finitely generated graded module over the ring S= k[x_0,…, x_n] is a classical problem in both commutative algebra and algebraic geometry. Most of the content of the Hilbert polynomial can be translated into tables called Betti diagrams which only depend on the given module. In this talk, we will discuss how free resolutions give rise to Hilbert polynomials and then discuss applications and techniques for calculating these polynomials.

• 02-18-2021 - Andrew Soto Levins (UNL) - The Injective Hull

Abstract: Do you think that injective modules are a mystery? When I was first learning about projective and injective modules, I thought "wow projective modules are a lot nicer than injective modules". The main reason for thinking this was that it is easy to write down a projective module (since every free module is projective), but difficult to write down an injective module. Recall that every module is contained in an injective module. A natural question to ask is: does there exist a "smallest" injective module containing a given module? The answer is yes and it is unique. It is called the injective hull. Over a Noetherian ring, the injective hull can be used to prove results about injective modules. In this talk I will define the injective hull and discuss some really cool results about injective modules over a Noetherian ring. It turns out that injective modules are nice.

• 02-25-2021 - Dylan Beck (KU) - Numerical Semigroup Rings

Abstract: Even though they are simple objects to describe, numerical semigroup rings provide an interesting class of one-dimensional local Cohen-Macaulay rings. In this talk, we will define numerical semigroups and their corresponding numerical semigroup rings. Ultimately, we will compute the multiplicity of a numerical semigroup (ring), and we will discuss when a numerical semigroup (ring) is Gorenstein or a complete intersection.

• 03-04-2021 - Will Allbritian (MU) - Cellular Resolutions of Monomial Modules

Abstract: A common pursuit in commutative algebra is to find a (minimal) free resolution of a monomial ideal (or more generally a monomial module) in a polynomial ring over a field. In 1997 Bayer and Sturmfels defined a natural construction of such a free resolution by associating the module to a cell complex whose vertices correspond to the generators of the module. In this talk we will give a brief overview of this construction, called a cellular resolution, along with a few significant examples: the Taylor resolution, the hull resolution, and (time permitting) the Eliahou-Kervaire resolution of Borel-fixed ideals.

• 03-11-2021 - Eric Walker (UArk) - Jet and arc spaces from a commutative algebra point of view

Abstract: (Subtitled: Everything will be affine until it's not.) In this talk, we'll explore the definition and compute some examples of jets and arcs of k-algebras. Doing so motivates several commutative algebra/algebraic geometry tools, like representability and gluing schemes via affine covers. The majority of the talk will be through a commutative algebra lens, but time permitting at the end we'll discuss the topic from the geometric point of view.

• 03-18-2021 - Dillon Lisk (MU) - Zero Divisors, (Weakly) Associated Primes and Depth

Abstract: In this talk we will examine the theory of associated primes--particularly as it relates to proving Rees' characterization of depth in terms of Ext functors--and discuss how much of the theory goes through in the non-Noetherian case.

• 03-25-2021 - Michael DeBellevue (UNL) - Computing Ext 3 Different Ways

Abstract: A common theme in algebra is to first try to understand simple objects, and then understand how they fit together to form more complicated ones. More precisely, modules M and N "fit together" if there is an exact sequence between them. In this talk, I'll introduce the Ext functor as equivalence classes of exact sequences. I will then discuss how Ext is typically computed in practice, using projective resolutions. Finally, specializing to the case of Ext(k,k) for k the residue field of a local ring, I will discuss how Ext(k,k) is generated by a certain Lie subalgebra known as the homotopy lie algebra.

• 04-08-2021 - Navaneeth Chenicheri Chathoth (MU) - Introduction to Valuation Rings

Abstract: Although not usually Noetherian, valuation rings have several other interesting properties which make them a useful class of rings to consider while working in non-Noetherian setting. In this talk we will define the concept of a valuation on a field and discuss some interesting properties of valuation rings! Only a basic knowledge of graduate algebra is assumed for this talk.

