2023/2024 academic year
Nikola Kuzmanovski (UNL)
97 years after MacaulayAbstract: In 1927, Macaulay's paper on lex ideals was published. Macaulay proved that for every homogeneous ideal in a polynomial ring, there exists a nicely structured ideal (called a lex ideal) with the same Hilbert function. This is the fundamental result on Hilbert function and has influenced both algebra and combinatorics throughout the last century. Unfortunately, the fields that Macaulay founded and influenced have developed separately since the publication of the Clements-Lindström Theorem in 1969.
This talk will be about a connection between extremal combinatorics and commutative algebra that will put everything into a single framework. There are several byproducts from this connection. I will deduce old and new results in both fields that give answers to questions posed by Bezrukov, Hoefel, Leck, Mermin and Peeva. Some of these results include other rings in which Macaulay's lex ideal theorem holds, and a classification in this direction.
Past talks 2023/2024
Sasha Pevzner (University of Minnesota)
Symmetric group fixed quotients of polynomial ringsAbstract: Let the symmetric group act on the polynomial ring S in n variables via variable permutation. We consider the quotient module M which sets a monomial equal to all of its images under the action. This is a module over the ring of invariants, with relatively little known about its structure. When using integer coefficients, we can embed M as an ideal inside the ring of symmetric polynomials. Doing so gives rise to a family of ideals - one for each n. Localizing at a prime p reveals striking behavior in these ideals, which stay stable (in a sense) as n grows, but jump in complexity each time n equals a multiple of p. In this talk, we will discuss the construction of this family of ideals, as well as a result and conjecture on its structure. The talk will end with a path forward towards solving the conjecture.
Lauren Cranton Heller (UNL)
Resolutions of truncations of multigraded modulesAbstract: Several independent groups of researchers have recently shown the existence of short virtual resolutions for sheaves on a smooth projective toric variety X--free complexes over the coordinate ring of X which give sheaf resolutions of length at most the dimension of X. In the case of products of projective spaces my collaborators and I have shown that these short virtual resolutions can be constructed by resolving truncations of multigraded modules. I will discuss that result and what is known about the general case.
Jordan Barrett (UNL)
Toric Varieties & Zariski-Nagata Type TheoremsAbstract: The Zariski-Nagata theorem is a classical result which expresses the nth symbolic power of a radical ideal I in a polynomial ring over a perfect field in terms of the nth regular powers of the maximal ideals in mSpec(I). In this talk I will discuss projective space and homogeneous coordinates and I will state a well known Zariski-Nagata type theorem for projective varieties. I will also give a brief crash course on toric varieties and generalized homogeneous coordinates and I will discuss my work on developing a Zariski-Nagata theorem for this class of varieties.
Tài Hà (Tulane University)
Regularity of graded families of homogeneous idealsAbstract: We will survey a number of research directions and recent results on the study of the (asymptotic) regularity of graded families of homogeneous ideals.
Zach Nason (UNL)
Gorenstein Differential Graded AlgebrasAbstract: In homological algebra, a fundamental object of study is a chain complex of modules over a ring. Since chain complexes consist of modules, there is no natural multiplicative structure on these objects. In order to endow chain complexes with a multiplicative structure, it's necessary to define a differential graded algebra (DGA) and differential graded modules over DGAs. In my talk, I'll be going over some basic properties of DGAs and their associated DG-modules, and will generalize the concept of a Gorenstein ring to define a Gorenstein DGA. I'll then sketch a result of Frankild, Iyengar, and Jorgensen that characterizes finite local Gorenstein DGAs using the behavior of Ext.
Cheng Meng (Purdue University)
h-function of local rings of characteristic pAbstract: For a Noetherian local ring R of characteristic p, we will study a multiplicity-like object called h-function. It is a function of a real variable s that estimates the asymptotic behavior of the sum of ordinary power and Frobenius power. The h-function of a local ring can be viewed as a mixture of the Hilbert-Samuel multiplicity and the Hilbert-Kunz multiplicity. In this talk, we will prove the existence of h-function and the properties of h-function, including convexity, differentiability and additivity. If time permits, I will also mention how h-function covers other invariants in characteristic p.
Uli Walther (Purdue University)
Matroidal PolynomialsAbstract: We introduce a class of polynomials that can be attached (meaningfully) to a matroid. The main structure properties model those of matroidal operations: deletion, contraction, etc. Over the complex numbers we investigate their jet spaces. This leads to the conclusion that matroidal polynomials have rational singularities. This includes the polynomials defining the integrands of Feynman diagrams, in Lee—Pomeransky form. Some natural cases will be discussed where matroidal polynomials enjoy two additional properties, homogeneity and duality. For these, we show that they are in positive characteristic strongly F-regular. This is joint work with Dan Bath.
