Commutative Algebra Seminar at Purdue

Commutative Algebra Seminar at Purdue
Time: Wednesday, 4:30pm--5:30pm
Location: SCHM 309

Spring 2023 Speakers / Abstracts

January 18: Jenna Tarasova (Purdue University)
Title: Linkage and F-Regularity of Determinantal Rings
Abstract: In this talk, we prove that the generic link of a generic determinantal ring defined by maximal minors is F-regular. In the process, we strengthen a result of Chardin and Ulrich. They showed that the generic residual intersections of a complete intersection ring with rational singularities again have rational singularities. We show that they are, in fact, F-regular in positive prime characteristic. Hochster and Huneke showed that generic determinantal rings are F-regular. However, their proof is quite involved. Our techniques allow us to give a new and simple proof of the F-regularity of generic determinantal rings defined by maximal minors.

January 25: Siamak Yassemi (Purdue University)
Title: Cohen-Macaulayness in a fixed codimension
Abstract: A concept of Cohen-Macaulay in codimension t is defined and characterized for arbitrary finitely generated modules and coherent sheaves by Miller, Novik, and Swartz in 2011. Soon after, Haghighi, Yassemi, and Zaare-Nahandi defined and studied CM_t simplicial complexes, which is the pure version of the abovementioned concept and naturally generalizes both Cohen–Macaulay and Buchsbaum properties. The purpose of this talk is to survey briefly recent results of CM_t simplicial complexes.

February 1: Adam LaClair (Purdue University)
Title: Invariants of Binomial Edge Ideals via Linear Programs
Abstract: The log-canonical threshold and the F-threshold are important invariants associated to the singularities of a variety in characteristic $0$ and characteristic $p$, respectively, and it is an interesting problem to give formulae for these invariants when the defining ideal is particularly nice. With the goal to give a combinatorial interpretation of these invariants for the class of binomial edge ideals, we associate to every graph a linear program for packings of vertex disjoint paths and prove that the optimal value of the linear program computed over the rational numbers computes these invariants when the graph is a block graph or of K\"onig type. We also show, surprisingly, that the primal and dual optimal values of this linear program computed over the integers recovers the binomial grade and height of the binomial edge ideal, respectively. We use this to give a new characterization of graphs of K\"onig type and shed light on the graph being traceable.

February 22: Matthew Weaver (University of Notre Dame)
Title: Jouanolou duality and the equations of Rees algebras
Abstract: One of the classical problems within algebraic geometry is to determine the implicit equations defining the closed image and the graph of a rational map between projective spaces. Algebraically, this corresponds to determining the defining equations of the Rees algebra of the ideal whose generators define such a map. In this talk, we recount some recent developments in this area and how a classical technique can be combined with modern methods to assist in the search for these equations.

March 1: Alexandra Seceleanu (University of Nebraska)
Title: Principal symmetric ideals
Abstract: Consider a homogeneous polynomial $f$ in variables $x_1, \ldots, x_n$. The set of polynomials obtained from $f$ by permuting the variables in all possible ways generates an ideal, which we call a principal symmetric ideal. What can we say about the graded Betti numbers of a principal symmetric ideal? I will answer this question reporting on joint work with Megumi Harada and Liana Sega.

March 29: Ilya Smirnov (Basque Center for Applied Mathematics)
Title: On generic freeness of local cohomology (talk cancelled)
Abstract: I will talk about a recent joint work with Yairon Cid-Ruiz. A classic result asserts generic freeness, over the base ring B, of a module finitely generated over a B-algebra. While it is often missing for infinitely generated modules, we establish generic freeness for local cohomology of smooth algebras in equal characteristic. Our proof builds on already known finiteness properties, for example, in characteristic 0 we exploit the D-module structure by proving a generic freeness theorem for the algebra of differential operators. In the talk, I will explain the structure of the proof in this case with all relevant ideas.

March 29: Gregor Kemper (Technical University of Munich)
Title: Distance geometry, algebra and drones
Abstract: Is it always possible to reconstruct a point configuration in the plane from the unlabeled set of mutual distances between the points? This and other questions translate to invariant theory and ultimately to problems in computational commutative algebra. As it turns out, the following question is related: Can a drone, or a ground-based vehicle, reconstruct the shape of a room from hearing echoes of a sound bouncing off the walls? What if it is unknown at what time the sound was emitted? Attacking these questions again leads to commutative algebra and requires massive computer algebra computations. The talk presents results by Mireille Boutin and the speaker, some of them old and some recent.

