## A Zoom Special Session on DG Methods in Commutative Algebra and Representation Theory May 2nd and 3rd, 2020

Differential graded techniques have proven to be powerful and clarifying in both commutative algebra and representation theory. Authors in these fields have been exploiting the higher structure hidden in DG algebras to great effect. Together with the ubiquity of these structures, this has resulted in an explosion of interest in these tools. We hope that this Zoom special session, in lieu of a physical meeting, will serve as an opportunity for researchers in commutative algebra and representation theory to share their DG ideas with each others.

Organised by Ben Briggs, Janina Letz, and Josh Pollitz

Thanks for coming. Talks will appear here soon.

US Pacific Time Saturday May 2nd Sunday May 3rd
9:00-9:20am Sean Sather-Wagstaff Pedro Tamaroff
9:30-9:50am David Jorgensen Hamid Rahmati
10:00-10:20am Nicholas Packauskas Claudia Miller
10:30-10:50am Luigi Ferraro Mohsen Gheibi
11:00-11:20am Pablo Ocal Rachel Diethorn
1:00-1:20pm Cris Negron Hugh Roberts Geller
1:30-1:50pm John Myers Pinches Dirnfeld
2:00-2:20pm Mark Walker Lars Winther Christensen
2:30-2:50pm James Cameron Liana Şega
3:00-3:20pm Julia Pevtsova

Abstracts
James Cameron (UCLA)
Stratification for cochain algebras on Borel constructions of G-spaces

For G a compact Lie group and X a finite G-CW complex I will discuss how the the Borel equivariant cohomology ring of X with $$F_p$$ coefficients controls the structure of the derived category of the cochain algebra of the Borel construction on X.
I will also indicate how generalizing from finite groups to compact Lie groups and G-spaces allows one to study some of the categories that appear in modular representation theory via categories that have structure that doesn’t appear at the purely algebraic level.

Lars Winther Christensen (Texas Tech University)
Dimension and codimension of homologically finite complexes

Let R be a commutative noetherian ring. In the 1970s Foxby and Iversen extended the notions of dimension, depth, and codimension from R-modules to R-complexes. While their notions agree up to a normalization, the notion of codimension used in Bruns and Herzog’s treatment of the homological conjectures (now mostly theorems) does not quite fit in; in fact, it is not even “obviously” homological. I will discuss how to align these invariants.

Rachel Diethorn (Syracuse University)
Generators of Koszul homology

One approach to understanding Koszul homology is to find the generators. In the first part of this talk, I will provide explicit formulas for the generators of Koszul homology on the minimal generators of an ideal J with coefficients in what we call a J-closed module. This generalizes work of Herzog and of Corso, Goto, Huneke, Polini, and Ulrich. In the second part of the talk I will demonstrate the utility of such formulas. In particular, I will discuss an application to the Koszul homology algebra of quotients by certain edge ideals, and I will give an answer, for such rings, to a question of Avramov about the Koszul homology algebra of a Koszul algebra.

Pinches Dirnfeld (University of Utah)
Base Change Along the Frobenius Endomorphism and the Gorenstein Property

Let R be a local ring of positive characteristic and X a complex with nonzero finitely generated homology and finite injective dimension. We prove that if derived base change of X via the Frobenius (or more generally, via a contracting) endomorphism has finite injective dimension then R is Gorenstein.

Luigi Ferraro (Wake Forest University)
The Taylor Resolution over a Skew Polynomial Ring - with Frank Moore and Desiree Martin

Let $$I$$ be a monomial ideal in the polynomial ring $$R = k[x_1, ... , x_n]$$ over a field $$k$$. In her thesis, Taylor introduced a complex which provides a multi-graded free resolution for $$R/I$$ as an $$R$$-module. Later, Gemeda provided a differential graded structure on this complex while Avramov showed that this DG algebra admits a divided power structure. We generalize these results to monomial ideals $$J$$ in a skew polynomial ring $$S$$. As an application we show that if one fixes the number of generators of the ideal $$J$$, then there are finitely many isomorphism classes for $$\pi^{\geq 2}(S/J)$$, where $$\pi(S/J)$$ is the homotopy color Lie algebra of $$S/J$$, an invariant which was introduced and studied by the first and last author in a different work. As a result it follows that there are finitely many possibilities for the Poincaré series of $$k$$ over $$S/J$$, if the number of generators of $$J$$ is fixed.

