# Seminars in Algebra (Spring 2023)

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

- Talks will be about 45 minutes, with time for questions afterwards.

If you have any questions or comments about the seminar, please contact Mads Sandøy.

## Upcoming seminar talks

(Previous 2022 & 2021 talks can be found below.)

## *Next talk:* Wednesday the 9th of February at 14:15 o'clock in room 656 (Simastuen), SBII

**Speaker:** Amit Shah

**Title**: Characterising Jordan-Hölder extriangulated categories via Grothendieck monoids

**Abstract**: The notions of composition series and length are well-behaved in the context of abelian categories. And, in addition, each abelian category satisfies the so-called Jordan-Hölder property/theorem. Unfortunately, these ideas are poorly behaved for triangulated categories. However, with the introduction of extriangulated categories, it is interesting (at least for me and my collaborators Thomas Brüstle, Souheila Hassoun and Aran Tattar) to see what sense we can make of these concepts for extriangulated categories.

I’ll present a result that characterises Jordan-Hölder, length extriangulated categories using the Grothendieck monoid of an extriangulated category. This is motivated by the exact category setting as considered by Enomoto, in which it becomes apparent that Grothendieck monoid is more appropriate to look at than the Grothendieck group. I’ll present some examples coming from stratifying systems. In fact, developing stratifying systems for extriangulated categories was the original motivation of our article.

## *Next talk:* Wednesday the 15th of February at 14:15 o'clock in room 656 (Simastuen), SBII

**Speaker:** Raphael Bennett-Tennenhaus

**Title**: Corner replacement for Morita contexts

**Abstract:** Abstract: A celebrated result of Morita characterises equivalences between categories of modules over unital rings. Bass exposed these results in terms of a more general situation, known today as a Morita context. Since then various authors have extended the setting of so-called Morita theory to cater for rings which need not be unital. For example, Ánh and Márki developed the theory for unital modules over rings with local units.

I will begin by explaining how Morita contexts with a common ring can be composed, excised and ligated back together. Specifying, I will then highlight the way in which replacing a corner subring is compatible with Morita equivalences (which I suspect was `folklore’), and give applications to representations of species. Time permitted, I will consider a potential connection to locally finitely presented additive categories. This talk is based on a recent arxiv article 2301.09518.

## Previous 2022 seminar talks

## *Next talk:* Monday, January 23rd, 2023, at 11:15 in room 656 (Simastuen), SBII

**Speaker:** Henning Krause

**Title**: Central support for triangulated categories

**Abstract:** Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.

## Monday, January 23rd, 2023, at 12:15 in room 656 (Simastuen), SBII

**Speaker:** Hugh Thomas

**Title**: An analogue of X-cluster variables for finite-dimensional algebras

**Abstract:** Let A be a finite-dimensional algebra with n isoclasses of simples. To an indecomposable module (or shifted indecomposable projective) M, we can associate a certain rational function u_M in Q(y_1,…y_n). In the case that A is the path algebra of a Dynkin quiver, the u_M are very closely related to a subset of the X-cluster variables, in the sense of Fock and Goncharov. Using a result of Domínguez and Geiss, we show that, when A is of finite representation type, the u_M satisfy some beautiful relations. When A is not of finite type, I will discuss analogues of the finite representation type relations which we expect to hold in the ring of formal power series. We are able to establish these relations in the case of gentle algebras. I will attempt to say a little something about the motivation for this problem, which is drawn from the physics of scattering amplitudes. This is joint work with Nima Arkani-Hamed, Hadleigh Frost, Pierre-Guy Plamondon, and Giulio Salvatori.

## Previous 2022 seminar talks

## Tuesday, March 15th, 2022, at 13:15 in room 734

**Speaker:** Hipolito Treffinger

**Title**: Two characterisations of higher torsion classes

**Abstract:** The notion of torsion classes in higher homological algebras was introduced by Jørgensen in 2016. In this talk, after recalling the definition, we will give two different characterisations of higher torsion classes: one intrinsic and another using the classical torsion classes of the ambient abelian category. Moreover we will discuss some combinatorial consequences of these results.

