## 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 \( (R,\mathfrak m,k) \):

*quasi-fiber product rings*— rings for which there exists a regular sequence \(\underline x\) in \(\mathfrak m\) such that \(\mathfrak m/(\underline x) \) decomposes into a nontrivial direct sum of ideals of \( R/(\underline x) \). We show that quasi-fiber product rings satisfy a sharpened form of (ARC) and we make some observations related to (HWC).

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.

**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.