Juan Migliore (Notre Dame)
Weak Lefschetz Property for Complete Intersections - Partial ResultsAbstract: Let R be a polynomial ring over an algebraically closed k of characteristic zero. Let M be a finitely generated graded R-module. We say that M has the Weak Lefschetz Property (WLP) if multiplication by a general linear form from any component to the next necessarily has maximal rank (i.e. is either injective or surjective). Our focus in this talk is the case that \(M = R/I\) is an artinian graded complete intersection. One of the most important open problems in Lefschetz theory is whether such \(R/I\) necessarily has the WLP, and it has been conjectured by several authors that the answer is yes. This is known to be true for 2 or 3 variables, but beyond this the results are sparse. I will focus on the case of 4 variables and review what's known. Then I'll describe some recent joint work with Mats Boij, Rosa María Miró-Roig and Uwe Nagel for the case where in addition we assume that all four of the generators of the ideal have the same degree \(d\). Our approach is somewhat surprising: we produce a smooth curve in \(\mathbb{P}^3\) and read the WLP from the Hilbert function of its general hyperplane section. We give an outline of a proof for any \(d\), but are not able to carry it out completely. We get a partial result for all \(d\) that improves what's known, and we prove the full WLP for \(d = 2,3,4,5\).
Ryan Watson (UNL)
A Classification of CI using DGAsAbstract: A main idea in math is that we can study objects better if we endow additional structures on them. For instance, putting a ring structure on the integers, or defining an inner product on vector spaces lets us better understand these objects. In this talk, I will introduce differential graded algebras (DGAs), which are essentially chain complexes endowed with a multiplicative structure. In particular, I'll go over how to construct DGA resolutions, a connection between DGAs and the BEH (Buchsbaum-Eisenbud, Horrocks) conjecture, as well as a classification of complete intersections using DGAs.
Juliann Geraci (UNL)
Simplicial Resolutions and the Scarf ComplexAbstract: Let M be a monomial ideal in the polynomial ring S over a field k. I am interested in finding a minimal free resolution of S/M over S. First, I'll introduce a method for resolving S/M by making use of a simplicial complex. Unfortunately, these resolutions are rarely minimal. Thus the Scarf complex is introduced. This complex is always contained in the minimal free resolution of a monomial ideal. We will discuss the ideals for which the Scarf complex provides the minimal free resolution.
Andrew Soto Levins (UNL)
An Auslander Bound for ComplexesAbstract: The Auslander bound of a module can be thought of as a generalization of projective dimension. For a finitely generated module \(M\) with finite projective dimension over a Noetherian local ring R, \(\textrm{Ext}_{R}^{n}(M,N)=0\) for \(n>\textrm{pdim}_{R}M\) and \(\textrm{Ext}_{R}^{\textrm{pdim}_{R}M}(M,N)\neq 0\) for all finitely generated modules \(N\). We say that the Auslander bound of \(M\) is finite if for all finitely generated modules \(N\), there exists an integer \(b\) that only depends on \(M\) so that \(\textrm{Ext}_{R}^{n}(M,N)=0\) for \(n>b\) whenever \(\textrm{Ext}_{R}^{n}(M,N)=0\) for \(n\gg 0\). The Auslander bound is the least such \(b\). In this talk we give an introduction to the Auslander bound of a module and share some related results. We will then see that we can define an Auslander bound for complexes and extend the module results to complexes.
Taylor Murray (UNL)
Graded Local Cohomology and *Bass NumbersAbstract: Local Cohomology modules have been an indispensable and powerful tool since being introduced by Alexander Grothendieck. However, with great power comes great (big) modules; these modules are rarely non-zero and finitely generated. Therefore, we may ask: when do local cohomology modules have finite Bass numbers? First, we will survey a few results in the literature that address this question. Then, in the case R is a standard graded, finitely generated K-algebra, we investigate how to utilize the graded structure to obtain information about the Bass numbers of Veronese submodules of local cohomology modules.
Christine Berkesch (University of Minnesota)
Differential operators of toric face ringsAbstract: Toric face rings, introduced by Stanley, are simultaneous generalizations of Stanley--Reisner rings and affine semigroup rings, among others. We use the combinatorics of the fan underlying these rings to inductively compute their rings of differential operators. Along the way, we discover a new differential characterization of the Gorenstein property for affine semigroup rings. This is joint work with C-Y. Jean Chan, Patricia Klein, Laura Matusevich, Janet Page, and Janet Vassilev.