March 31: Alessandra Costantini (Oklahoma State University)
(Note: Special day, 4:30pm-5:30pm)
Title: Residual intersections and the core of modules
Abstract: Since the pioneering work of Northcott and Rees in 1954, the minimal reductions of an ideal I have been a powerful tool to study both the powers and the integral closure of I. In fact, a beautiful theorem of Briançon and Skoda relates the I-adic filtration with the integral closure filtration of I through the core of I, which is the intersection of all minimal reductions of I. Determining the core of an arbitrary ideal is usually difficult, as this is a priori an infinite intersection of ideals. However, work of Corso, Polini and Ulrich provides explicit formulas to calculate the core of ideals with "good" residual intersections. With geometric motivations in mind, one analogously wishes to determine the core of a module. In this new setting, one faces the additional challenge that several techniques used to study the problem for ideals (including residual intersections) become inapplicable, hence new methods need to be developed. In this talk I will report on joint work with Louiza Fouli and Jooyoun Hong, where we extend some of the results of Corso, Polini and Ulrich to modules.

April 5: Vaibhav Pandey (Purdue University)
Title: Counting geometric branches using the Frobenius map
Abstract: We give a formula to count the number of geometric branches of a curve in positive prime characteristic using the theory of tight closure. This formula readily gives a characterization F-nilpotent curves. Further, we show that reduced and excellent F-nilpotent rings have a single geometric branch; in particular, they are domains. We then give a recipe to create F-nilpotent rings and show that the usually tedious tight closure calculations simplify immensely for F-nilpotent affine semigroup rings. This is joint work with Hailong Dao and Kyle Maddox.

April 12: Ayah Almousa (University of Minnesota)
Title: Koszulity and Stirling representations
Abstract: Stirling numbers of the first (resp. second) kind count permutations (resp. partitions) of {1,2,...,n} with a fixed number of cycles (resp. blocks). Those of the first kind also give the Hilbert function for two graded algebras coming from type A reflection arrangements, or cohomology of configuration spaces: the Orlik-Solmon (OS) algebra and the graded Varchenko-Gelfand (VG) algebra. The representations of the symmetric group on these algebras are very interesting and well-studied. Since type A reflection arrangements are supersolvable, their OS and VG algebras have quadratic initial ideals and are Koszul algebras, as observed by Peeva (for OS) and by Dorpalen-Barry (for VG). We study here their Koszul dual algebras, whose Hilbert functions are given by Stirling numbers of the second kind. These dual algebras also have quadratic initial ideals, with pleasant standard monomial bases, and carry interesting representations of the symmetric group. We give descriptions of some of these representations, and branching rules, along with some intriguing conjectures. This is a preliminary report on joint work with Vic Reiner and Sheila Sundaram.

April 19: Lisa Seccia (University of Genoa)
Title: On binomial edge ideals of weakly-closed graphs and determinantal facet ideals
Abstract: Determinantal facet ideals are ideals of maximal minors associated with pure simplicial complexes. If the defining simplicial complex is one-dimensional (i.e., it is a graph), the corresponding ideal is called binomial edge ideal and many of its algebraic properties have been extensively studied. This talk will cover some results on binomial edge ideals of closed and weakly-closed graphs, and their generalizations to higher dimensions. Additionally, we will discuss some open problems on F-singularities of determinantal facet ideals in prime characteristic. Part of this talk is based on a joint work with Bruno Benedetti and Matteo Varbaro.

April 26: Qiurui Li (Purdue University)
Title: On Generalized Deformation Problems for F-singularities
Abstract: Let $(R,m)$ be a Noetherian local ring, $I$ an $R$-ideal with finite projective dimension. If $R/I$ satisfies some property $\mathcal{P}$, then it is natural to ask if $R$ would also satisfy the property $\mathcal{P}$. This is called the generalized deformation problem. Motivated by Aberbach's work, we show that if every ideal of $R/I$ generated by a maximal regular sequence on $R/I$ is Frobenius closed, then every ideal generated by a maximal regular sequence on $R$ is Frobenius closed. We also give out a tight closure version of the above statement. Thus we can show F-injectivity holds in Cohen-Macaulay ring case and F-rationality holds in excellent ring case for the generalized deformation problem.

