Algebra Seminar at Georgia State University

Spring 2024, Tuesdays 4pm--5pm, Room 1441 25 Park Place

January 16, Florian Enescu, Georgia State
The Combinatorial Nullstellensatz and Erdös-Ginzburg-Zvi Theorem.

Abstract I will discuss an approach to the famous Erdös-Ginzburg-Zvi theorem based on Alon Nullstellensatz.

January 30, Thomas Polstra, University of Alabama
On the uniform Izumi-Rees theorem.

Abstract. Let $R$ be a commutative Noetherian normal domain, $K$, the field of fractions of $R$, and $I\subseteq R$ an ideal. The Rees valuations of $I$ correspond to valuation rings $V\subseteq K$, containing $R$, with the property that an element $f$ belongs to the integral closure of $I$ if and only if the element $f$ belongs to ideal generated by $I$ in each of the valuation rings $V$. The Izumi-Rees Theorem is a striking generalization of Zariski’s Main Theorem and informs that if $V_1$ and $V_2$ are Rees valuation rings of an ideal $I$ with common center in $R$, then the power of the maximal ideal of $V_1$ and $V_2$ for which an element of $R$ can belong to are of linear bounded difference. The linear bound is dependent upon the ideal $I$. We introduce a Uniform Izumi-Rees Property which provides a uniform linear bound for the Rees valuations of all prime ideal centered on that prime: There exists $E\in\mathbb{N}$ so that if $\mathfrak{p}$ $ is a prime ideal, if $\nu_1,\nu_2 $ are Rees valuations of $\mathfrak{p}$ centered on $\mathfrak{p}$ $, then for all $x\in R$, $\nu_1(x)\leq E\nu_2(x)$. We show that rings which are essentially of finite type over a field enjoy the Uniform Izumi-Rees Property and provide applications, including applications to symbolic powers of ideals.

February 6, Kyle Maddox University of Arkansas
F-singularities of amalgamated algebras along an ideal.

Abstract. Work in this talk is joint with Souvik Dey. Amalgamated algebras along an ideal are large and interesting class of examples of rings formed from the data of a ring homomorphism and an ideal of the codomain. These rings also recover a variety of classical constructions such as Nagata idealizations. In this talk, I will describe ongoing work to classify and provide examples of when these rings have several different classes of singularity types defined in terms of the Frobenius endomorphism when the underlying rings are of positive prime characteristic, including F-rational, F-injective, and F-nilpotent singularities.

February 20, Michael K Brown Auburn University
Periodicity of ideals of minors in free resolutions.

Abstract. I will discuss results on periodicity of ideals of minors in minimal free resolutions over complete intersection and Golod rings. This is joint work with Hailong Dao and Prashanth Sridhar.

March 5, Abu C Thomas Georgia State University, Perimeter College
Ideals in multi-projective space

Abstract The famous Hilbert's Nullstellensatz gives a one-to-one correspondence between closed varieties in $\mathbb{P}^n$ over an algebraically closed field $k$ and homogeneous radical ideals of the polynomial ring $k[x_0,\dots,x_n]$. The Waldschmidt constant of a homogeneous ideal $I \subset k[x_0,\dots,x_n]$ is very useful in the study of ideal containment problems, that is, knowing for what pairs $(m, r) \in \mathbb{N}^2$ does $I^{(m)} \subseteq I^r$, where $I^{(m)}$ is the $m$-th symbolic power of the ideal $I$. Given a finite set of points $Z \subset \mathbb{P}^n$, it also plays a very important role in determining the lower bounds on the least degree of a hypersurface containing $Z$ with a fixed multiplicity $m \in \mathbb{N}$. Motivated by this, we define and compute the Waldschmidt set of $0-$dimensional schemes in a multi-projective space.

April 2, William D. Taylor Tennessee State University
Monomial Ideals Fixed by Differential Operators

Abstract In this talk we will look at monomial ideals in polynomial rings, or more generally semigroup rings, and differential operators on those rings. We say an ideal is compatible with an operator if the operator sends the ideal to itself, and an ideal is fixed by an operator if in addition the ideal is the image of itself under the operator. We will ask the following questions: Which differential operators fix some monomial ideal, and which monomial ideals are fixed by some differential operator? These questions are motivated by analogies between Cartier maps in positive characteristic and differential operators in any characteristic, and by similar results on fixed ideals of Cartier maps. We will give a complete, and satisfyingly constructive, answer to both questions in the case of a homogeneous differential operator on a normal complex semigroup ring. Our main tools will be a characterization of such operators due to Saito and Traves and identifying the monomials of the ring with a lattice of points in Euclidean space.