2022/2023 academic year
Carlos Espinosa Valdéz (CIMAT)
Regularity index of the generalized minimum distance functionAbstract: We show that the generalized minimum distance function is non-increasing as the degree varies for reduced standard graded algebras over a field. This allows us to define its regularity index and its stabilization value. The stabilization value is computed for every case. We study how the regularity index varies as the number of polynomials increases, and use this to give bounds for it.
Pedro Ramírez Moreno (CIMAT)
Connectedness dimension and graphs modulo a parameterAbstract: The connectedness dimension of a variety is a numerical invariant that measures how connected this space is. Given a ring, there is a family of graphs related to the connectedness dimension of its spectrum. In this talk we will discuss the behavior of this dimension and these graphs when we go modulo a parameter. Finally, we will show some consequences related to initial ideals and Gröbner deformations. This talk is based on joint work with Lilia Alanís-López and Luis Núñez-Betancourt.
Josh Pollitz (University of Utah)
Frobenius push forwards and generators for the derived categoryAbstract: By now it is quite classical that one can understand singularities in prime characteristic commutative algebra through properties of the Frobenius endomorphism. The foundational result illustrating this is a celebrated theorem of Kunz characterizing the regularity of a noetherian ring (in prime characteristic) in terms of whether a Frobenius push forward is flat. In this talk, I'll discuss a structural explanation of the theorem of Kunz, that also recovers it, and other theorems of this ilk. Namely, I’ll discuss recent joint work with Ballard, Iyengar, Lank, and Mukhopadhyay where we show that over an F-finite noetherian ring of prime characteristic high enough Frobenius push forwards generate the bounded derived category.
Sylvia Wiegand (UNL)
Auslander-Reiten and Huneke-Wiegand conjectures over quasi-fiber ringsA quasi-fiber ring is a commutative local ring \((R,m,k)\) containing a regular sequence \(x=x_1,\ldots, x_n\) such that \(R/(x)\) has decomposable maximal ideal \(m/(x)\); that is, \(m/(x)=I/(x)\oplus J/(x)\), with nontrivial summands. We give some results concerning homological properties of quasi-fiber rings.
Mahrud Sayrafi (Minnesota)
Short resolutions of the diagonal and a Horrocks-type splitting criterion in Picard rank 2Abstract: In 1964, Horrocks proved that a vector bundle on a projective space splits as a sum of line bundles if and only if it has no intermediate cohomology. Then in 2015, Eisenbud-Erman-Schreyer used the BGG correspondence for products of projective spaces to prove a version of this criterion under an additional hypothesis. This talk is about the key ingredient for proving a Horrocks-type splitting criterion for vector bundles over a smooth projective toric variety X of Picard rank 2: a short resolution of the diagonal sheaf consisting of finite direct sums of line bundles. I'll discuss the construction via a variant of Weyman's "geometric technique," as well as additional properties and applications. This is joint work with Michael Brown.
David Lieberman (UNL)
Linear Simplicity for Some Rings of Differential OperatorsAbstract: In the realm of differential operators over a ring, an exceptionally useful result is Bernstein's Inequality. This inequality puts a lower bound on the dimension of D-modules (where D is the ring of differential operators). One way to go about showing the inequality holds for a particular ring is via linear simplicity. In this talk, we will investigate the property of linear simplicity for the differential operators of some specific singular rings. Time permitting, we will also see some of the useful consequences (besides Bernstein's Inequality) of having the property of linear simplicity for a ring of differential operators.
Matthew Bachmann (UNL)
Examining the higher Herbrand differenceAbstract: Let \(R\) be a codimension \(c\) complete intersection with isolated singularity and let \(M\) and \(N\) be finitely generated \(R\)-modules. We examine a pairing defined by Celikbas and Dao using the modules \(\textrm{Ext}_R^i(M,N)\) called the higher Herbrand difference, \(h_c^R(M,N)\) (a generalization of Buchweitz's notion for \(c=1\)). In particular, we show that \(h_c(M,N)=(-1)^{d+c}h_c(N,M)\) where \(d\) is the Krull dimension of \(R\).
Ray Heitmann (University of Texas at Austin)
Number of generators of perfect idealsAbstract: This talk will explore bounds on the number of generators of perfect ideals \(J\) in regular local rings \( (R,\mathfrak{m}) \). If \(J\) is sufficiently large modulo \(\mathfrak{m}^n \), a bound is established depending only on \(n\) and the projective dimension of \(R/J\). More ambitious conjectures are also introduced with some partial results.