Fall 2022 Speakers / Abstracts

August 31: Vaibhav Pandey (Purdue University)
Title: When are the natural embeddings of determinantal rings split?
Abstract: Over an infinite field, a generic determinantal ring is the fixed subring of an action of the general linear group on a polynomial ring; this is the natural embedding of the title. If the field has characteristic zero, the general linear group is linearly reductive, and it follows that the invariant ring is a split subring of the polynomial ring. We determine if the natural embedding is split in the case of a field of positive characteristic. Time permitting, we will address the corresponding question for Pfaffian and symmetric determinantal rings. This is ongoing work with Mel Hochster, Jack Jeffries, and Anurag Singh.

Septermber 14: Hunter Simper (Purdue University)
Title: Ext and Local Cohomology of Thickenings of Ideals of Maximal Minors
Abstract: Let $R$ be the ring of polynomial functions in $mn$ variables with coefficents in $\mathbb{C}$, where $m>n$. Set $X$ to be the matrix in these variables and $I$ the ideal of maximal minors of this matrix. I will discuss the R-module structure of certain Ext and local cohomology modules arising from the rings $R/I^t$. In particular, for $i$ equal to the cohomological dimension of $I$, I will discuss the embedding of $Ext^i_R(R/I^t,R)$ into $H_\frak{m}^{mn}(R)$, explicitly describing this embedding when $X$ is size $n \times (n-1)$. More generally for $X$ of arbitrary size I will describe the annihilator of $Ext^i_R(R/I^t,R)$ and thereby completely determine the $R$-module structure of $H_\frak{m}^{mn-i}(R)$.

Septermber 21: Swaraj Pande (University of Michigan)
Title: The F-signature function of the ample cone of a globally F-regular variety
Abstract: The F-signature of a strongly F-regular local ring R is an interesting invariant of its singularities. In this talk, we will discuss this invariant when R is the normalized homogeneous coordinate ring of a projective variety. In particular, we study how the F-signature varies as we vary the embedding of a fixed projective variety X into various projective spaces. For this purpose, we will introduce the F-signature function, a real valued function on the ample cone of X, and discuss its continuity properties. We will also present some analogies and comparisons to the well-known volume function, which records the Hilbert-Samuel multiplicities. This is joint work with Seungsu Lee.

October 5: Wenbo Niu (University of Arkansas)
Title: Multiplier ideals on varieties and local properties
Abstract: In this talk, we discuss the notion of Mather-Jacobian ideals defined on an arbitrary variety. It was introduced by Ishii-Ein-Mustata and de Fernex-Docampo extending the notion of multiplier ideals on normal varieties. We also discuss local syzygies of MJ-multiplier ideals, extending the work of Lazarsfeld-Lee and Lazarsfeld-Lee-Smith. This is a joint work with Ulrich.

October 12: Rabeya Basu (Indian Institute of Science Education and Research)
Title: On the completion of unimodular rows and its applications in commutative algebra and classical K-theory
Abstract: The transitivity action of the elementary subgroups of the general linear groups on unimodular rows helps to study the problems related to Serre's problem on projective modules. In this seminar, we will discuss the basic properties of unimodular rows over general rings, and its applications in commutative algebra and classical K-theory.

October 19: Alapan Mukhopadhyay (University of Michigan)
Title and Abstract : see here

November 2: Javier Carvajal-Rojas (KU Leuven)
Zoom talk: https://purdue-edu.zoom.us/j/98129072188?pwd=cHpRWnIwUHRQZXZySUI2Wm5XYmNNdz09, Meeting ID: 981 2907 2188, Passcode: 812577
Title: Centers of F-purity and their behavior under finite covers
Abstract: I'll discuss how the spectrum of centers of F-purity of a Frobenius splitting behaves with respect to finite extensions of the base ring. I'll explain why if the extension is tamely ramified then the centers of F-purity upstairs are exactly those prime ideals contracting to centers of F-purity downstairs. I'll also discuss the converse statement. This is ongoing joint work with Anne Fayolle (University of Utah).