Hugh Roberts Geller (Clemson University)
DG-Structures for Fiber Products

A construction of Tate shows that every algebra over a ring R possess a DG-algebra resolution over R. These resolutions are not always minimal and Avramov even shows that certain algebras cannot have a minimal resolution with a DG-algebra structure. This talk gives an explicit construction of a DG-structure for certain fiber products and criteria for determining when the structure is a DG-module, DG-algebra, or minimal DG-algebra.

Mohsen Gheibi (University of Texas at Arlington)
Quasi-Projective Dimension - with David A. Jorgensen and Ryo Takahashi

In this talk, I will introduce a homological invariant namely quasi-projective dimension, which is a generalization of the projective dimension. I will discuss the basic properties of the quasi-projective dimension and compare it with other homological dimensions. In particular, I will show that the modules with finite quasi-projective dimension, in many cases, behave similarly as modules of finite complete intersection dimension. I also will address some open problems about the quasi-projective dimension.

David Jorgensen (University of Texas Arlington)
Asymptotic Behavior of Ext over graded complete intersections - with Liana Şega and Peder Thompson

It is well known that for a pair of modules over a complete intersection, when all higher Ext modules have finite length, then the lengths of these higher Ext modules grow polynomially. In fact, the lengths are determined by two polynomials, one giving the lengths of the even Ext, and one giving the lengths of the odd Ext. In this talk we will discuss conditions when these two polynomials have the same leading coefficient. This is equivalent to the vanishing of the Herbrand difference, or Hochster’s Theta invariant, a phenomenon studied by several authors in recent papers.

Claudia Miller (Syracuse University)
Transferring algebra structures on complexes, part 2 - with Hamid Rahmati

We discuss how to use the homotopy on the tautological Koszul complex given by the weighted de Rham map to build a concrete permutation invariant dg-algebra structure on a well-known resolution.

John Myers (SUNY Oswego)
Three algebras and three definitions of Koszulness

Let R be a standard graded commutative algebra over a field k, let K be the Koszul complex on a minimal set of generators of the irrelevant ideal of R, and let H be the homology of K. Recall that R is said to be \textit{Koszul} if k has a linear free resolution over R. We adapt this definition to apply to K (viewed as a DG algebra) and then to H (viewed as a bigraded algebra). We describe how these three Koszul properties transfer back and forth between the three algebras R, K, and H, and we give several examples of classes of algebras R for which H is Koszul.

Cris Negron (University of North Carolina)
Support data for Hopf algebras via noncommutative hypersurfaces - with Julia Pevtsova

In recent work with J. Pevtsova, we develop an approach to support theory for Hopf algebras via noncommutative hypersurfaces. As a starting point, one considers a Hopf algebra $$u$$ which admits a smooth deformation $$U\to u$$ by a Noetherian Hopf algebra $$U$$ of finite global dimension. One uses this deformation to produce a rank variety for $$u$$ which takes values in an associated projective space. Our work is inspired by earlier contributions of Avramov and Buchweitz, which concerned support for (commutative) local complete intersections. I will discuss some modular examples, functions on finite group schemes and Drinfeld doubles of infinitesimal group schemes, and also quantum groups over the complexes. I will discuss how one can use this hypersurface approach to address the tensor product property in certain “solvable” examples.

Pablo Ocal (Texas A&M University)
On the Gerstenhaber bracket in relative Hochschild cohomology of associative algebras

We will begin by briefly presenting the inception of the Gerstenhaber bracket in Hochschild cohomology, which helped in capturing in an algebraic way the infinitesimal information stored in the cohomology of the algebra. On the first uses of the bracket by Gerstenhaber and Schack, they essentially claimed that everything that can be done on Hochschild cohomology can also be done in relative Hochschild cohomology. However, they required a separability condition to obtain relative projective resolutions when working with diagrams of algebras. This additional requirement motivates contextualizing our work to relative homological algebra. This is a less general context but it has multiple advantages: we can remove the separability condition, proofs are approachable, computations can be carried out, and an there is an interpretation of the bracket as a dg Lie algebra structure on a complex. We will also comment on recent results by Kaygun, who constructed a Jacobi-Zariski long exact sequence, and by Cibils, Lanzilotta, Marcos, Schroll, and Solotar, who described aspects of the Hochschild cohomology of bounded quiver algebras using relative cohomological tools. This strongly suggests that this context may be adequate for a better understanding of the cohomology of associative algebras.