Jack Jeffries (UNL)
Local cohomology of invariant rings in bad characteristicAbstract: Invariant rings of finite groups are well-studied objects and arise in many situations. Lots of great theorems are known about them, but many of the best hold only when the order of the group is invertible. We investigate the top local cohomology module of the ring of invariants — a module that contains lots of good information about the rings — in the bad case where the order of the group may not be invertible. This is based on joint work with Kriti Goel and Anurag K. Singh.
Keller VandeBogert (Notre Dame)
Arithmetic Cobar Complexes and Sheaf CohomologyAbstract: An arithmetic cobar complex is the result of bundling together infinitely many ``arithmetic" Koszul complexes, which are objects arising in the computation of cohomology of certain types of line bundles on flag varieties. Combining all of these complexes has the advantage that this cobar complex and hence its cohomology algebra admit the structure of a multigraded (DG)-algebra. This begs the question: can one present this cohomology algebra explicitly, in every characteristic? The answer to this question is yes, and in this talk we will see the surprising uniformity of the answer. This reduces the problem of computing cohomology of line bundles on flag varieties (a very hard, characteristic-dependent problem) to enumerating the multigraded pieces of a very simple k-algebra. This is joint work with Claudiu Raicu.
Mark Walker (UNL)
The total rank conjecture in characteristic 2
Abstract: This is joint work with Keller VandeBogert. Let R be a local ring of characteristic 2 and dimension d, and assume M is a non-zero \(R\)-module of finite length and finite projective dimension. Then the sum of the ranks of the modules occurring in the minimal free resolution of M is at least \(2^d\). I will give the proof of this in the special (but key) case when R is regular. The proof uses the theory of simplicial modules, which I will introduce.
Roger Wiegand (UNL)
Decompositions of pure projective modules over local ringsAbstract: An \(R\)-module is termed pure projective if it is projective relative to pure-exact sequences. There is a useful structure theorem: A module is pure projective if and only if it is a direct summand of a direct sum of finitely presented modules. Assume now that \(R\) is a Noetherian local ring. Is every pure-projective module actually a direct sum of finitely generated modules? The answer is "no", even when the module is a direct summand of \(M^{\omega}\) for some finitely generated module \(M\). We discuss this and the related question of which modules over the completion of \(R\) are extended from \(R\)-modules. This is joint work with Dolors Herbera (Barcelona) and Pavel Prihoda (Prague).
Alexandra Seceleanu (UNL)
Principal symmetric idealsAbstract: 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 Betti numbers of a principal symmetric ideal? I will give a general answer in this talk.
Nawaj KC (UNL)
The length conjectureAbstract: Suppose \( (R, \mathfrak{m}) \) is a noetherian local ring. The length conjecture of Iyengar-Ma-Walker contends that if \(I \subseteq R\) is an \(\mathfrak{m}\)-primary ideal of finite projective dimension, \(\ell_R(R/I) \geqslant e(R)\), where \(e(R) := e(\mathfrak{m}, R)\) is the Hilbert Samuel multiplicity of the maximal ideal. In this talk, we will introduce this innocuous open problem. We will also present our solution to this problem, joint with Andrew Soto Levins, in the case \(\textrm{dim} (R) =2\). Our new idea is to use a result of Skalit in local intersection theory and a notion of lifting modules.
Jake Kettinger (UNL)
New Perspectives on Geproci-nessAbstract: The geproci property is a recent development in the world of geometry. We define the notion of the geproci set and discuss new developments in the characterization of the geproci property. These developments come from analyzing configurations of points in new settings, such as in positive characteristic and in the non-reduced setting. These developments come with exciting new connections to combinatorics and group theory. Finally, we will discuss natural next questions these developments can lead to, especially with regard to investigating the combinatorial connection.
Kriti Goel (University of Utah)
Hilbert-Kunz multiplicity of powers of an idealAbstract: 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.
Eloísa Grifo (UNL)
Bounding the dimension of cohomological support varietiesAbstract: Given a complex of R-modules M, we can associate to M a projective variety containing homological information about M; this is the cohomological support variety of M. When R is a complete intersection, any projective variety can be realized as the cohomological support of some M. However, when R is Cohen-Macaulay but not a complete intersection, this realizability question has a negative answer; to show this, we give bounds on the dimension of any cohomological support variety over R. We will also discuss a connection between cohomological support varieties and the homotopy Lie algebra of R. This is joint work with Ben Briggs and Josh Pollitz.