• 04-15-2021 - Alexander Duncan (Uark), David Smith (Uark), Eric Walker (Uark), and Michael DeBellevue (Nebraska) - Macaulay2 Panel

• 04-22-2021 - Souvik Dey (KU) - Hilbert function associated to Coherent functors

Abstract: The starting point of the theory of classical Hilbert functions and Hilbert polynomials is the fact that given a module M and an ideal I, if M/IM has finite length, then the lengths of the modules M/I^nM agrees with a polynomial for all large n. If one thinks of M/I^nM as the tensor product of M and R/I^n, then one is naturally led to ask: What happens if we replace this tensor product by some other functors? Like Hom or Ext s and Tor s in general, or even some composition of these functors? Some results on the polynomial behaviour of the length of some of these functors were proved by V. Kodiyalam and E. Theodorescu (independently) by different methods. Recently, it has been shown by A. Banda and L. Melkersson that all such results fall under the common umbrella of so called "Coherent functors". In this talk, we will define the notion of Coherent functors, give examples and general recipes for constructing such functors and prove the polynomial behaviour of Hilbert functions associated to such functors. Time permitting, we will briefly venture into what can be said about the degree and leading coefficient of the corresponding polynomials in some special cases.

• 04-29-2021 - Justin Lyle (UArk) and Kyle Maddox (KU) - Discussion on Applying for Jobs in Academia

• 09-15-2021 - Alexander Duncan (UArk) - Why Koszul Complexes are the First Homological Technique One Should Learn

Abstract: When asked which homology theory is the easiest to understand, most mathematicians will likely say homology of spaces as this theory is usually the first homological theory one encounters in their career. I aim to change that. In this talk, I will motivate the desire to study the Koszul complex by introducing depth (a topic we will likely see in future talks) and then show why Koszul cohomology is the most natural homological technique to learn *first*. Expect many examples and a few applications in the form of exercises form Eisenbud’s classic “Commutative Algebra with a View Towards Algebraic Geometry”. The beginning of the talk will review the fundamentals of homological algebra.

• 10-12-2021 - Eric Walker (UArk) - The Module of Kähler Differentials

Abstract: (Subtitled: Calculus 1, but spicy) While the derivative of a function at a point is defined via the limit of its difference quotient, calculus students greatly appreciate all the differentiation rules which allow you to compute derivatives easily, like the sum, power, product rules etc. If we instead start with these formal rules as our foundation, how can we generalize them algebraically? What information do we keep on the geometric side? In this talk, we'll define a formal notion of a derivation of a k-algebra, build a universal module representing all such derivations, and play with some examples, hopefully also discovering some geometry on finite type k-algebras along the way.

• 10-19-2021 - John Portin (KU) - Random Monomial Ideals

Abstract: Probabilistic methods have long been used in graph theory and combinatorics to study expected behavior. Recently, these methods have become popular in the study of monomial ideals. In this talk we will survey some models and results from the theory of random monomial ideals. We will see how these models have been used to make broad claims about basic properties of ideals such as Krull dimension, Betti numbers, projective dimension, and the Cohen-Macaulay property.

• 10-26-2021 - Patrick Lank (USC) - Regularity in prime characteristic via Kunz

Abstract: Within this talk, a characterization of regularity in a Noetherian ring of prime characteristic is exhibited. We will walk through a proof by a well-known theorem of Kunz that states such a ring is regular if, and only if, the Frobenius endomorphism is flat. Since this initial result, there have been other equivalent characterizations of regularity, but our primary focus will be the classical proof, and there will be only a brisk mention of these recent ideas. This result has motivated a wealth of work regarding the homological properties of the Frobenius endomorphism, but the requirement of Noetherianity cannot be dropped, and this will be also explored in some detail.

• 11-09-2021 - Alessandra Costantini (OSU), Justin Lyle (UArk), and Kyle Maddox (KU) - Panel discussion on writing good talks

• 11-16-2021 - Srishti Singh (MU) - Numerical Semigroups and Wilf’s Classification

Abstract: In this talk the concept of numerical semigroups is introduced. The simplicity of this concept makes it possible to state problems that are easy to understand but far from being trivial. After looking at an application to algebraic geometry, we discuss a crucial question of Wilf and construct an important class of numerical semigroups that give a positive answer to it.

