Algebra Seminar Archive

Schedule of Talks Fall 2024:

• August 30th: Henry Potts-Rubin (Syracuse University), Resolving the Module of Derivations of an n x (n+1) Determinantal Ring

  • Abstract: We use the construction of the relative bar resolution via differential graded structures to obtain the minimal graded free resolution of Der_{R| k}, where R is a determinantal ring defined by the maximal minors of an n x (n+1) generic matrix and k is its coefficient field.  Along the way, we compute an explicit action of the Hilbert-Burch differential graded algebra on a differential graded module resolving the cokernel of the Jacobian matrix whose kernel is Der_{R| k}.  As a consequence of the minimality of the resulting relative bar resolution, we get a minimal generating set for Der_{R| k} as an R-module, which, while already known, has not been obtained via our methods. 

• September 6th: Shah Roshan-Zamir (University of Nebraska-Lincoln), Interpolation problem in weighted projective space

• Abstract: Over an algebraically closed field, the double point interpolation problem asks for the vector space dimension of the projective hypersurfaces of degree d singular at a given set of points. After being open for 90 years, a series of papers by J. Alexander and A. Hirschowitz in 1992–1995 settled this question in what is referred to as the Alexander-Hirschowitz theorem. In this paper we primarily use commutative algebra to lay the groundwork necessary to prove analogous statements in the weighted projective space, a natural generalization of the projective space. We prove the Hilbert function of general simple points in any n-dimensional weighted projective space exhibits the expected behavior. We also introduce an inductive procedure for weighted projective space, similar to that originally due to A. Terracini from 1915, to demonstrate an example of a weighted projective plane where the analogue of the Alexander-Hirschowitz theorem holds without exceptions and prove our example is the only such plane. Furthermore, Terracini’s lemma regarding secant varieties is adapted to give an interpolation bound for an infinite family of weighted projective planes.

• September 20th: Gabe Sosa (Colgate College), Algebraic Invariants of Rees and multi-Rees algebras of principal (pure) Lexsegment ideals

  • Abstract: Using tools from computational and combinatorial commutative algebra such as Gröbner basis, initial ideals, the Stanley Reisner ideal and the Alexander dual, we determine the exact values of algebraic invariants such as the regularity, the reduction number and the analytic spread of the special fiber, and associated ideal/module, of a Rees (or multi-Rees) algebra of a direct sum of copies of a fixed principal (pure) Lexsegment ideal. Our results expand on bounds that Jonathan Montaño had determined for the reduction number of a principal Lexsegment ideal in his 2015 thesis and have implications regarding the core of monomial ideals. Our methods also allow us to characterize when these Rees algebras are Gorenstein. This is joint work with Alessandra Costantini and Kuei-Nuan Lin.

• September 27th: Marty Bishop (Northwestern University), How Chow rings connect algebra and geometry

  • Abstract: One of the central objects in algebraic geometry is the Chow ring, an algebraic object (a ring) which encodes the geometric data of how subspaces of a given space intersect each other. We will discuss the history and motivation of Chow rings, as well as give several techniques for their computation. We will conclude with discussing the Chow ring of the moduli stack of elliptic curves.

• October 11th: Kaiyue He (Syracuse University), On the Behavior of Syzygies and Invariants over Artinian Rings

  • Abstract: We definte a new invariant $\gamma_I(M)$ over an Artinian ring and investigate its relations with the betti numbers of the module M. Several fundamental properties of $\gamma_I(M)$ are established, including conditions for vanishing and inequalities related to Betti numbers. We further explore the behavior of $\gamma_I(M)$ in short exact sequences and provide bounds for its value in relation to the syzygy modules of $M$. Additionally, we prove a quadratic relation involving $\gamma_I(M)$ under vanishing Tor conditions, leading to a new vanishing result inspired by Huneke, Sega, and Vraciu’s work on Ext and Tor over Cohen-Macaulay local rings. Finally, we examine the implications for modules with finite projective dimension, highlighting scenarios where modules must be free or possess nontrivial syzygies. 

