Syracuse Algebra Seminar

Schedule of Talks Spring 2025:

• January 31: Jen Biermann (Hobart and William Smith Colleges), Algebraic invariants of weighted oriented graphs

  • Abstract: Monomial ideals associated to (unweighted, unoriented) graphs have been studied extensively since the introduction of edge ideals by Villarreal in 1990.  More recently, motivated by applications in coding theory, interest has arisen in a generalization of edge ideals to weighted, oriented graphs.  In this talk I will review basic definitions and what is known about the regularity of the edge ideals of unweighted, unoriented graphs before discussing the regularity of edge ideals of weighted, oriented paths and cycles.

• February 7: Des Martin (Syracuse University), An introduction to DG ideals

  • Abstract:  Let G be a tree and I_G the edge ideal of G. In this talk we use discrete Morse theory to construct the minimal free resolution of k[x_1,.., x_n]/I_G from the Taylor resolution. We then explore when the minimal free resolution obtained from this process inherits the differential graded (dg) algebra structure from the Taylor resolution. In joint work with Hugh Geller and Henry Potts-Rubin.

• February 14: Henry Potts-Rubin (Syracuse University), DG-sensitive pruning for squarefree monomial ideals

  • Abstract: Given a differential graded (dg) algebra resolution F of k[x_1,...,x_n]/I, what kinds of processes can we apply to F to obtain a new resolution (possibly of a different module, possibly over a different ring) that also admits the structure of a dg algebra?  In this talk, we describe one such process, a "pruning process" introduced by Boocher in 2012, which we prove is "DG-sensitive" when I is a squarefree monomial ideal.  In the language of combinatorics, our result states that when the minimal free resolution of the facet ideal of a simplicial complex ∆ admits the structure of a dg algebra, then so does the minimal free resolution of the facet ideal of each facet-induced subcomplex of ∆ (over the smaller polynomial ring).  We will also discuss the existence of a dg algebra structure on the minimal free resolution of the edge ideal of a diameter-four tree, which we combine with pruning and results discussed in the previous week’s seminar to completely classify the trees and cycles with edge ideal minimally resolved by dg algebras.  This is joint work with Hugh Geller and Des Martin.

• February 21: Nawaj KC (University of Nebraska-Lincoln), Loewy lengths of modules of finite projective dimension

  • Abstract: If M is a module of finite length over a local Noetherian ring (R, m), its Loewy length is the smallest non-negative integer i such that m^iM = 0. I will sketch a proof of the following result: if R is a strict Cohen-Macaulay ring, the Loewy length of a module of finite projective dimension is at least as big as the Loewy length of a quotient by sufficiently general linear system of parameters on R. This is joint work with Josh Pollitz. 

• February 28: Liana Șega (University of Missouri-Kansas City), General symmetric ideals

  • Abstract: Given a set of general homogeneous polynomials, we consider the ideal generated by all polynomials obtained from the original ones by permuting the variables. We describe explicitly the Hilbert function and the betti numbers of these ideals. We also address asymptotic stability problems, by describing the dependence of the invariants on the number of variables, when the given polynomials are embedded in polynomial rings with larger number of variables. This is joint work with Alexandra Seceleanu.

• March 7: Eloísa Grifo (University of Nebraska-Lincoln), Cohomological support varieties and monomial ideals

  • Abstract: Given a ring R and an R-module M, the cohomological support variety of M is a geometric object that encodes homological information about M. Even restricting to the case when M=R, we can obtain information about the singularities of R, such as whether R is complete intersection. In this talk, we will consider the problem of whether R has an embedded deformation, and how cohomological support varieties might be used to detect this property when R is defined by monomials. This is joint work with Ben Briggs and Josh Pollitz.

• March 21: Keller VandeBogert (Fields Institute), Stable cohomology and Koszul duality

  • Abstract: The cohomology of certain families of vector bundles on projective space exhibit a striking stability phenomenon, the proof of which was motivated by similar phenomena for line bundles on flag varieties. In this talk, I'll talk about how this "stable" cohomology is secretly computing derived invariants on the category of polynomial functors, a connection which we can exploit to prove a conjectured representation stability phenomenon for line bundle cohomology. On the other hand, this bridge allows us to apply commutative-algebraic techniques to tackle (and solve) problems in polynomial functor theory. This is based on joint work with Claudiu Raicu.

