Commutative Algebra Seminar at UNL

The seminar organizers are Alexandra Seceleanu (aseceleanu at unl.edu) and Mark Walker (mark.walker at unl.edu).

Seminars are held in Avery 351 and on Zoom on Wednesdays 3:30-4:45 pm. The Zoom link will be sent through the seminar mailing list.

Seminar schedule

 

Spring 2021

March 24, 2021

          Robert Walker (Wisconsin)

Title: Uniform Asymptotic Growth of Symbolic Powers of Ideals

Abstract: Algebraic geometry (AG) is a major generalization of linear algebra which is fairly influential in mathematics. Since the 1980's with the development of computer algebra systems like Mathematica, AG has been leveraged in areas of STEM as diverse as statistics, robotic kinematics, computer science/geometric modeling, and mirror symmetry. Part one of my talk will be a brief introduction to AG, to two notions of taking powers of ideals (regular vs symbolic) in Noetherian commutative rings, and to the ideal containment problem that I study in my thesis. Part two of my talk will focus on stating the main results of my thesis in a user-ready form, giving a "comical" example or two of how to use them. At the risk of sounding like Paul Rudd in Ant-Man, I hope this talk will be awesome.

 

March 31, 2021

          Paolo Mantero (Arkansas)

Title: Interpolation problems.

 

Abstract: Homogeneous polynomial interpolation problems are interesting problems in Algebraic Geometry. They ask for information regarding hypersurfaces of given degrees passing with given multiplicity through a given set X of points in a projective space. A theorem by Zariski and Nagata allows one to translate these problems into questions about symbolic powers of the ideal I_X defining the set of points. In the present talk we will state classical and recent results about some interpolation problems, and discuss open problems and conjectures.

April 14, 2021

          Erica Musgrave (UNL)

Title: Free complexes over the exterior algebra with small homology

 

April 21, 2021

          Luis Núñez-Betancourt (CIMAT, Mexico)

Title: TBA

 

Fall 2020

October 28, 2020

          Laila Awadalla (UNL)

Title: Levels, ghost maps, and Gorenstein dimension

October 14, 2020

          Eloísa Grifo (UC Riverside)

Title: Constructing non proxy-small modules

October 7, 2020

          Jack Jeffries (UNL)

Title: Bernstein’s inequality for some singular rings

 

September 23, 2020

          Amadeus Martin (UNL)

Title: Curved BGG correspondence

 

September 16, 2020

          Eric Hopkins (UNL)

Title: N-fold matrix factorizations

 

September 9, 2020

          Alexandra Seceleanu (UNL)

Title: Surprising examples involving Nagata idealizations

 

August 19, 2020

          Mark Walker (UNL)

Title: On lim-Ulrich modules

Spring 2020

 

March 11-12, 2020

          Hamid Rahmati (UNL)

Title: Weak Complete intersection ideals

 

Abstract: We introduce the notion of weak complete intersection ideals; these are the ideals with the property that every differential in their minimal free resolution can be represented by a matrix whose entries are in the ideal itself. We provide a family of these ideals and present some applications. We also discuss algebra structures on free reclusions of such ideals. This talk is based on joint works with Claudia Miller, Janet Striuli and Zheng Yang.

 

February 26-27, 2020

          Justin Lyle (University of Kansas)

Title 1: Extremal growth of Betti numbers and rigidity of (co)homology

 

Abstract 1: Let R be a commutative Noetherian ring. Several open conjectures and questions on the vanishing of Ext and Tor have attracted interest in recent years. For instance, the longstanding Auslander-Reiten conjecture poses that the vanishing of ExtiR(M,M+R) for all i>0 should force M to be projective, and a recently introduced dual version of Şega asks whether the vanishing TorRi(M,M) for i>0 should imply the same. We study these and related questions via various asymptotic invariants associated to M. As a consequence, we provide several new cases of these conjectures. This is based on joint work with Jonathan Montaño and on joint work with Jonathan Montaño and Sean Sather-Wagstaff.  

Title 2: Maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum

Abstract 2: We say that a Cohen-Macaulay local ring has finite CM+-representation type if there exist only finitely many isomorphism classes of indecomposable maximal Cohen-Macaulay modules that are not locally free on the punctured spectrum. In this talk, we consider finite CM+-representation type from various points of view, relating it with several conjectures on finite/countable Cohen-Macaulay representation type. We prove in dimension one that the Gorenstein local rings of finite CM+-representation type are exactly the local hypersurfaces of countable CM+-representation type; under mild hypotheses these are exactly the hypersurfaces of type A and D. We also consider partial results and obstacles in higher dimension. This is joint work with Toshinori Kobayashi and Ryo Takahashi.