Nicholas Packauskas (SUNY Cortland)
Quasi-Polynomial Growth of Betti Sequences - with Luchezar Avramov and Nicholas Packauskas

Let $$Q$$ be a regular local ring and $$I$$ an ideal generated by a regular sequence of $$c$$ elements in the square of the maximal ideal. It is known that over the complete intersection $$R = Q/I$$ that any finitely generated module $$M$$ has Betti numbers eventually given by quasi-polynomial of degree less than $$c$$. That is, there are integer-valued polynomial functions $$p^M_+$$ and $$p^M_-$$ with the same leading term such that $$\beta_{2i}^R(M)=p^M_+(2i)$$ and $$\beta_{2i+1}^R(M)=p^M_-(2i+1)$$ for $$i$$ sufficiently large. We will show that if $$q$$ is the height of the ideal generated by the quadratic initial forms of $$I$$ in the associated graded ring of $$Q$$, then the degree of $$p^M_+-p^M_-$$is less than $$c − q − 1$$.

Julia Pevtsova (University of Washington)
Support theory for Elementary supergroups - with Dave Benson, Srikanth Iyengar and Henning Krause

Elementary supergroup schemes arise as a detecting family in the theory of supports for finite supergroup schemes. As such, they play a similar role to finite supergroup schemes as elementary abelian p-groups play for finite groups, as known from the classical work of Quillen and Chouinard. In this talk I’ll describe the theory of varieties, the calculation of the Balmer spectrum and the Benson-Iyengar-Krause stratification for the singularity category of an elementary supergroup scheme. An interesting and novel feature of the theory is that it combines the π-point approach of Friedlander-Pevtsova with the hypersurface approach of Avramov-Iyengar as an attempt to construct Carlson’s rank varieties in the super context.

Transferring algebra structures on complexes, part 1 - with Claudia Miller

We discuss a homological method for transferring algebra structures on complexes along suitably nice homotopy equivalences. We also show how one can get such homotopy equivalencies, from old ones, using a homological tool called the perturbation lemma.

Sean Sather-Wagstaff (Clemson University)
Ascent of Test Modules

We investigate modules for which vanishing of Tor-modules implies finiteness of projective dimension. In particular, we answer a question of O. Celikbas and Sather-Wagstaff about ascent properties of such modules over residually algebraic flat local ring homomorphisms. To accomplish this, we consider ascent and descent properties over local ring homomorphisms of finite flat dimension, and for flat extensions of finite dimensional differential graded algebras.

Liana Şega (University of Missouri)
Free resolutions over compressed artinian level algebras of socle degree three - with Rasoul Ahangari Maleki

We will discuss rationality of Poincaré series and linearity of free resolutions in the case of compressed artinian level algebras of socle degree three.

Pedro Tamaroff (Trinity College Dublin)
The non-commutative calculus of fields and forms through dg-resolutions

To every smooth manifold we may assign its Cartan calculus consisting of multi-vector fields $$\Theta(M)$$ with the NS bracket and the $$\Theta(M)$$-module of differential forms $$\Omega(M)$$ on $$M$$ with the de Rham differential, and the same can be done for a commutative algebra $$A$$, in which case the pair of multi-vector fields $$\bigwedge_A{\rm Der}(A)$$ and differential forms $$\Omega_A$$ on $$A$$ carry the same kind of structure. In case $$A$$ is smooth, these spaces are in fact Hochschild (co)homology groups of $$A$$, which gives a Cartan calculus on the pair $$({\rm HH}^*(A),{\rm HH}_*(A))$$. The non-commutative analogue of this Cartan calculus ---which exists for an arbitrary associative algebra, commutative or not--- is called the Tamarkin--Tsygan calculus of $$A$$. It consists of the Gerstenhaber algebra $${\rm HH}^*(A)$$ and the $${\rm HH}^*(A)$$-module $${\rm HH}_*(A)$$ along with its Connes differential. In this talk I will explain how to use dg resolutions in associative algebras to compute this calculus, and give some examples of computation for some monomial examples, which in particular gives streamlined computations of the Gerstenhaber bracket of associative algebras through dg (as opposed to projective) resolutions.