This talk is based on a joint work with J. Asadollahi, P. Jørgensen and S. Schroll and a joint work in progress with J. August, J. Haugland, K. Jacobsen, S. Kvamme and Y. Palu.

## Previous Fall 2021 seminar talks

## Tuesday, November 16th, 2021, at 13:15 in room 734

**Title:** The Higher Auslander—Solberg correspondence for exact categories.

**Speaker:** Jacob Fjeld Grevstad

**Abstract**: The Auslander correspondence is a classical result which establishes a bijection between representation finite algebras, and so called Auslander algebras up to Morita equivalence. This has also been generalized to the Higher Auslander correspondence, between n-cluster tilting objects and n-Auslander algebras (Iyama ‘07), and to the Higher Auslander—Solberg correspondence, between n-precluster tilting objects and n-minimal Auslander—Gorenstein algebras (Iyama—Solberg ‘18).

Recently the Auslander correspondence and the Higher Auslander correspondence has been generalized to exact categories (Henrard—Kvamme—Roosmalen ’20, Ebrahimi—Isfahani ‘21). We give a definition for n-precluster tilting subcategories, and for n-minimal Auslander—Gorenstein categories, and establish a correspondence between them. We also recover some of the results of Iyama—Solberg in the exact setting.

## Tuesday, November 9th, 2021, at 13:15 in room 734

**Title:** Constructions of n-cluster tilting subcategories.

**Speaker:** Laertis Vaso

**Abstract:** In higher-dimensional Auslander–Reiten theory, a central role is played by n-cluster tilting subcategories. However, there are still many unanswered questions about such subcategories. One reason for that may be that finding them is not an easy task. In this talk I will present a classification for n-cluster tilting subcategories in the cases of radical square zero algebras and Nakayama algebras with homogeneous relations and explain how we can use representation-directed algebras to obtain many more examples with different properties. I will also explain how these results answer some of these questions.

## Tuesday, November 2nd, 2021, at 13:15 in room 734

**Title:** Two-term silting and \(\tau\)-cluster morphism categories

**Speaker:** Erlend Due Børve

**Abstract:** Cluster morphism categories, defined by Igusa—Todorov, are useful in the study of picture groups and picture spaces.
Going beyond the hereditary representation finite case, Buan—Marsh (and Buan—Hanson) define \(\tau\)-cluster morphism categories in the context of \(\tau\)-tilting theory. However, a passage via two-term silting is occasionally needed to prove the necessary results.

We give a definition purely in terms of two-term (partial) silting objects in the bounded derived category. When proving that the composition rule is associative, we use the properties of truncation in t-structures.

Paths in \(\tau\)-cluster morphism categories are signed \(\tau\)-exceptional sequences, or just signed exceptional sequences in the hereditary representation finite case. With our definition, these paths correspond to particularly nice bases of the Grothendieck group of the bounded homotopy category.

The talk will be based on the preprint arXiv:2110.03472.

## Previous Spring 2021 seminar talks

## January 25th, 2021, at 13:15 (digital only on Zoom)

**Title:** Canonical join and meet representations in lattices of torsion classes

**Speaker:** Eric Hanson (NTNU)

**Abstract:** Let A be a finite-dimensional associative algebra over a field. A family of subcategories of A-modules (known as torsion classes) are known to form a lattice under inclusion. In several recent papers, this lattice has been studied using "brick labeling", a method of associating a special type of module (called a brick) to each cover relation in the lattice. A collection of these bricks labels the "downward" (resp. "upward") cover relations of some element of the lattice if and only if there are no nontrivial morphisms between them. In this talk, we consider the bricks labeling both "downward" and "upward" cover relations at the same time. More precisely, if we are given two sets of bricks D and U, we formulate necessary and sufficient algebraic conditions for there to exist a torsion class T so that the bricks in D label cover relations of the form S less than T and the bricks in U label cover relations of the form T less than R. This is based on joint works with Emily Barnard and Kiyoshi Igusa.