• October 18th: Michael DeBellevue (Syracuse University), Asymptotic Structure of Syzygies and Ideals of Minors over (Quasi)-Fiber Product Rings

  • Abstract: Over singular rings, there will be infinite minimal resolutions.  Certain concrete questions like which entries appear in differential matrices and which modules occur as syzygies are difficult to answer in general.  Fiber product are, intuitively, the smallest local ring built from two other local rings which have a common residue field k.  Recent research has shown that such rings enjoy a number of properties, such as the Auslander-Reiten conjecture.  Using an explicit resolution due to Moore, we show that for a module M over any fiber product ring, the ideals generated by minors of differential matrices appearing in the minimal free resolution of M must asymptotically be powers of the maximal ideal (outside of one exceptional case).  We also provide a description of the syzygies of M over the fiber product in terms of syzygies of M and k over the factors of the fiber product.  Time permitting, we will discuss how these results can be extended to quasi fiber product rings.  This work is joint with Souvik Dey.

• October 25th:  Michael DeBellevue (Syracuse University), Simplicial Structures in Macaulay2: Algebra resolutions and Andre-Quillen Cohomology, and the Dold-Kan Correspondence

  • Abstract: Simplicial structures provide a general way of computing homotopy information and possess many features which are desirable from a theoretical perspective. In commutative algebra, two prominent applications are the simplicial algebras used to compute derived functors of the derivations functor, and simplicial modules, which can be used to extend functors from ordinary modules to complexes in a way that preserves homotopy equivalence.  Despite their theoretical benefits, the sizes of the objects involved make computation especially challenging.  In this talk I will discuss recent work on implementing simplicial algebras and modules in Macaulay2, including the upcoming package "SimplicialModules".

• November 8th: Courtney Gibbons (Hamilton College),  Toward understanding semigroup properties under the tensor product(s)

  • Abstract: In this talk, I will present some results that the late Nick Baeth and I proved about tensor products of [your favorite adjective here] semigroups. I will also outline Nick's grand vision and our intended future directions for those interested in carrying on in Nick's footsteps. Our journey start with the results on tensor products due to Fulp, Grillet, and Head from the 1960s (groovy!) and maybe ends with... you?!

• November 15th: Selvi Kara (Bryn Mawr College), Degree of h-polynomials of edge ideals

  • Abstract: This talk will focus on the degree of h-polynomials of edge ideals for finite simple graphs, focusing on combinatorial formulas for various fundamental graph classes, including paths, cycles, and bipartite graphs. To the best of our knowledge, this work represents the first effort to interpret this algebraic invariant combinatorially. Moreover, we provide a characterization of all connected graphs where the sum of the Castelnuovo-Mumford regularity and the degree of the h-polynomial of an edge ideal attains its maximum, which is the number of vertices in the graph.

• November 22nd: Hugh Geller (CNA Corporation), Minimal Free Resolutions of Neural Codes

• • Abstract:  In their 2013 publication, Curto, Itskov, Jeffs, and Youngs introduced algebraic geometry to neuroscience. They did this by viewing neural codes (a collection of strings of 0’s and 1’s representing neuron firing) as varieties over F_2. From there, given a neural code C, they define the neural ring of a code to be the coordinate ring F_2[C] and define the neural ideal to be a particular subideal of the vanishing ideal I(C). Their work was later translated to the context of monomial ideals by Güntürkün, Jeffries, and Sun via the use of polarization.

In this talk, we discuss preliminary results classifying a specific family of neural codes known as Inductively Pierced Codes. Our work utilizes results on the Betti numbers of monomial ideals and fiber products. This is joint work with Rebecca R.G. and Nora Youngs.

Schedule of Talks Spring 2024:

• February 2nd: Steven Diaz (Syracuse University), Eigenvalues and Eigenvectors of Tensors

  • Abstract: This is joint work with Adam Lutoborski. We present a possible extension of the concepts of eigenvalues and eigenvectors from matrices to tensors of higher order. Our goal is to find an extension that preserves as much as possible the concept of diagonalization. We present what can be considered to be an extension of the theorem that a real symmetric matrix can be orthogonally diagonalized.

• March 15th: Luigi Ferraro (University of Texas Rio Grande Valley), Trimming five generated Gorenstein ideals

  • Abstract: Let R be a regular local ring of dimension 3 with maximal ideal m. Let I be a Gorenstein ideal of R of grade 3. Buchsbaum and Eisenbud proved that there is a skew-symmetric matrix of odd size such that I is generated by the sub-maximal pfaffians of this matrix. Let I be the ideal obtained by multiplying some of the pfaffian generators of I by m; we say that J is a trimming of I. In a previous work, A. Hardesty and I constructed an explicit free resolution of R/J and computed a DG algebra structure on this resolution. We used the products on this resolution to study the Tor algebra of such trimmed ideals. Missing from our result was the case where I is five generated. In this talk, which is based on a joint work with F. Moore, we will address this case after covering the necessary background information.

