# Seminars in Algebra (Spring 2024)

Here are a few notes about our algebra seminar this semester:

- Talks will be about 45 minutes, with time for questions afterwards.
- Usually we will have coffee/tea and cake at the lunchroom after a talk.

If you have any questions or comments about the seminar please contact Mads Sandøy or Laertis Vaso. To subscribe to the algebra seminar mailing list please contact one of them.

## Upcoming seminar talks

(Previous talks can be found below and by clicking on the menu on the left.)

## Monday, September 16th, 2024, at 13:00 in room 656 (Simastuen), SBII

**Speaker:** Eric Hanson (North Carolina State University)

**Title**: Mutation of tau-exceptional sequences

**Abstract:** By the work of Crawley-Boevey and Ringel, the set of complete exceptional sequences over a finite-dimensional hereditary algebra admits a transitive braid group action. This can also be viewed as a "mutation theory" for exceptional sequences. In this talk, we discuss recent joint work with Aslak Buan and Bethany Marsh which extends this into a mutation theory for (complete) tau-exceptional sequences over an arbitrary finite-dimensional algebra. In addition to giving the formulas for this mutation, we discuss the existence of non-mutable sequences, the problem of transitivity, and the (lack of) braid relations. This talk serves as a sequel to Bethany Marsh's September 12 seminar.

## Fall 2024 seminar talks

## Thursday, September 12th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** Bethany Marsh (University of Leeds)

**Title**: An introduction to tau-exceptional sequences

**Abstract:** Exceptional sequences in module categories over hereditary algebras (e.g. path algebras of quivers) were introduced and studied by W. Crawley-Boevey and C. M. Ringel in the early 1990s, as a way of understanding the structure of such categories. They were motivated by the consideration of exceptional sequences in algebraic geometry by A. I. Bondal, A. L. Gorodontsev and A. N. Rudakov.

Exceptional sequences can also be considered over arbitrary finite-dimensional algebras, but their behaviour is not so good in general: for example, complete sequences do not always exist. Tau-exceptional sequences offer an alternative generalisation to the non-hereditary case resolving this issue. They were introduced in joint work with A. B. Buan, motivated by the tau-tilting theory of T. Adachi, O. Iyama and I. Reiten, and signed exceptional sequences in the hereditary case defined by K. Igusa and G. Todorov.

This talk, while self-contained, will also serve as an introduction to E. Hanson’s upcoming talk on joint work with A. B. Buan and B. Marsh on mutation of tau-exceptional sequences.

## Friday, September 13th, 2024, at 13:15 in room Kjl22, Kjelhuset

**Speaker:** Erlend Due Børve

**Title**: Silting reduction using generalised concentric twin cotorsion pairs

**Abstract:** Given a triangulated category \(\mathcal{T}\) and a rigid subcategory \(\mathcal{R}\), Iyama–Yang put forward mild technical conditions that let us compute the Verdier quotient \(\mathcal{T}/\mathrm{thick}(\mathcal{R})\) explicitly. There are good reasons to extend Iyama–Yang's work to extriangulated categories, but one has to grapple with a more complicated theory of localisation. In this talk, we propose a generalisation of Iyama–Yang's work. More precisely, given an extriangulated category \(\mathcal{C}\) and a rigid subcategory \(\mathcal{R}\) satisfying a certain condition involving generalised concentric twin cotorsion pairs, we show that the Veridier quotient \(\mathcal{C}/\mathrm{thick}(\mathcal{R})\) can be expressed as an ideal quotient. If time permits, we argue that this condition is very mild in practice, especially for 0-Auslander extriangulated categories, e.g. the two-term category \(K^{[-1,0]}(proj(\Lambda))\) of a finite-dimensional algebra \(\Lambda\).

## Spring 2024 seminar talks

## Wednesday, January 17th, 2024, at 13:00 in room 656 (Simastuen), SBII

**Speaker:** Claire Amiot (Institut Fourier)

**Title**: Linear Invariants for poset representations

**Abstract:** Let \(P\) be a poset. An \(\mathbb Z\)-invariant on \(\rm mod P\) is a \(\mathbb Z\)-linear map \(K_0^{\rm sp}(\rm mod P)\to \mathbb Z^N\). In this talk, I will discuss several examples of invariants that appear in persistence theory coming either from relative exact structures or from order embeddings. This is a joint work with Thomas Brüstle and Eric Hanson.

## Monday, January 22th, 2024, at 11:00 in room 656 (Simastuen), SBII

**Speaker:** Dag Oskar Madsen (Nord universitet)

**Title**: The classification of Higher Auslander algebras among Nakayama algebras

**Abstract:** Together with Marczinzik and Zaimi we proved that for any \(n \leq r \leq 2n-2\) there exists a unique Nakayama algebra with \(n\) vertices and global dimension \(r\) that is a higher Auslander algebra. In this talk I will discuss this result and progress made by other authors in the more complicated case \(r < n\).

## Tuesday, February 13th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** Endre Sørmo Rundsveen (NTNU)

**Title**: \(\tau_d\)-tilting theory for Nakayama Algebras

**Abstract:** Tilting theory has been a great tool in representation theory since its inception in the 1970’s. It was generalized to the concept of support \(\tau\)-tilting in 2014, closing a gap introduced by mutation, and providing a bijection with functorially finite torsion classes. In higher AR-theory there has recently been several proposed generalization of support \(\tau\)-tilting (see e.g. Jacobsen-Jørgensen, Martínez–Mendoza and Zhou–Zu) with differing points of departure. In this talk our point of departure is in the generalization of torsion classes to \(d\)-torsion classes.