 

February 5-6, 2020

          Alexandra Seceleanu (UNL)

Title: Symmetric shifted ideals

 

Abstract: Shifted ideals are a class of monomial ideals with remarkable homological properties. In particular their Betti numbers are completely determined by numerical information easily read off the generators. We introduce symmetric shifted monomial ideals, a class of monomial ideals fixed by the action of the symmetric group on the polynomial ring, which have properties resembling those of shifted ideals. We completely determine their Betti numbers and further properties of their homology algebras.

 

January 23, 2020

          Jan Trlifaj (Charles University, Prague)

Title: Test sets for factorization properties of modules, and transfinite extensions of simple artinian rings

 

Abstract: The Baer Criterion of injectivity implies that injectivity of a module is a factorization property w.r.t.  a single monomorphism. We will discuss generalizations and dualizations of factorization properties in dependence on the algebraic structure of the underlying ring R and on additional set-theoretic hypotheses. For R commutative noetherian of Krull dimension 0< d <∞, we will show that the assertion “Projectivity is a factorization property w.r.t. a single epimorphism” is independent of ZFC. We will also consider an old problem of Carl Faithon the validity of the Dual Baer Criterion (DBC) for non-perfect rings.  We will construct a class of transfinite extensions of simple artinian rings for which the problem is undecidable in ZFC.

 

 

Fall 2019

November 20-21, 2019

          Rachel Diethorn (Syracuse University)

Title 1:  Koszul homology of quotients by edge ideals

 

Abstract 1:  We show that the Koszul homology algebra of a quotient by the edge ideal of a forest is generated by the lowest linear strand. This provides a class of Koszul algebras whose Koszul homology algebras satisfy this property. We also recover a result of Roth and Van Tuyl on the graded Betti numbers of quotients of edge ideals of trees.

 

Title 2:  Generators of Koszul homology with coefficients in a J-weak complete intersection module

 

Abstract 2:  We introduce a class of modules, called J-weak complete intersection modules, inspired by the weak complete intersection ideals studied by Rahmati, Striuli, and Yang. We present explicit formulas for the generators of Koszul homology with coefficients in such modules. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. We use these formulas to study connections between J-weak complete intersection ideals and weak complete intersection ideals.

 

November 7, 2019

          Alessandra Costantini (University of California Riverside)

Title: Cohen-Macaulay property of special fiber rings

 

Abstract: Let R be a Noetherian local ring. The special fiber ring, or fiber cone, F(E), of a module E is defined as the tensor product of the Rees algebra with the residue field.  In this talk, I will discuss a general technique to study the Cohen-Macaulay property of F(E), and provide sufficient conditions for F(E) to be Cohen-Macaulay. The latter generalize previous work of Corso-Ghezzi-Polini-Ulrich and Montaño for the special fiber ring of an ideal.

 

October 30, 2019

          Ben Drabkin

            Title:   Freiman cover ideals

 

Abstract: For a monomial ideal I in a polynomial ring, it is difficult in general to find lower bounds on the number of generators of I^2.  However, Freiman's Theorem, a result in additive number theory, gives rise to such a lower bound for certain types of monomial ideals in terms of the number of generators of I and the analytic spread of I.  Ideals which meet this bound are called Freiman.  This talk covers an attempt to determine when cover ideals (a class of squarefree monomial ideals arising from graphs) are Freiman.

 

October 23-24, 2019

          Mark Walker

            Title: How short can a module of finite projective dimension be?

 

Abstract: This is very much in progress work, done in collaboration with Srikanth Iyengar and Linquan Ma. I’ll describe what we know so far about the question in the title.

October 9-10, 2019

          Lucho Avramov

            Title: Monoids of Betti tables over graded algebras. The case of short Gorenstein algebras

 

Abstract: These talks, based joint work with Courtney Gibbons and Roger Wiegand, I will explore a reflection of representation theory in homological algebra.  Some general approaches will be discussed in the first talk.  In the case of graded Gorenstein algebras R with R_i=0 for i>2 the picture is complete; it will be presented in the second talk.