• March 29th: Janina Letz (Bielefeld University/UCLA), Generation time for biexact functors and Koszul objects in triangulated categories

  • Abstract: The derived category of modules over a commutative ring captures many properties of the ring. One approach is to study its triangulated structure through finite building. An object X finitely builds an object Y, if Y can be obtained from X by taking cones, suspensions and retracts. The X-level measures the number of cones required in this process. I will explain the behavior of level with respect to tensor products and other biexact functors. This results makes use of the fact that the derived category is an enhanced triangulated category. I will further present applications to Koszul objects, which generalize Koszul complexes. This is joint work with Marc Stephan.

• April 5th: Ben Briggs (University of Copenhagen/SLMath), Koszul duality tricks for (seemingly) non-Koszul rings

  • Abstract: This is a talk about a relative version of Koszul duality in commutative algebra, and how it can tell us about the asymptotic homological algebra of (seemingly) non-Koszul local rings. It’s all joint work with James Cameron Janina Letz and Josh Pollitz.

A homomorphism f of commutative local rings, say S to R, has a derived fibre F (a differential graded algebra over the residue field k of R) and we say that f is Koszul if F is formal and its homology H(F) (the Tor algebra of R and k over S) is a Koszul algebra in the classical sense. I’ll explain why this is a very good definition and how it is satisfied by many many examples. The main application is the construction of explicit free resolutions over R in the presence of a Koszul homomorphism. This construction simultaneously generalises the resolutions of Priddy over a Koszul algebra, the resolutions of Shamash and Eisenbud over a complete intersection ring, and the bar resolutions of Iyengar and Burke over a Golod ring.

If you think you’ve seen similar talks before - don’t be fooled. I’ll try to give a different perspective this time.

• April 12th: Ritvik Ramkumar (Cornell University), Cartwright-Sturmfels ideals and their moduli

  • Abstract: Cartwright-Sturmfels ideals, CS-ideals for short, are multigraded ideals whose generic initial ideals are radical. First studied by Cartwright and Sturmfels, some examples include the ideals of maximal minors of a matrix of linear forms, binomial edge ideals, closure of linear spaces, and multiview ideals. In this talk, I will discuss the geometry of CS-ideals inside the multigraded Hilbert scheme, with a particular focus on bigraded CS-ideals. This is joint work with Alessio Sammartano.

• April 26th: Hamid Rahmati (Syracuse University), Comparison of the Bar and Eagon Resolutions

  • Abstract: I will begin by introducing the constructions of the classical bar and Eagon resolutions and then discuss their relationship. This is joint work with Ben Briggs, Claudia Miller and Zheng Yang. 

Schedule of Talks Fall 2023:

• September 8th: Josh Pollitz (Syracuse University), Frobenius pushforwards and generators for the derived category

  • Abstract: By now it is quite classical that one can understand singularities in prime characteristic commutative algebra and algebraic geometry 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.

• September 15th: Michael DeBellevue (Syracuse University),  k-Summands of Syzygies of Burch Rings via the Bar Resolution

  • Abstract: In recent work, Dao and Eisenbud have defined the notion of a Burch ring. They have shown that for any module over a Burch ring of depth zero, the n'th syzygy contains direct summands of the residue field for all n>6. In joint work with Claudia Miller, we investigated how this behavior is explained by the bar resolution. I will define Burch rings and the bar resolution, and show how it can be used to describe some cycles which generate k-summands of syzygies of an R-module M.  When M is a Golod module, we show that the number of these elements grows exponentially as the homological degree increases.

• September 29th: Richard Bartels (Syracuse University), Generalized Loewy length of Cohen-Macaulay local and graded rings

  • Abstract: In his 1994 paper, Ding proved that for a Gorenstein local ring (R,m,k) with Cohen-Macaulay associated graded ring and infinite residue field, the generalized Loewy length gll(R) and index of R are equal. However, if k is finite, equality may not hold. In this talk, I will prove a relation between gll(R) and index(R) that generalizes Ding’s theorem for one-dimensional Cohen-Macaulay local rings with finite index. In particular, this generalization applies when the residue field is finite or infinite. I will then show that for certain one-dimensional hypersurfaces, this relation gives a formula for the generalized Loewy length. Finally, I will introduce a graded version of the generalized Loewy length, and determine its value for numerical semigroup rings.