Tom Marley (UNL)
Using contracting endomorphisms to detect ring singularitiesAbstract: Let \(R\) be a local ring with maximal ideal \(m\). A endomorphism \(\phi:R\to R\) is called contracting if \(\phi^i(m)\subseteq m^2\) for some \(i>0\). An important example of a contracting endomorphism is the Frobenius endomorphism in the case \(R\) is a ring of prime characteristic. We discuss ways in which actions of such an endomorphism detect when the ring is regular, a complete intersection, or Gorenstein. We review classical results of Kunz and Rodicio, as well as more recent results by Avramov, Hochster, Iyengar, Yao, myself and Brittney Falahola (a recent PhD student), and others. We focus in particular on a recent result of Pinches Dirnfeld which characterizes Gorenstein rings by the action of Frobenius on a dualizing complex for \(R\), answering a question by Falahola and myself.
Levi Heath (UNL)
Enumerative geometry in physics and quantum Serre duality for quasimapsAbstract: Let X be a smooth variety and let Z be a complete intersection in X defined by a section of a vector bundle E over X. Gromov-Witten and quasimap invariants of X are integrals over Kontsevich's moduli space of stable maps to X and Ciocan-Fontanine and Kim's moduli space of stable quasimaps respectively. Originally proposed by Givental, quantum Serre duality refers to a precise relationship between the Gromov-Witten invariants of Z and those of the dual vector bundle \(E^\vee\). In this talk, we motivate the study of Gromov-Witten and quasimap invariants, which have applications in theoretical physics, and present a quantum Serre duality statement for quasimap invariants. We describe how working with quasimaps allows us to obtain a comparison that is simpler, and that also holds in greater generality than previous quantum Serre duality results in Gromov-Witten theory. This is joint work with Mark Shoemaker.
Paolo Mantero (University of Arkansas)
Formulas for symbolic powers of ideals.Abstract: In this talk we provide a couple of formulas to compute symbolic powers of unmixed, generically complete intersection ideals in Cohen-Macaulay rings. For instance, we give a multiplicity-based characterization for the \(m\)th symbolic power \( I^{(m)} \) of an ideal I, a formula to obtain \(I^{(m)}\) as the saturation of \(I^{m}\) with respect to an explicit ideal only depending on \(I\) (and not \(m\)), and effective bounds for the exponent achieving the saturation. We plan to discuss the connection with a conjecture by Eisenbud and Mazur on \(\textrm{ann}(I^{(m)}/I^m)\), which we prove in a generalized form, and a couple of applications of the formulas.
Craig Huneke (University of Virginia)
Title: A monomial licci problemAbstract: This talk is based on ongoing work with Claudia Polini and Bernd Ulrich. An ideal is licci if it is in the linkage class of a complete intersection. We began with a basic question: if I is a licci and square-free monomial ideal in a polynomial ring in n-variables, must the number of generators of I be at most n? This simple question has strong consequences if it has a positive answer. Several positive answers will be given, some of which unexpectedly use Boij-Soderberg theory.
Joshua Rice (Iowa State University)
Numerics of Koszul Algebras and Generic Collections of Lines in Projective SpaceAbstract: We briefly introduce Koszul algebras and discuss some of their interesting homological and numerical properties. We then prove that if M is a generic collection of lines in projective space, then, depending on the size of M, we can guarantee that the coordinate ring is Koszul. This theorem partially generalizes a result of Conca, Trung, and Valla. Furthermore, if the collection M is too large, the coordinate ring is not Koszul; this is achieved using the numerical properties of Koszul algebras.
Javier González Anaya (University of California, Riverside)
Finite generation of symbolic Rees algebras from a geometric perspectiveAbstract: The problem of finite generation of symbolic Rees algebras of space monomial curves has received renewed interest in the past few years because of the essential role it plays in Castravet and Tevelev's proof that the compactified moduli space of genus zero curves with N marked points is not a Mori Dream Space as soon as N is larger than 133. Subsequent work of Karu-Gonzalez and Hausen-Keicher-Laface improved the result to N larger than 9 using toric geometry. In this talk I'll explain how this geometric perspective offers a powerful tool to study the problem of finite generation, how it recovers and expands classical results in the area, and present the main results by Gonzalez, Karu and myself.
Mark Walker (UNL)
BGG from a topological point of viewAbstract: The Berstein-Gel'fand-Gel'fand (BGG) correspondence refers to an equivalence of categories that relates modules over a polynomial ring and modules over an exterior algebra in the same number of variables. In this (mostly expository) talk, I will channel my inner topologist and explain the BGG correspondence from the point of view of topological spaces equipped with continuous torus actions. I'll also discuss the To*** Rank Conjectures, were *** = ral or *** = tot.