• 11-30-2021 - Shah Roshan Zamir (UNL) - Hilbert's Fourteenth Problem and Introductory Geometric Invariant Theory

Abstract: In 1900, Hilbert published a list of 23 problems, later increased to 24, some of which remain unsolved. His 14th problem asked: Is the ring of invariants of an algebraic group acting on a polynomial ring always finitely generated? Although the first counterexample was constructed by M. Nagata in 1959, the efforts in answering it and its generalized versions generated a wealth of research. I intend to state (non)constructive proofs and examples of a positive answer to this problem e.g. Gordon-Hilbert's theorem. I will further translate this question in the category of algebraic varieties and how to interpret a positive answer in that category. My talk is "mainly" based on Chapters 3 and 4 of Igor Dolgachev's "Lectures in Invariant Theory". The talk is accessible to anyone with basic familiarity of algebraic concepts such as group actions, algebras, varieties etc.

• 02-04-2022 - Trevor Arrigoni (KU) - F-Regularity - A quick way to show Cohen-Macaulayness
  Abstract: Recall that a local ring is Cohen-Macaulay if every system of parameters is a regular sequence. One can transcribe the properties of being a regular sequence in terms of a containment of certain colon ideals. In general, this is a hard property to show for an arbitrary local ring. However, if we place some mild hypotheses on our ring, one being positive characteristic, we can uncover that Cohen-Macaulayness is intimately related to the condition that ideals be tightly closed. In the late 80's and early 90's, this observation led Hochster and Huneke to define and develop the theory of F-Regularity. In this talk, we will define F-Regularity, show some basic results, and give an intuitive way of thinking about this property.

• 02-11-2022 - Michael DeBellevue (UNL) - Using Divided Powers to Fix Problems with Derivations in Positive Characteristic
  Abstract: Derivations are maps of algebras satisfying the familiar Leibnitz rule from calculus. They are used in the study of algebraic varieties as a means of defining tangent spaces. In positive characteristic it can be shown that derivatives of prime powers of elements are zero. This issue can be resolved using a construction called divided powers. In this talk we’ll introduce divided powers, discuss some of their properties, and ultimately show that the divided power algebra is the dual of the polynomial ring.

• 02-18-2022 - Seungsu Lee (Utah) - Strong F-regularity and F-split rings
  Abstract: In characteristic p commutative algebra, Kunz's theorem plays a crucial role in determining rings' regularity. When the ring is a Noetherian local ring over an F-finite field, the theorem suggests that the extent of R^1/p being close to free measures a singularity of the ring. With this point of view, we will define F-split and strongly F- regular rings in this talk and see several examples of F-split/F-regular rings. In particular, we will discuss Fedder's criterion that determines when the ring becomes F-split.

• 02-25-2022 - Dillon Lisk (Mizzou) - A Hopefully Not Too Scary Look at Derived Categories
  Abstract: In this talk we will introduce the idea of the derived category of an abelian category, discussing both why the structure of these categories can be difficult to understand in general and why the familiar case of the category of R-modules is much more tractable. If time permits, we will explain how derived functors fit into this setting. Here is an example of two non-quasi-isomorphic chain complexes which have isomorphic homology.

• 03-04-2022 - Sudipta Das (NMSU) - Tight Closure and F-Singularities
  Abstract: In this talk, we will introduce the idea and a short history of tight closure theory with basic properties and a few examples. If time permits, we will discuss classical results of singularities of algebraic varieties over a field of characteristic 0, for example, Rational Singularities are Cohen Macaulay: Boutot's Theorem, and Brianson Skoda theorem, and then show analogous results in positive characteristics using the theory of tight closure.