October 3, 2019

          Roger Wiegand

          Title: Vanishing of Tor over fiber products

Abstract: Let  (S,m,k)  and  (T,n,k)  be local rings, and let  R   denote their fiber product over their common residue field  k.  Inspired by work of Nasseh and Sather-Wagstaff, we explore consequences of vanishing of  Tor^R_m(M,N)  for various values of m, where  M  and  N  are finitely generated  R-modules.  For instance, assume that neither  S  nor  T  is a discrete valuation domain and that  Tor^R_m(M,N) = 0 for some  m > 5.  Then at least one of  M, N  has projective dimension at most one.   This is joint work with Thiago Freitas, Victor Hugo Jorge Pérez, and Sylvia Wiegand.

September 26, 2019

          Tomasz Szemberg (Pedagogical University of Cracow)

          Title: Unexpected hypersurfaces and their geometry

Abstract: In the ground-breaking article of Cook II, Harbourne, Migliore and Nagel, the notion of unexpected curves has been introduced. It has been generalized in the subsequent article to unexpected hypersurfaces. The subject is dynamically growing ever since and shows unexpected ties to other areas of algebra and geometry. I will report on some of recent developments.

September 25, 2019

          Justyna Szpond (Pedagogical University of Cracow)

          Title: Fermat-type configurations in geometry and algebra

Abstract: A Fermat arrangement of lines in the complex projective plane is given by linear factors of the polynomial (x^n-y^n)(y^n-z^n)(z^n-x^n) for some n≥3. Singular points of these arrangements have appeared recently in commutative algebra, more precisely in the containment problem between symbolic and ordinary powers of homogeneous polynomials and in algebraic geometry in the theory of linear systems and unexpected hypersurfaces. I will explain these two appearances and present considerable generalizations which led to substantial, new results in both fields.

September 18-19, 2019

          Eloísa Grifo (University of California Riverside)

          Title 1: Symbolic powers and the containment problem

Abstract 1: The n-th power of an ideal is easy to compute, though difficult to describe geometrically; in contrast, symbolic powers are difficult to compute while having a natural geometric description. The Containment problem tries to compare these two notions, and it relates to other questions such as the degree of polynomials vanishing on a given variety. Harbourne conjectured that a famous answer to the containment problem, by Ein--Lazersfeld--Smith, Hochster--Huneke and Ma--Schwede, could be improved. In this talk, we will introduce the containment problem and Harbourne's Conjecture, and discuss how despite the fact that the conjecture does not hold in full generality, many interesting versions of it do hold.

Title 2: Two versions of Harbourne's Conjecture

Abstract 2: Harbourne's Conjecture is a statement depending on n that unfortunately has been disproved for particular values of n. In this talk, we will discuss two different versions of this conjecture that do hold. The first is joint work with Linquan Ma and Karl Schwede, showing that even over certain singular rings, Harbourne's Conjecture holds for a large class of ideals. The second is joint work with Craig Huneke and Vivek Mukundan, towards a stable version of Harbourne's Conjecture that may hold for all ideals -- where we replace all n with n large enough.

September 5, 2019

          Brian Harbourne

          TitleAn open problem motivated by algebra, geometry and combinatorics

Abstract: I'll discuss how a problem involving the Strong Lefschetz Property (algebra) is equivalent to a vanishing problem involving points in the plane (geometry), which in a special case concerns open problems involving line arrangements (combinatorics) that recent progress has been made on (by Ben Green and Terence Tao a few years ago, and Krishna Hanumanthu, me and Alex Dimca more recently).

 

 

Spring 2019

January 16, 2019

          Ralf Schiffler (University of Connecticut)

          TitleFrieze varieties: A characterization of the finite-tame-wild trichotomy for acyclic quivers

Abstract: Given a quiver (=oriented graph) without oriented cycles, one can construct its frieze variety. It is defined in an elementary recursive way by constructing a set of points in affine space. From a more conceptual viewpoint, the coordinates of these points are specializations of cluster variables in the cluster algebra associated to the quiver. We give a new characterization of the finite-tame-wild trichotomy for acyclic quivers in terms of their frieze varieties. We show that an acyclic quiver is representation finite, tame, or wild, respectively, if and only if the dimension of its frieze variety is 0,1 or >1, respectively. This is a joint work with Lee, Li, Mills and Seceleanu.

