## 2021/2022 academic year

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.

Claudia Miller (Syracuse University)

TBAAbstract: TBA

note the unusual day of the week, time, and location

Uli Walther (Purdue University)

TBAAbstract: TBA

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 \( 0 \leqslant n \leqslant \operatorname{pd}_{R}(M) \)?

David Lieberman (UNL)

TBAAbstract: TBA

## Past talks

**on zoom**

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.