• 03-11-2022 - Monalisa Dutta (KU) - An Introduction to Gorenstein Dimension
  Abstract: In this talk, we will introduce the notion of totally Reflexive Modules and classical Gorenstein dimension (also called G-dimension) for finitely generated modules, along the lines of Auslander and Bridger. We will describe that this G-dimension is a refinement of projective dimension, and explain the philosophy that what projective dimension is to regular local rings, G-dimension is to Gorenstein local rings. Unlike projective dimension however, G-dimension cannot be detected by Ext or Tor vanishing, and to get a "functorial criteria" for G-dimension, we will show how one needs to upgrade to complexes. Time permitting, we will also discuss how for characteristic p F-finite local rings, Gorenstein ness can be detected by finite G-dimension of the Frobenius.

• 03-25-2022 - Annie Giokas (Utah) - The Cohen Structure Theorem
  Abstract: The goal of this talk is to explain The Cohen Structure Theorem addresses the existence of the coefficient field of a local complete ring. It guarantees that if a local complete ring R contains a field then it contains an exact copy of the residue field of R. This property is important because using this fact, we are able to show that every local complete ring that contains a field is a homomorphic image of a formal power series ring over a field.

• 04-01-2022 - Trung Chau (Utah) - D-modules and an application
  Abstract: Local cohomology modules are important objects to study in commutative algebra. However, most basic commutative algebra results (like Nakayama's lemma) only work with finitely generated modules, and local cohomology modules are not. However, they are finitely generated when considered as modules over the ring of differential operators, or D-modules. In this talk we will go through the very basics of the theory of D-modules and as an application, prove that local cohomology modules have finitely many associated primes.

• 04-15-2022 - Patrick Lank (USC) - Dimension and generation for derived categories
  Abstract: Within this talk, I will discuss a method for constructing objects within the derived category of complexes of finitely generated modules from a single object, called a classical generator, and three operations (i.e. cones, shifts, and direct summands). This gives one a meaningful understanding to the structure of the bounded derived category by isolating an object and studying it thru these operations. From this, we can determine whether or not there is a uniform bound on the number of steps required to build any object from a given classical generator. In the case that there is a finite number, we call such objects strong generators. This introduces the question, what is the smallest number of steps required to build an object? We answer these questions and explore further notions of dimension which have been introduced for derived categories.

• 04-22-2022 - Panel discussion on getting started with research in commutative algebra

• 04-29-2022 - Shah Zamir (UNL) - Nagata's Counterexample to Hilbert's 14th Problem
  Abstract: In 1900 David Hilbert published a list of 23 open problems. The 14th problem concerns the finite generation of the ring of invariants of a group acting on a finitely generated K-algebra. In 1959 Nagata provided a negative answer to Hilbert's 14th problem. In this talk, I intend to sketch a proof of this counter example. Our proof will follow a simplified version by R. Steinberg. I only assume basic knowledge of algebraic objects such as polynomial rings, algebraic varieties, and groups.

• 09-26-2022 - Dillon Lisk (MU) - The Universal Property of Derived Functors
  Abstract: We study the ramification theory of local rings in the context of a finite dimensional (f.d.) Galois extension. More explicitly, if (R,m,k) is a normal local domain, k* is a f.d. Galois extension of k, then we will study the splitting group of a given local ring lying over R. It turns out that if you intersect a local ring R* lying over R with the splitting field of R* over R, calling this ring S, then S is unramified over R.

• 10-03-2022 - Stephen Landsittel (MU) - The Galois Theory of Local Rings
  Abstract: In this talk we will characterize derived functors by a universal property involving derived categories, focusing on the case of the Tor functors. Prerequisites regarding derived categories are minimal: if you understand how the Tor functors are constructed, that should be sufficient.

• 10-17-2022 - Alexander Duncan (UofA) - Boij Soderberg Theory
  Abstract: The graded Betti diagram of a graded module M is a table of generators of a graded free resolution of M. For certain modules it is possible to decompose their Betti diagrams into linear Q-combinations of pure diagrams which are “as simple as possible”. The algorithm is known as the Boij Soderberg decomposition. We will be going over the main results and examples while filling in details from Gunner Floystad's renowned introduction of the theory.