November 9: Takumi Murayama (Purdue University)
Title: The cancellation problem, steadfastness, and deformation
Abstract: Let A and B be rings. The cancellation problem asks whether an isomorphism A[X1,X2,...,Xn] -> B[X1,X2,...,Xn] implies an isomorphism A -> B. While Hochster answered the cancellation problem in the negative, various versions of the cancellation problem remain open. In this talk, I will focus on a version of the cancellation problem considered by Abhyankar, Eakin, and Heinzer. Hamann introduced the class of steadfast rings as the rings for which this version of the cancellation problem holds. I will describe joint work with Alexander Bauman, Havi Ellers, Gary Hu, Sandra Nair, and Ying Wang showing that the class of steadfast rings is stable under deformations (for reduced Noetherian local rings) and adjoining formal power series variables (for rings that are either reduced or Noetherian). Our results give new ways to construct steadfast rings.

November 16: Kriti Goel (University of Utah)
Title: Hilbert-Kunz multiplicity of powers of an ideal
Abstract: P. Monsky proved the existence of Hilbert-Kunz multiplicity in 1983. Since then, it has been extensively studied, partly because of its connections with the theory of tight closure and its unpredictable behaviour. Unlike the Hilbert-Samuel multiplicity, the Hilbert-Kunz multiplicity need not be an integer. In this talk, we consider Hilbert-Kunz multiplicity of powers of an ideal, in an attempt to write it as a function of the power of the ideal. This involves a surprising connection with the Hilbert-Samuel coefficients of Frobenius powers of an ideal.

November 21: Alessandra Costantini (Oklahoma State University)
(Note: Special day, 12:30-1:30)
Title: The powers of a symmetric strongly shifted ideal
Abstract: Symmetric strongly shifted ideals are a special class of monomial ideals, which are invariant under the action of the symmetric group by permutation of the variables. In this talk, I will describe how the combinatorial structure of an ideal of this kind dictates the algebraic properties of its powers and of its Rees algebra. The content of this talk is part of joint work with Alexandra Seceleanu, sponsored by the 2021 AWM Mentoring Travel Grant.

November 30: Jennifer Kenkel (University of Michigan)
Title: Lengths Of Local Cohomology Using Some Surprising Hilbert Kunz Functions
Abstract: We investigate the lengths of certain local cohomology modules over polynomial rings. By fixing the degree component, and using the fact that the length of an Artinian ring is the same as that of its injective hull, we transform this into a question about rings of the form $k[x_1, \dots, x_n]/(x_1^{k_k}, \dots, x_n^{k_n})$ and the annihilator of $x_1 + \cdots + x_n$ therein. We in particular use refinements of functions introduced by Han and Monsky. This was motivated by questions about behavior of the length of local cohomology with support in the maximal ideal of thickenings, that is, $R/I^t$ as $t$ grows. This is joint work with Mel Hochster.

December 7: Omar Colon Reyes (University of Puerto Rico)
Zoom talk: https://purdue-edu.zoom.us/j/97529988974?pwd=dDYxYUcvenhnZ0Vjd2xDalNYTU50UT09, Meeting ID: 975 2998 8974, Passcode: 911991
Title: Obtaining the transient of discrete dynamical systems
Abstract: An open problem in the theory of discrete dynamical systems is linking the structure of a system with its dynamics. This paper contains such a link for a family of nonlinear systems over the field with two elements. For a family of systems that can be described by monomials (including Boolean AND systems), one can obtain information about the transient of the system from the structure of the monomials. In particular, this work contains a formula for the transient of monomial systems that only has fixed points as limit cycles. This condition depends on the cycle structure of the dependency graph of the system.

December 14: Pham Hung Quy (FPT University)
Title: Tight Hilbert Polynomial and F-rational local rings
Abstract: I will define the Buchsbaum property for tight closure. After that we discuss tight Hilbert polynomial, Hilbert coefficients. The talk is based on the paper A Buchsbaum theory for tight closure with Linquan Ma, and Tight Hilbert Polynomial and F-rational local rings with Saipriya Dubey, and Jugal Verma.