Download talk slides here.

## February 15th, 2021, at 13:00 (digital on Zoom)

**Title:** (Co)compact objects and duality in triangulated categories

**Speaker:** Torkil Stai (NTNU)

**Abstract:** The study of compact objects in triangulated categories with coproducts, has been a success. Compact objects are plentiful in many categories of interest, and their presence allows for far-reaching results.

Unfortunately, the dual theory is rather empty: Many of the categories we care about, do not have any cocompact objects. In ongoing work with Steffen Oppermann and Chrysostomos Psaroudakis, we consider a weakened version of cocompactness, called 0-cocompactness.

In this talk we will try to justify the study of this new notion, in particular by providing real-life examples of 0-cocompact objects. We will also see how versions of (co)compactness are linked via two types of duality, and moreover come up in connection with almost split triangles.

Download talk slides here.

## February 22nd, 2021, at 13:00 (digital on Zoom)

**Title:** "Model categories of quiver representations" (report on joint work with Henrik Holm)

**Speaker:** Peter Jørgensen (Aarhus University)

**Abstract:** Let R be a k-algebra. Given a cotorsion pair (A,B) in Mod(R), Gillespie's Theorem shows how to construct a model category structure on C(Mod R), the category of chain complexes over Mod(R). There is an associated homotopy category H.

If (A,B) is the trivial cotorsion pair (projective modules, everything), then H is the derived category D(Mod R).

Chain complexes over R are the Mod(R)-valued representations of a certain quiver with relations: Linearly oriented A double infinity modulo the composition of any two consecutive arrows. We show that Gillespie's Theorem generalises to arbitrary self-injective quivers with relations, providing us with many new model category structures.

Zoom meeting ID: 924 5793 8256, contact Peder directly for the password (or click the link on the Monday morning email). The link is the same each week.

Recording can be viewed here.

## March 1st, 2021, at 13:00 (digital on Zoom)

**Title:** Functors and subcategories of n-exangulated categories

**Speaker:** Johanne Haugland (NTNU)

**Abstract:** Herschend, Liu and Nakaoka introduced n-exangulated categories as a higher dimensional analogue of extriangulated categories. Natural examples are given by n-exact and (n+2)-angulated categories in the sense of Jasso and Geiss–Keller–Oppermann. In this talk, we give an introduction to n-exangulated categories and explain how we can understand their subcategories in terms of subgroups of the associated Grothendieck group. We also discuss functors between such categories. This is based on joint work in progress with R. Bennett-Tennenhaus, M. H. Sandøy and A. Shah.

## March 8th, 2021, at 13:00 (digital on Zoom)

**Title:** The wall and chamber structure of an algebra: a geometric approach to tau-tilting theory

**Speaker:** Hipolito Treffinger (University of Bonn)

**Abstract:** The notion of stability condition for representation of algebras was introduced by King in the '90s as an application of Mumford's Geometric Invariant Theory. In this talk we will show an explicit and surprising relation between King's stability conditions and tau-tiling theory, an homological theory introduced by Adachi, Iyama and Reiten at the beginning of the last decade. Time permitting, we will show how the wall and chamber structure of an algebra can be used to solve a tau-tilting version of the first Brauer-Thrall conjecture.

Some of the results presented in this talk are the result of collaborations with Thomas Brüstle, David Smith and Sibylle Schroll.

Download slide talks here.

## March 15th, 2021, at 13:00 (digital on Zoom)

**Title:** Mutation and minimal inclusions of torsion classes

**Speaker:** Rosanna Laking (Università degli Studi di Verona)

**Abstract:** Torsion pairs are fundamental tools in the study of abelian categories, which contain important information related to derived categories and their t-structures. In this talk we will consider the lattice of torsion classes in the category of finite-dimensional modules over a finite-dimensional algebra, with a particular focus on the minimal inclusions of torsion classes.