April 9, Saeed Nasseh Georgia Southern University
Lifting theory of differential graded modules

Abstract Lifting theory was studied intensively by Auslander, Ding, and Solberg for modules and by Yoshino for complexes. This notion has several applications in deformation theory, representation theory, commutative algebra, and related fields. Further progress on this theory has been obtained recently by Nasseh, Ono, Sather-Wagstaff, and Yoshino in the context of differential graded (DG) modules -- a notion from rational homotopy theory -- in order to obtain a clearer insight on some major problems in commutative algebra. One of these problems that originates from representation theory of Artin algebras is called the Auslander-Reiten Conjecture. In this talk, I will survey recent developments on the lifting theory of DG modules and describe the relationship between this notion and the Auslander-Reiten conjecture.

Algebra Seminar at Georgia State University

Fall 2023 Mondays 10:00am--10:50 am Room 1441 25 Park Place

September 11, Florian Enescu, Georgia State
Buchberger's Algorithm (I).

AbstractThe existence of Grobner bases for ideals of polynomials with coefficients in a field is a fundamental fact and establishes an important computational tool in non-linear algebra. Buchberger's algorithm is the first and the most important way to obtain a Grobner basis for a given ideal. I will present this algorithm..

September 18, Florian Enescu, Georgia State
Buchberger's Algorithm (II).

Abstract. The existence of Grobner bases for ideals of polynomials with coefficients in a field is a fundamental fact and establishes an important computational tool in non-linear algebra. Buchberger's algorithm is the first and the most important way to obtain a Grobner basis for a given ideal. I will present this algorithm. (This is a continuation of the talk from September 11)

September 25, Florian Enescu, Georgia State
Buchberger's Algorithm (III).

Abstract. The existence of Grobner bases for ideals of polynomials with coefficients in a field is a fundamental fact and establishes an important computational tool in non-linear algebra. Buchberger's algorithm is the first and the most important way to obtain a Grobner basis for a given ideal. I will present this algorithm. (This is a continuation of the talk from September 11 and 18.)

October 2, Yongwei Yao, Georgia State
The Wedderburn-Artin Theorem.

Abstract. The Wedderburn-Artin theorem states that a semisimple ring is isomorphic to a direct product of matrix rings over division rings. We will go over its proof and some of its consequences.

October 9, Yongwei Yao, Georgia State
The Wedderburn-Artin Theorem (II).

Abstract. The Wedderburn-Artin theorem states that a semisimple ring is isomorphic to a direct product of matrix rings over division rings. We will go over its proof and some of its consequences (part two of the talk).

October 16, Yongwei Yao, Georgia State
The Wedderburn-Artin Theorem (III).

Abstract. The Wedderburn-Artin theorem states that a semisimple ring is isomorphic to a direct product of matrix rings over division rings. We will go over its proof and some of its consequences (part three of the talk).

October 30, Florian Enescu, Georgia State
The Elimination Theorem.

Abstract. I will present the Elimination Theorem for Groebner bases and show some of its applications.

November 16, Justin Lyle, University of Arkansas
Endomorphism Algebras Over Commutative Rings and Torsion in Self Tensor Products

Abstract. Let R be a commutative Noetherian local ring. In this talk, we study tensor products involving a finitely generated R-module M through the natural left module structure of its endomorphism ring. In particular, we study torsion in self tensor products of M over its endomorphism ring E in the case when E has an R*-algebra structure. We will see that, under mild hypotheses, such tensor products must always have torsion when M is indecomposable. We will present a detailed example at the end contrasting tensor products over E with those over R.

December 4, Florian Enescu, Georgia State
The Geometric Extension Theorem and Consequences.

Abstract. I will present geometric version of the Extension Theorem for Groebner bases and show some of its applications, including connections to Hilbert Nullstellensatz.