January 30, 2019

Matt Mills (Michigan State University)

          Title: Cluster and Poisson structures on quasi-affine varieties 

Abstract: One large appeal of the study of cluster algebras is that the ring of regular functions for many varieties can be equipped with a cluster algebra structure. In this talk we will describe sufficient conditions for a quasi-affine variety to admit a cluster algebra structure. Furthermore, we will follow the work of Gekhtman, Shapiro, and Vainshtein to show when these cluster structures can be realized as Poisson algebras with a Poisson bracket compatible with the cluster structure. Finally, we present an open question about cluster variables in Poisson-Lie groups. 

February 6, 2019

          Alessandro De Stefani

Title: Cohomologically full rings

Abstract: The class of cohomologically full rings has been recently introduced by Hailong Dao, Linquan Ma and myself. This class includes nice singularities such as Cohen-Macaulay rings, F-pure rings in characteristic p, Stanley Reisener rings, etc. The goal of this talk is to present some of the motivations behind the definition, and some of the properties that cohomologically full rings enjoy. Among them, upper bounds on projective dimensions, and on the Castelnuovo-Mumford regularity in the graded case.

February 13, 2019

          Alessandro De Stefani

Title: Weak and strong F-regularity for Gorenstein rings

Abstract: I will present a proof due to Huneke and Leuschke of the fact that weak and strong F-regularity are equivalent for Gorenstein rings. Whether the same statement holds in the general case is one of the major open problems in the area of singularities in characteristic p>0. The aim is to make the talk largely accessible to graduate students (in particular those who have been showing up for Math 918 this semester).

February 20-21, 2019

          Hailong Dao (University of Kansas)

Title 1: Mysterious ideals, part 1: Golod ideals

Abstract 1: In these talks I will introduce some joint work with recent Nebraska postdocs. One common theme is that the problems reduce to very concrete, but surprisingly stubborn problems about ideals, even in polynomial rings of few variables. In the first talk, I will describe my work with Alessandro De Stefani, focusing on the question: when is the product of two homogenous ideals Golod

 

Title 2: Mysterious ideals, part 2: linearly presented ideals

Abstract 2: A homogenous ideal is linearly presented if the presentation matrix of it has only linear entries. They have attract increasing attention in recent years, due to connections to regularity, Hirsch-type bound on diameter of polytopes, and so on. I will introduce these questions and then focus on new results, obtained with Thanh Vu, on regularity bound for such linearly presented monomial ideals.

 

February 27, 2019

Ben Drabkin

Title: Growth of the Symbolic Defect

Abstract: Let R be a commutative Noetherian ring, and let I be an ideal in R. The symbolic defect is a numerical measurement of the difference between the symbolic and ordinary powers of I. In the case that I has sufficiently well-behaved symbolic powers (i.e. its symbolic Rees algebra is finitely generated) we prove that the symbolic defect grows eventually quasi-polynomially

March 1, 2019 at 10:30 am in 351 Avery (please note unusual date/time)

June Huh (IAS Princeton)

Title: Standard conjectures for finite vector spaces

Abstract: I will build a commutative ring that satisfies "standard conjectures", starting from a finite field. What is this ring? What does it say about the finite field? This talk will be elementary: No background beyond the first year graduate algebra will be necessary. Joint with Mats Boij, Bill Huang, and Greg Smith.

March 6, 2019

William Trok (University of Kentucky)

            Title:  The Geometry of Point Configurations and Hyperplane Arrangements

 

Abstract: Given a finite set of points, Z, in a projective space, P(V), we can associate to Z a dual set, A(Z), of hyperplanes in the dual space, P(V^*). We ask, "How does the geometry of Z relate to the geometry of A(Z)?". In this talk, building off of work of Faenzi and Valles, as well as Cook, Harbourne, Migliore and Nagel, we relate the module of derivations, D(A(Z)) of A(Z), to the homogeneous ideal I(Z). For instance, we show that D(A(Z)), gives bounds on the regularity of I(Z). Along the way we study ideals, I(Z), whose intersection with powers of ideals of general linear subspaces have larger than expected intersection.

March 13-14, 2019

Daniel Smertnig (University of Graz and Waterloo)

          Title 1: The role of transfer Krull monoids in studying non-unique factorizations

Abstract 1: In studying factorizations of elements in domains (or cancellative monoids; e.g. when studying direct sum decompositions of modules), surprisingly often one is able to reduce the problem to one over a Krull monoid. In this case, factorization questions typically reduce to problems in combinatorial and additive number theory, many of which have been well studied. I will recall this technique and survey some recent results.