It was shown by Adachi, Iyama and Reiten that minimal inclusions of functorially finite torsion classes correspond to irreducible mutations of associated two-term silting complexes in the category of perfect complexes. In this talk we will see how minimal inclusions of arbitrary torsion classes correspond to irreducible mutations of associated two-term cosilting complexes in the unbounded derived category.

This talk will be based on joint work with Lidia Angeleri Hügel, Jan Stovicek and Jorge Vitoria.

Download the notes from the talk here.

## March 22nd, 2021, at 13:00 (digital on Zoom)

**Title:** Representing hearts of t-structures as Serre quotients

**Speaker:** Jan Stovicek (Charles University in Prague)

**Abstract:** If D is a triangulated subcategory with a t-structure induced by a silting object T, then the heart of the t-structure is equivalent to mod-End(T). We generalize this well-known fact to the situation where T is what we call a t-generator - in such a case the heart is a Serre quotient of mod-End(T).

The notion of t-generating subcategory is inspired by Lurie's study of aisles of t-structures in the context of spectral algebraic geometry, and the problem is also very closely related to the more classical task of decribing the relation of A to mod-End(G), where A is an abelian category and G is a generator of A. My main motivation, however, was to get criteria for when the heart of a t-structure in a compactly generated triangulated category is a Grothendieck category. I will try to explain these connections.

This talk will be based on joint work with Manuel Saorín, arXiv:2003.01401.

## April 12th, 2021, at 13:00 (digital on Zoom)

**Title:** Grassmannian categories of infinite rank and rings of countable Cohen-Macaulay type

**Speaker:** Eleonore Faber (University of Leeds)

**Abstract:** We construct a categorification of the coordinate rings of Grassmannians of infinite rank in terms of graded maximal Cohen-Macaulay modules over a hypersurface singularity. This gives an infinite rank analogue of the Grassmannian cluster categories introduced by Jensen, King, and Su. We show that there is a structure preserving bijection between the generically free rank one modules in a Grassmannian category of infinite rank and the Plücker coordinates in a Grassmannian cluster algebra of infinite rank. In a special case, when the hypersurface singularity is a curve of countable Cohen-Macaulay type, our category has a combinatorial model by an `infinity-gon`

and we can determine triangulations of this infinity-gon. This is joint work with Jenny August, Man-Wai Cheung, Sira Gratz, and Sibylle Schroll.

## April 19th, 2021, at 13:00 (digital on Zoom)

**Title:** Tilting preserves finite global dimension

**Speaker:** Henning Krause (Bielefeld University)

**Abstract:** The talk is about a result (joint with Bernhard Keller) which one would expect in any text book on tilting:
Given a tilting object of the bounded derived category of an abelian category of finite global dimension,
there is (under suitable finiteness conditions) a bound for the global dimension of its endomorphism ring.

## April 26th, 2021, at 13:00 (digital on Zoom)

**Title:** Triangulated categories and topological models

**Speaker:** Claire Amiot (Institut Fourier, Grenoble Alpes University)

**Abstract:** This is a survey talk on different triangulated categories, such as derived categories and cluster categories, arising in representation theory that are strongly linked with some topological data. The idea is to encode the category via some topological object, typically a surface with marked points. I will explain how one can use this topological data to deduce results on the categories.

## May 3rd, 2021, at **15:00** (digital on Zoom)

**Title:** Simples in the heart

**Speaker:** Lidia Angeleri Hügel (Università degli Studi di Verona)

**Abstract:** The lattice tors(A) formed by the torsion classes in the category of finite dimensional modules over a finite dimensional algebra A can be studied in terms of (possibly infinite dimensional) cosilting modules and their mutations. Any cosilting module C gives rise to a t-structure in the derived category of A. Its heart is a locally coherent Grothendieck category whose injective cogenerator is determined by C and whose simple objects are related to the brick labelling of the Hasse quiver of tors(A).

In this talk, we focus on hearts associated to cotilting modules. Using tools from model theory, we describe the indecomposable pure-injective modules that correspond to injective envelopes of simple objects in the heart. This sheds new light on the process of mutation.

The talk is based on joint work with Ivo Herzog and Rosanna Laking.