Brian Harbourne (UNL)
Algebraic Geometric Concepts (weakly) Motivated by Inverse ScatteringAbstract: Studying inverse scattering problems has led to remarkable advances in science and technology, from computed tomography to the determination of the double helix structure of DNA. Here I will discuss current research which applies this idea to classification problems in algebraic geometry. In inverse scattering problems (ISP), one tries to discern an object's structure from structure in projected or reflected data. We carry this idea over to algebraic geometry and commutative algebra by asking to classify objects based on the structure of projected images. As a main focus, I will discuss work to classify sets of points in projective space whose projection from a general point is a complete intersection in a hyperplane.
2021/2022 academic year
Srikanth Iyengar (University of Utah)
Congruence modules and Wiles defect for local rings over discrete valuation ringsAbstract: Andrew Wiles, as part of his proof of Fermat’s theorem, discovered a criterion for a map between local rings over a fixed discrete valuation ring, and of relative dimension zero, to be an isomorphism of complete intersections. This result has been subsequently generalized by Lenstra, Diamond, Fakhruddin, Khare, and others, for it has applications in number theory, around the problem of modularity lifting. I will present some results from an ongoing joint collaboration with Khare and Manning that extends these results to higher relative dimension. The focus of the talk will be on the commutative algebra aspects but I will try to indicate why these results are of interest to researchers in number theory.
Past talks
Daniel Duarte (Universidad Autónoma de Zacatecas, Mexico)
Nash blowups in positive characteristicAbstract: Given a normal variety over a field of positive characteristic, we show that its Nash blowup is an isomorphism if and only if the variety is non-singular. This result is obtained by combining general properties of a suitable Grassmanian together with recent developments on derivations and differential operators. Joint work with Luis Núñez Betancourt.
Jack Jeffries (UNL)
Are determinantal rings direct summands of polynomial rings?Abstract: Over any infinite field, the generic determinantal rings are known to be fixed subrings of the action of the general linear group on a polynomial ring. Since the general linear group is linearly reductive in characteristic zero, these generic determinantal rings are direct summands of polynomial rings, which explains many of their good properties in this case. In positive characteristic, these determinantal rings have many of the same good properties even though the general linear group is no longer linearly reductive. In this talk we investigate if these determinantal rings continue to be direct summands of polynomial rings in characteristic p>0. We will also encounter some interesting varieties related to linear algebra along the way.
This is joint work with Mel Hochster, Vaibhav Pandey, and Anurag Singh.
Ben Briggs (MSRI)
The higher cotangent modulesAbstract: The cotangent complex is an important but difficult to understand object associated to a ring homomorphism. It connects closely with some more familiar commutative algebra invariants: you can see the module of differentials, the conormal module, and the first Koszul homology as the first few syzygies inside the cotangent complex. In general, these syzygies are known as the cotangent modules.
Quillen conjectured (for maps of finite flat dimension) that the cotangent complex can only be bounded for complete intersection homomorphisms. This was proven by Avramov in 1999. I will explain how to get a new proof (of a stronger result) by paying attention to the cotangent modules. This is all joint work with Srikanth Iyengar.
Sylvia Wiegand (UNL)
Ideals in a local ring under small perturbationsAbstract: Consider the following properties for a commutative Noetherian local ring \( (R,\mathfrak m,k) \):
Luís Duarte (University of Genoa)
Ideals in a local ring under small perturbationsAbstract: Let I be an ideal of a Noetherian local ring R. We study how properties of I change for small perturbations, that is, for ideals J that are the same as I modulo a large power of the maximal ideal. In particular, assuming that J has the same Hilbert function as I, we show that the Betti numbers of J coincide with those of I. We also compare the local cohomology modules of R/J with those of R/I.
Shah Roshan Zamir (UNL)
Nagata's theorem on the ring of invariantsAbstract: 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. Nagata's theorem gives a positive answer for geometrically reductive groups, a large class of groups, acting on finitely generated K-algebras. In this talk, we will prove the graded case of Nagata's theorem. It is notable that in 1959 Nagata also provided a negative answer to Hilbert's 14th problem for arbitrary groups.
David Lieberman (UNL)
The Dimension of D-modules and Bernstein's InequalityAbstract: Given a module M over an A-algebra R, one can construct the ring of differential operators on M over R. This ring is a subset of the A-linear maps from M to M, and is a rich source of study for commutative algebra. In particular much work has been done in studying modules over the ring of differential operators, which we call D-modules. In this talk, we will investigate the notion of dimension for D-modules and prove a hallmark property on the lower bound of dimension. This somewhat surprising result is known as Bernstein's inequality, which we will prove for D-modules over the ring of differential operators on a polynomial ring in characteristic zero.