In ongoing work August, Haugland et.al. have shown that functorially finite torsion classes can be injectively sent to what we call strongly maximal \(\tau_d\)-tilting pairs. The aim of the talk is to classify these for truncated (acyclic) Nakayama Algebras when \(d>2\) or \(l=2\). Despite having a different point of origin, we also show that there is a clear relationship between strongly maximal \(\tau_d\)-tilting pairs and the \(\tau_d\)-tilting modules investigated by Martínez–Mendoza through \((d+1)\)-silting complexes. This is joint work with Laertis Vaso.

## Tuesday, February 27th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** Jacob Fjeld Grevstad (NTNU)

**Title**: Derived representation type and G-equivariant spectra

**Abstract:** The problem of classifying indecomposable representations is one of the earliest problems in representation theory. The famous Tame-Wild dichotomy loosely says that algebras split into two groups: those where classifying modules are relatively easily, and those where a classification is completely hopeless. In 2003 a similar dichotomy was proven for perfect complexes in the derived category.

There is a close relationship between the homotopy category of G-equivariant spectra and the derived category of cohomological MacKey functors. In this talk we look at which groups give rise to derived tame and which give rise to derived wild cohomological MacKey functors.

(Joint work in progress w/ Clover May)

## Tuesday, March 12th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** Håvard Utne Terland
(NTNU)

**Title**: Some combinatorics of tau-exceptional sequences

**Abstract:** Tau-exceptional sequences were introduced by Buan and Marsh as a generalization of exceptional sequences, which Crawley-Boevey had shown to among other things have nice combinatorial properties. The definition of tau-exceptional sequences uses a lot of machinery in tau-tilting theory and is sometimes difficult to study directly. To assist in studying tau-exceptional sequences Treffinger and Mendoza introduced TF-admissible orders of tau-tilting modules. It is the combinatorics of such TF-orders and how they interact with the combinatorics of tau-exceptional sequences which is the topic of ongoing work with Maximilian Kaipel.

## Tuesday, May 14th, 2024, at 14:05 in room 656 (Simastuen), SBII

**Speaker:** Maximilian Kaipel (University of Cologne)

**Title**: τ-cluster morphism categories of factor algebras

**Abstract:** For a finite-dimensional algebra, its τ-cluster morphism category encodes the structure of torsion classes and wide subcategories of its module category. It has been defined and studied algebraically using τ-tilting and silting theory, as well as geometrically using the g-vector fan. We introduce a new lattice-theoretic approach and define the category from the lattice of torsion classes.

This point-of-view allows us to define functors between the tau-cluster morphism category of an algebra A and that of its factor algebras A/I. Additionally, we characterise categorical properties of these functors, like when they are full, faithful and adjoint.

## Thursday, June 6th, 2024, at 13:15 in room 656 (Simastuen), SBII

**Speaker:** Peder Thompson (Mälardalen University)

**Title**: Periodic cotorsion pairs and one-sided Gorenstein rings

**Abstract:** A basic homological invariant attached to a ring is the left global dimension: this is the supremum of the projective dimensions of all left modules. A natural analogue of this is the left Gorenstein global dimension, defined as the supremum of the Gorenstein projective dimensions of all left modules, and rings where this invariant is finite are known as left Gorenstein. In this talk we will introduce the notion of periodic cotorsion pairs, and use them to characterize left Gorenstein rings. In particular, this characterization refines a result of Beligiannis and Reiten. This talk is based on joint work with Christensen, Estrada, Liang, and Wang.

## Thursday, June 6th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** David A. Jorgensen (The University of Texas at Arlington)

**Title**: A construction of Shamash revisited

**Abstract:** Avramov and Iyengar have recently proven that if \(f\) and \(g\) are regular elements in an ideal \(I\) in a commutative local ring \((P,n,k)\) such that \(f-g\in nI\), then for any \(P/I\)-module \(M\)

\(\mathrm{Tor}^{P/(f)}_i(M,k)\cong\mathrm{Tor}^{P/(g)}_i(M,k)\) for all \(i\ge 0\). We give an alternate proof of this fact using an embellishment of a construction of Shamash that yields a \(P/(f)\)-free resolution of \(M\) from a \(P\)-free resolution of \(M\). This is part of some joint work with Petter Bergh.

## Tuesday, June 11th, 2024, at 14:15 in room 656 (Simastuen), SBII

**Speaker:** Joseph Winspeare (Université Grenoble Alpes)

**Title**: From derived category of gentle algebras to 1-periodic Fukaya categories of marked surfaces

**Abstract:** In recent years, a strong link was discovered between Fukaya categories of surfaces with boundary and derived categories of gentle algebras. However, the numerical invariant for derived equivalence of gentle algebras of Amiot, Plamondon and Schroll shows the need for a more "general" category. During this talk, I will give motivations for this category and show one way it can be constructed. I will finish by giving basic properties of this new category and explain why it can be called the 1-periodic Fukaya category of the marked surface.