• October 6th: No seminar-Fall Break Weekend

• October 13th: James Cameron (University of Utah), Koszul ring maps and transferring resolutions in local algebra

  • Abstract: I will discuss what it means for a map of commutative local rings to be Koszul, and how a Koszul ring map allows homological information to be transferred from the source to the target. This gives, in many cases, constructions of “universal resolutions” of modules over a non-regular local ring in terms of a finite amount of data. I will discuss these universal resolutions, and how the construction of these resolutions generalizes other transfer-of-resolutions results, such as the Eisenbud-Shamash construction. A key ingredient in these results are A-infinity algebra techniques, and I will give a brief introduction to A-infinity algebras and their role as a helpful generalization of dg-algebras. All this work is joint with Benjamin Briggs, Janina Letz, and Josh Pollitz.

• October 27th: Gordana Todorov (Northeastern University), Higher Auslander Algebras

  • Abstract: A celebrated theorem of Maurice Auslander about artin algebras describes the correspondence between {algebras A of representation finite} and {algebras B, with gl.dim.B ≤ 2 ≤ dom.dim.B} which are now called Auslander algebras. Higher Auslander algebras were introduced by O. Iyama as algebras C, with {gl.dim.C ≤ k ≤ dom.dim.C} and it was also shown that there is correspondence between {higher representation finite algebras} and {higher Auslander algebras} (up to certain equivalences). A large part of my talk is on recent work of Emre Sen. In addition, Emre Sen, Shijie Zhu and I obtained another method of constructing higher Auslander algebras in the family of Nakayama algebras. Using this method we get a complete characterization of higher Auslander Nakayama algebras.

• November 3rd: Dave Jorgensen (University of Texas-Arlington), Variations on a classical dimension inequality

  • Abstract: For a pair of finitely generated modules M and N over a codimension c complete intersection ring R with M⊗N having finite length over R, we pay special attention to the inequality dim(M)+dim(N) ≤ dim(R) +c. In particular, we develop an extension of Hochster's theta invariant whose nonvanishing detects equality. In addition, we consider a parallel theory where dimension and codimension are replaced by depth and complexity, respectively.  This is joint work with Petter Bergh and Peder Thompson.

• November 10th:  Des Martin (Syracuse University), On the splitting of the 2nd exterior power of the conormal module from Tor 

  • Abstract: Let R be a local Noetherian ring and I an ideal of R. In “The Syzygies of the Conormal Module,” Simis and Vasconcelos give an embedding map, and a somewhat cryptic remark that this map causes a splitting of the 2ndexterior power of the conormal module from Tor_1(I,R/I). In this talk we utilize differential graded algebra resolutions to define an explicit splitting map and show that if I/I^2 is free (or has enough free summands), then the 2nd exterior power of the conormal module (or its free summands) will split from Tor_1(I,R/I). 

• November 17th: Torkil Stai (NTNU), Perhaps Nakayama Algebra

  • Abstract: After semisimple ones, Nakayama algebras are the best understood algebras. But while their representation theory is under control, basic questions regarding their homological nature remain unanswered. We will look at both these features, insisting on accessibility for non-experts. Time permitting, we will also mention recent work with Fosse and Oppermann.

• November 24th: No seminar-Thanksgiving Break

• December 1st: Alexandra Seceleanu (University of Nebraska-Lincoln), Principal symmetric ideals

  • Abstract: A ubiquitous theme in mathematics is that general members in a family of mathematical objects have nice properties. We focus on the family of principal symmetric ideals, that is, ideals generated by the orbit of a homogeneous polynomial under the action of the symmetric group. In joint work with Megumi Harada and Liana Sega we determine the minimal free resolutions for a general member of this family, as well as other notable properties it satisfies.

• December 8th: David Lieberman (University of Nebraska-Lincoln), Making Sandwiches of Functional Equations: Bernstein's inequality, Bernstein-Sato polynomials, and Sandwich Bernstein Sato Polynomials

  • Abstract: For a polynomial ring over a field K, the ring of differential operators is known as the Weyl algebra: the non-commutative K-algebra whose generators are the "multiply by a variable" maps and the partial derivatives with respect to each variable. In this setting a powerful result is Bernstein's Inequality, which puts bounds on the dimension of modules over the Weyl Algebra. Another closely related object of study is the Bernstein-Sato polynomial for an element of the polynomial ring. In this talk, I will share some recent developments in proving Bernstein's Inequality for the ring of differential operators over some singular rings, as well as a "Sandwich" version of Bernstein-Sato polynomials that gives a new method for showing the inequality in novel settings. This work is a collaboration with my advisor Jack Jeffries.