Uli Walther (Purdue University)
Lyubeznik and Cech-de Rham numbersAbstract: If Y is an affine variety inside \(\mathbb{C}^n\) cut out by the ideal \(I\) inside, and \(m\) a distinguished maximal ideal of, \(\mathbb{C}[x_1,...,x_n]\), one can attach two sets of numbers to them, either by applying the de Rham functor the D-module \(H^t_I(R)\), or the D-module restriction functor for the inclusion \(\textrm{Spec}(R/m) \hookrightarrow \textrm{Spec} (R)\). It turns out that these numbers are in fact functions of Y and not of the embedding into an affine space. In the talk we discuss known facts as well as some recent insights on these double arrays of numbers. This will include some general vanishing results, as well as a discussion on when the associated spectral sequence for which these arrays are the \(E_2\)-page, collapses.
Andrew Soto-Levins (UNL)
A Rigidity Theorem for ExtAbstract: The goal of this talk is to present the following theorem: if R is an unramified hypersurface, if M and N are finitely generated R modules, and if \( \operatorname{Ext}_{R}^{n}(M, N) = 0 \) for some \( n \leqslant \operatorname{grade} M \), then \( \operatorname{Ext}_{R}^{i}(M, N) = 0 \) for \( i \leqslant n\). A corollary of this says that \( \operatorname{Ext}_{R}^{i}(M, M) \neq 0\) for \(i \leqslant \operatorname{grade} M \). This gives a partial answer to a question of Jorgensen: if \( (R, m, k) \) is a complete intersection and if M is a nonzero finitely generated module of finite projective dimension, then must \( \operatorname{Ext}_{R}^{n}(M, M) \) be nonzero for \( 0 \leqslant n \leqslant \operatorname{pd}_{R}(M) \)?
Claudia Miller (Syracuse University)
Torsion in exterior powers of differentials over complete intersection ringsAbstract: In this talk, after a short review of the definition of and facts about Kaehler differentials, I will give some history behind the classic Lipman-Zariski Conjecture and the generalized Lipman-Zariski questions of Graf. Then I’ll give some results on the torsion and cotorsion of exterior powers of the module of Kaehler differentials over complete intersection rings, and say how these are used to prove a generalized Lipman-Zariski result under certain conditions. This is joint work with Sophia Vassiliadou.
Alexandra Seceleanu (UNL)
Axial constants of homogeneous idealsAbstract: In commutative algebra, generic initial ideals are monomial ideals used for estimating important homological features of arbitrary ideals of a polynomial ring. In this talk we introduce new invariants termed axial constants, which can be read off the generic initial ideal, and we explain how the axial constants relate to better known algebraic and homological invariants. This is joint work with Michael DeBellevue, Shah Roshan Zamir, and the members of the Polymath Jr group on computational algebra.
Brian Harbourne (UNL)
The concept of geproci subsets of P3: a timelineAbstract: Interest in geproci sets grew out of work on unexpected hypersurfaces. We define the notion of a geproci set, we give examples and we discuss some recent results, all in the context of a timeline of relevant events.
Eloísa Grifo (UNL)
Macaulay2Abstract: We will talk about Macaulay2: what is it, what one can do with it, and how one can learn how to use it.
Jack Jeffries (UNL)
A Jacobian Criterion in Mixed CharactersticAbstract: The classical Jacobian criterion is an important tool for finding singular points on a variety over a (perfect) field. How can we find the singular locus over the p-adics or over the integers? In this talk, I'll discuss a new analogue of the Jacobian criterion that gives a simple description of the singular locus in this setting. This criterion uses a curious notion of differentiation by a prime number called p-derivations. If time permits, we will also discuss an extension of the theory of Kahler differentials to this mixed characteristic setting. This is based on joint work with Melvin Hochster.
Michael K. Brown (Auburn University)
Minimal free resolutions of differential modulesAbstract: A differential module is a pair \( (M, d) \), where \( M \) is a module, and d is an endomorphism of \( M \) that squares to \(0\). This is a generalization of a familiar notion: when \(M\) is graded and d has degree \(-1\), \( (M, d) \) is a chain complex. In this talk, I'll discuss a theory of minimal free resolutions of differential modules over local rings. This is joint work with Daniel Erman.
Josh Pollitz (University of Utah)
Cohomological support in local algebraAbstract: Cohomological supports have been integral in revealing structural information in commutative algebra. They were imported from modular representation theory to local algebra by Avramov, and were put on centerstage by Avramov and Buchweitz in 2000 during their investigations of cohomology modules over complete intersection rings. In the past twenty years this theory has been further developed, extended and applied by Avramov-Iyengar, Burke-Walker, Jorgensen, P-, and many others. In this talk, classical and recent applications of cohomological support will be surveyed and I will discuss two new support theories that have been developed in two separate collaborations: one joint with Briggs and Grifo, and the other joint with Briggs and McCormick.