          Title 2: Locally free cancellation for definite quaternion orders

Abstract 2: Steinitz's theorem can be generalized to describe finitely generated locally free modules over an order in a central simple algebra B (over a number field). However, when B is a definite quaternion order, generally locally free cancellation fails, and one only obtains a description up to stable isomorphism. I will talk about the classification of the finitely many definite quaternion orders that still possess the cancellation property. This property is also reflected in the factorization of elements. This is joint work with John Voight.

March 27-28, 2019

          Janina Letz (University of Utah)

          Title 1: Generation time and (co)ghost maps

          Title 2: Local to global principles for generation time over commutative rings

Abstract: In the derived category of modules over a noetherian ring a complex G is said to generate a complex X if the latter can be obtained from the former by taking finitely many summands and cones. The number of cones needed in this process is the generation time of X. I will present some
local to global type results for computing this invariant, and also discuss some applications.

April 3, 2019

          Taran Funk

          Title: Frobenius and Homological Dimensions of Complexes

Abstract: The Frobenius endomorphism has proven to be an effective tool for characterizing when a given finitely generated module over a commutative Noetherian local ring of prime characteristic has certain homological properties. In particular, many have used this endomorphism to detect when such a module has finite projective dimension. I will be discussing some results by Tom Marley and myself in our paper with the same title as this talk. We show, among other things, how to extend a few of these results to arbitrary modules. 

April 10-11, 2019

Claudia Miller (Syracuse University)

          Title: Resolutions and possible dg-algebra structures for compressed Artinian algebras

Abstract: We construct free resolutions of compressed Artinian graded algebra quotients of polynomial rings and give a method to reduce them to a minimal resolutions. Our result generalizes results of El Khoury and Kustin for Gorenstein algebras of even socle degree with a different proof. 

 

Then we use this to show current progress towards constructing dg-algebra structures in the Gorenstein case. For this we will discuss two general homological tools less known in the commutative algebra world, namely of transferring A structures (and dg-algebra structures in nice situations) along homotopy equivalences and a tool for creating new homotopy equivalences from old ones. 

 

This is joint work with Hamid Rahmati.

April 18, 2019

Claudia Miller (Syracuse University)

          Title: Resolutions and possible dg-algebra structures for compressed Artinian algebras (continued from the previous week)

April 24-25, 2019

Lorenzo Guerrieri (University of Catania and the Ohio State University)

          Title: Directed unions of local monoidal transform of regular local rings

Abstract: Let R be a regular local ring of dimension d >1. Recently, several authors studied the rings obtained as infinite directed union of iterated local quadratic transforms of R, and call them quadratic Shannon extensions. Here we study features of directed union of local monoidal transforms of a regular local ring (monoidal Shannon extensions) and more generally of directed unions of GCD domains. In particular we are interested in understanding when these rings are still GCD domains.

Claudia Miller (Syracuse University)

          Title: Resolutions and possible dg-algebra structures for compressed Artinian algebras (continued from the previous week)

May 1st, 2019 at 3:30 pm in 351 Avery (finals week)

Kurt Herzinger (United States Air Force Academy)

            Title: Using Numerical Semigroups to Study the Game of Sylver Coinage

 

Abstract: Sylver Coinage is a two-player game played on the positive integers described by John Conway in the book Winning Ways for Your Mathematical Plays.  The game is connected in a natural way to submonoids of the non-negative integers, known as numerical semigroups.  We will examine the connections between these two topics and demonstrate how numerical semigroups can be used to analyze winning and losing positions in Sylver Coinage.

May 20, 2019 at 3:30 pm in 11 Avery

Hamid Rahmati (Miami University)

            Title: Free Resolutions of Frobenius powers of the maximal ideal over a generic hypersurface in 3 variables

 

Abstract: We discuss the asymptotic behavior of the free resolutions of the bracket powers of the maximal ideal in the hypersurface ring $R=k[x,y,z]/(f)$, where $k$ is a field of positive characteristic and $f$ is chosen generically. We show that high enough Frobenius powers of the maximal ideal have identical Betti numbers. We also compute the Hilbert-Kunz function of such rings. This is joint work with Claudia Miller and Rebecca R.G.