• March 28:  Rachel Diethorn (Oberlin College),  Poincare series over almost complete intersection rings

  • Abstract: We study Poincare series of modules over almost complete intersection rings defined by n+1 quadrics in n variables, starting first with a particular example and then moving to the generic case.  We also study the Gorenstein rings that are linked to such almost complete intersections.  In particular we show that both such rings are Golod and provide explicit formulas for the Poincare series of their residue fields.

• April 4: Ryan Watson (University of Nebraska-Lincoln), Cohomological Support Varieties Along Surjections

  • Abstract: Cohomological support varieties give a way to assign a conical affine variety to a finitely generated module over a local ring. This theory was first introduced into commutative algebra by Luchezar Avramov for complete intersections in 1989 and by the work of many authors over the past couple decades, has been extended to encompass all local rings. The cohomological support variety of a module holds important homological information about the module (and ring) taken and has proven to be an important tool in local commutative algebra. In this talk I will go over what cohomological support varieties are, why they're useful, and a recent result of mine concerning how these varieties behave when restricting scalars along a surjective map.

• April 11: Oana Veliche (Northeastern University), On a construction of a minimal free resolutions of the residue field of a codepth 3 local ring

  • Abstract: In a paper with Van Nguyen we gave a description of a truncated minimal free resolution, up to homological degree five, of the residue field k over any local Noetherian ring (R,m,k)  of any codepth. This description relies on the graded commutative algebra structure of the Koszul homology A=H(K), where K is the Koszul complex of the maximal ideal. The free resolution is given as building blocks in terms of the Koszul complex K. In a subsequent paper we constructed the entire free resolution of the residue field over a complete intersection ring, using an iterated mapping cone construction. 

In my talk, I will discuss how the mapping cone construction in general can yield a minimal free resolution of k over other classes of rings. In particular, for local ring (R,m,k) of class T of codepth 3 (that is part of the Weyman and Avramov-Kustin-Miller classification), we construct a graded resolution of k over A and then from it we obtain the minimal free resolution of k over R. 

• April 18, Talk 1: Prashanth Sridhar (Auburn University), Differential Graded Noncommutative Geometry

  • Abstract: Pioneering work of Artin-Tate-Van den Bergh-Zhang extends important aspects of projective geometry to the noncommutative (nc) setting. In particular, the derived category of a nc scheme shares many features with the derived category of a classical one. In this talk, I'll discuss an ongoing program to extend classical and modern results in the theory of nc projective geometry to nc spaces associated to differential graded algebras. The focus will be on its applications to projective varieties: for instance, this approach results in a generalization of a celebrated theorem of Orlov concerning the derived category of a projective complete intersection. 

• April 18, Talk 2: Justin Lyle (Auburn University), Vanishing of Tor and Depth of Tensor Products Over Cohen-Macaulay Rings

  • Abstract: Let R be a commutative Noetherian local ring and let M,N be finitely generated-modules. We say R satisfies the condition (dep) if the depth formula depth_R(M \otimes_R N)+depth(R)=depth_R(M)+depth_R(N) holds whenever Tor^R_i(M,N)=0 for all i>0. Recent work of Kimura-Lyle-Otake-Takahashi shows this condition need not hold in general, and in fact characterizes the so-called AB property for Gorenstein rings of positive dimension. Replacing the equality in the above formula by either the inequality ≥ or the inequality ≤ the (dep) condition bifurcates into two conditions which we call (ldep) and (rdep), respectively. We show when R is Cohen-Macaulay that (ldep) is tied to the so-called uniform Auslander condition, and that (ldep) implies (rdep) for Gorenstein rings of positive dimension. We also describe how these conditions behave under standard operations in commutative algebra, such as localization, completion, and modding out by a regular sequence. This talk is based on joint work with Kaito Kimura and Andrew Soto-Levins.