Seminar notes can be downloaded here.

*Note the later time for seminar. This is to allow for those who would like to also attend the lecture series by Osamu Iyama, see here for more information: https://networkonsilting.wordpress.com/ *

## May 10th, 2021, at 13:00 (digital on Zoom)

**Title:** Characterising $\Sigma$-pure-injectivity in triangulated categories

**Speaker:** Raphael Bennett-Tennenhaus (Bielefeld Univeristy)

**Abstract:** The model theory of modules involves interpreting model theoretic notions in terms of module theory. For example, an injective module map is called pure if solutions to pp-formulas are reflected. A module is $\Sigma$-pure-injective provided any set-indexed coproduct of it is pure-injective: that is, injective with respect to pure embeddings. There are various well-known ways to characterise both pure-injective and $\Sigma$-pure-injective modules. In this talk I will begin by replacing the category of modules with a compactly generated triangulated category. The notions of purity in this setting were defined by Krause, and the canonical model theoretic language here was defined by Garkusha and Prest. I will then present some ways to characterise $\Sigma$-pure-injective objects here, analogous to the module category setting. Time permitting, I will try to say something about the proof, and motivate the introduction of endoperfect objects. This talk is based on the arxiv preprint 2004.06854.

## Tuesday May 18th, 2021, at 13:00 (digital on Zoom)

**Title:** “Continuous” Representation Theory

**Speaker:** Job D. Rock

**Abstract:** In this expository talk we'll cover several places where representation theory is done over continuous structures, such as the real line or the circle. We'll examine some properties that still hold in the continuum and mention a few that do not. Additionally, we’ll point out some interesting things that happen in only the continuum or similar structures. The presented work comes from several projects (some in preparation). The works are with: Eric J. Hanson, Kiyoshi Igusa, Karin M. Jacobsen, Maitreyee C. Kulkarni, Jacob P. Matherne, Kaveh Mousavand, Charles Paquette, Gordana Todorov, Emine Yıldırım, and Shijie Zhu.

Talk slides: download here.

Talk video recording: view here.

## Tuesday May 25th, 2021, at 13:00 (digital on Zoom)

**Title:** Some classification results for $n$-representation finite algebras and connections with higher almost Koszulity

**Speaker:** Mads Hustad Sandøy

**Abstract:** In 2004, Iyama presented a “higher dimensional” generalization of Auslander-Reiten theory. Within this theory, the so-called n-hereditary algebras play an important role. These algebras come in two flavors: n-representation finite and n-representation infinite. Based on joint work with Louis-Philippe Thibault, some classification results for the former class are presented. Then, based on joint work with Johanne Haugland, we introduce the notion of a higher almost Koszul algebra and discuss connections between this notion and that of n-representation finite algebras.

## Monday, May 31st, 2021, at 13:00 (digital on Zoom)

**Title:** The Jordan-Holder property for Quillen exact categories

**Speaker:** Souheila Hassoun

**Abstract:** Quillen exact categories generalise the important and widely used notion of abelian categories. We discuss ways to generalise the intersections and sums of subobjects to the realm of exact categories. The first way, using pushouts and pullbacks, leads to new characterisations of quasi-abelian and abelian categories, and motivates the general way. We introduce general notions of intersection and sum that allows us to consider the Artin-Wedderburn exact categories and generalise the famous Jordan-Hölder theorem.

This talk is based on two joint works, one with T.Brüstle and A.Tattar and another one with A.Shah and S-A.Wegner.

Download the talk slides or view the talk recording.

## Previous Fall 2020 seminar talks

## September 29th, 2020, at 14:15 in room S1

**Title:** Support varieties for finite tensor categories

**Speaker:** Petter Andreas Bergh

**Abstract:** Tensor categories appear naturally in several settings, such as group representations, representations of Hopf algebras (quantum groups), fusion categories and conformal field theory. This talk is a report on recent joint work with Julia Plavnik and Sarah Witherspoon, where we develop a theory of cohomological support varieties for finite tensor categories.