Roger Wiegand (UNL)
Vanishing of Tor over quasi-fiber product ringsAbstract: A fiber product ring is a local ring \( (R,\mathfrak{m},k) \) whose maximal ideal \(\mathfrak{m} \) decomposes as a direct sum of two non-zero ideals. (Local rings are always assumed to be commutative and Noetherian.) Equivalently, \(R\) is isomorphic to a fiber product \(S\times_k T\), where \( S \) and \( R \) are local rings, both different from \(k\). More generally, a quasi-fiber product ring is a local ring \(R\) such that \(R/(x_1, \ldots, x_r)\) is a fiber product ring for some regular sequence \( (x_1, ..., x_r) \) and some \(r \geqslant 0\). We study vanishing of \(\textrm{Tor}\) and \(\textrm{Ext}\) over these rings, with particular focus on the Auslander-Reiten and Huneke-Wiegand Conjectures. This is joint work with Thiago Freitas, Victor Jorge Pérez, and Sylvia Wiegand.
Michael DeBellevue (UNL)
Graded Deviations and the Koszul PropertyAbstract: The graded deviations \(\varepsilon_{ij}(R)\) of a graded ring \(R\) record the vector space dimensions of the graded pieces of a certain Lie algebra attached to the minimal resolution of the quotient of \(R\) by its homogeneous maximal ideal. Vanishing of deviations encodes properties of the ring: for example, \(\varepsilon_{ij}(R)= 0\) for \(i \geqslant 3\) if and only if \(R\) is complete intersection and, provided \(R\) is standard graded, \(\varepsilon_{ij}(R)\) whenever \(I\) is not equal to \(j\) implies R is Koszul. We extend this fact by showing that if \( \varepsilon_{ij}(R)=0 \) whenever \(j\) and \(i \geqslant 3\), then \( R \) is a quotient of a Koszul algebra by a regular sequence. This answers a conjecture by Ferraro.
Vaibhav Pandey (University of Utah)
Are determinantal rings direct summands of polynomial rings?Abstract: Over an infinite field, the generic determinantal rings are known to be fixed subrings of the action of the general linear group on a polynomial ring. Since the general linear group is linearly reductive in characteristic zero, it follows from a theorem of Hochster and Roberts that these generic determinantal rings are direct summands of polynomial rings (in characteristic zero). In this talk we investigate if these determinantal rings continue to be direct summands of the polynomial rings in which they naturally embed into by the above group action in characteristic p>0. Note that the general linear group is not linearly reductive in characteristic p>0! This is joint work with Mel Hochster, Jack Jeffries, and Anurag Singh.
Tom Marley (UNL)
Gorenstein projective dimensions and levels of complexesAbstract: We review (or introduce for some/many) the concept of the Gorenstein projective dimension of a module, and how it can be used to characterize Gorenstein local rings. We then show how this dimension can be generalized for complexes. We then define a related concept, called the Gorenstein level or G-level of a complex, and show how that is related to the Gorenstein projective dimension of a complex.
Eloísa Grifo (UNL)
A survey of Harbourne's ConjectureAbstract: Harbourne's conjecture on the containment problem for symbolic and ordinary powers of ideals is not true in its original form, but it has sparked a lot of different research avenues. We will discuss some of the known counterexamples but mostly focus on the different variations of the conjecture that are true or still open.
Mark Walker (UNL)
On the cone of Betti tables for a singular ringAbstract: This is joint work with Srikanth Iyengar and Linquan Ma. Let k be a field and R a standard graded k-algebra. When R is a polynomial ring, "Boij-Soderberg Theory" (developed by Boij, Eisenbud, Erman, Schreyer, Soderberg, and others) gives a description of the rational cone spanned by the Betti tables of finitely generated graded R-modules. We give extensions of these results to other graded rings. For instance, we prove the following: When the characteristic of k is prime and R is any Cohen-Macaulay standard graded k-algebra, then the cone of Betti tables of graded R-modules of finite length and finite projective dimension coincides with that for a polynomial ring of the same dimension. We also have results for complexes of graded modules, which include the case when R is not Cohen-Macaulay. Eisenbud and Erman have previously established results such as these under the assumption that the associated projective scheme Proj(R) admits an Ulrich sheaf. The central technique we use is the notion of a lim Ulrich sequence of graded R-modules. In prime characteristic, such sequences exist by a theorem of Ma.