## October 6th, 2020, at 14:15 in room S1

**Title:** Representations which are realizable over Set

**Speaker:** Steffen Oppermann

**Abstract:** A representation of a quiver (poset, category) in the category of sets can be turned into a linear representation by turning any set into a free vector space. In my talk I will discuss the question which linear representations arise in this way. This is part of an ongoing discussion with Ulrich Bauer, Magnus Botnan, and Johan Steen.

## October 13th, 2020, at 14:15 in room S1

**Title:** Support theory for triangulated categories in algebra and topology

**Speaker:** Drew Kenneth Heard

**Abstract:** We will survey the support theory of triangulated categories through the machinery of tensor-triangulated geometry. We will discuss the stratification theory of Benson—Iyengar—Krause for triangulated categories, the construction by Balmer of the spectrum of a tensor-triangulated category, and the relation between the two.

## October 20th, 2020, at 14:15, digitally on zoom (not in person)

**Title:** d-abelian categories are d-cluster tilting

**Speaker:** Sondre Kvamme

**Abstract:** In 2014 Jasso introduced d-abelian categories as an axiomatization of d-cluster tilting subcategories, and he showed that any projectively generated d-abelian category is d-cluster tilting. In this talk we will explain how any d-abelian category is d-cluster tilting, without the assumption on the projective objects. If time permits, we will also explain how the proof can be adapted to give axiomatization of more general subcategories of abelian categories.

This talk will be held digitally on Zoom (no in-person meeting). The Zoom meeting ID is **935 3915 4938**. You will need the password which will be emailed Tuesday morning (or just click the link there).

## October 27th, 2020, at 14:15 in room S1

**Title:** From Linear Algebra to Zero-Knowledge Proofs (and More)

**Speaker:** Jiaxin Pan

**Abstract:** In this talk, we will start with basic linear algebra facts and then, by combining them with computationally hard problems in number theory, we will survey some of the current advanced cryptography in a "simple" yet intuitive way. We will focus on zero-knowledge proof systems which is the goldmine of cryptography (and theoretical computer
science).

## November 3rd, 2020, at 14:15 in room S1

**Title:** Tau-exceptional sequences

**Speaker:** Aslak Bakke Buan

**Abstract:** I will review some classical results about exceptional sequences for hereditary algebras, and discuss a possible generalization of such to general finite dimensional algebras. This is (partly ongoing) joint work with Marsh. It builds on recent work by Igusa-Todorov on signed exceptional sequences and also on work by many, including Adachi-Iyama-Reiten and Jasso, on tau-tilting theory.

## November 10th, 2020, at 14:15, digital only on Zoom

**Title:** Cryptographic voting and lattices

**Speaker:** Kristian Gjøsteen

**Abstract:** Cryptographic voting systems often need some way to prove that a given collection of ballots is the decryption of a collection of ciphertexts, without revealing which ciphertext decrypts to which ballots. How to do this is well-known with classical methods, but these are not quantum-safe. We explain how this can be done using simple linear algebra and lattice-based cryptography.

## December 8th, 2020, at 14:15, digital only on Zoom

**Title:** Classification of d-representation-finite trivial extensions of Dynkin type algebras

**Speaker:** Tor Kringeland

**Abstract:** Given a Dynkin quiver $Q$, we consider for which values $d\ge 2$ the trivial extension algebra $T(kQ)$ is $d$-representation-finite.

Download talk slides here.

## December 15th, 2020, at 14:15, digital only on Zoom

**Title:** Morphism spectra in exact (\(\infty\)-)categories

**Speaker:** Erlend Due Børve

**Abstract:** We sketch the proof of a (rather old) result of Retakh: If \({\cal C}\) is an exact category in which \(A\) and
\(B\) are objects, then the extension categories \({\cal E}xt^n_{\cal C}(B,A)\) form an \(\Omega\)-spectrum. Time permitting, we will explain how this generalises to exact \(\infty\)-categories, as well as the connection to topological enhancements of extriangulated categories. This is a report on joint work in progress with Paul Trygsland.

Slides can be downloaded here