Seminar Representation Theory, SS 2022

Time and place: Wednesdays 10–12 in room U2-232

Organisers: Prof. Dr. William Crawley-Boevey, Prof. Dr. Henning Krause

The seminar covers different topics within the field of representation theory.

Schedule of Talks

Wednesday, 13 April
Janina Letz: $A_{\infty}$-algebras
Abstract: In this talk I give an introduction to $A_{\infty}$-algebras and talk about how an $A_{\infty}$ algebra structure on $A$ can be transferred to a semi-free resolution of $A$.

Wednesday, 20 April
Rudradip Biswas: Unbounded derived categories and the finitistic dimension (after Rickard), Talk I
Abstract: In this talk, we went through the proof of Rickard's main result from the "Unbounded derived categories and the finitistic dimension" paper which states that if injectives generate the derived unbounded category of modules over a finite-dimensional algebra, then that algebra satisfies the finitistic dimension conjecture. In addition to that, we proved Rickard’s other result of understanding the conjecture and injective generation in terms of perpendicular categories. We also showed how one can arrive at a similar connection between injective generation and the fintistic dimension for group algebras.

Wednesday, 27 April
Estanislao Herscovich: Unbounded derived categories and the finitistic dimension (after Rickard), Talk II
Abstract: We cover sections $5$ to $7$ of the previously mentioned article by J. Rickard. In particular we explain the proof of the facts that, if the injectives generate for a finite dimension algebra $A$, then the projectives cogenerate for the opposite algebra, which in turn implies the finitistic dimension conjecture for $A$. We then present the proof of one criterion to prove when a finite dimension module is in the localizing subcategory generated by all injectives, namely, if it has finite cosyzygy type. This allows to show that many families algebras satisfy that injectives generate for them, such as algebras of finite representation type, radical square zero algebras and monomial algebras.

No talk on Wednesday, 04 May

Wednesday, 11 May
Rudradip Biswas: Stable categories of discrete modules over TDLC groups with a cohomological finiteness condition
Abstract: Exploiting the recent progress made my Castellano and Weigel in studying the rational discrete modules over totally disconnected locally compact (TDLC) groups, we show how one can draw parallels with the behaviour of various cohomological invariants of discrete groups to construct well-behaved stable categories over TDLC groups, mirroring the constructions of a recent Mazza-Symonds paper. For this construction, the TDLC groups in question will have to satisfy certain cohomological finiteness conditions.

Wednesday, 18 May
Henning Krause: On Külshammer-Miemietz's "Uniqueness of exact Borel subalgebras and bocses" paper, Talk I
Abstract: The talk discusses for quasi-hereditary algebras the notion of an exact Borel subalgebra. Existence can be proved via the theory of bocses and A-infinity algebras, following work of Koenig, Külshammer, and Ovsienko from 2014. The recent paper of Külshammer and Miemietz is devoted to the uniqueness of such constructions, using the A-infinity structure of the Ext-algebra of the standard modules. This is parallel to the construction of the basic algebra via the the A-infinity structure of the Ext-algebra of the simple modules.

Wednesday, 25 May
Estanislao Herscovich: On Külshammer-Miemietz's "Uniqueness of exact Borel subalgebras and bocses" paper, Talk II
Abstract: Our aim is to discuss the different uniqueness results of the article in a little more detail. More precisely, we will first discuss the proof that if the right algebras of two basic directed bocses are Morita equivalent (as quasi-hereditary algebras) then their exact Borel subalgebras are isomorphic, and we will further deduce that the right algebras are isomorphic. This is done using Keller's higher multiplication theorem and some representation theoretic arguments. Secondly, we will focus on the far stronger result of the article stating that if the right algebras of two basic directed bocses are Morita equivalent (as quasi-hereditary algebras) then the corresponding bocses are isomorphic. The uniqueness results mentioned in the first part are then a direct corollary. The proof of the stronger result is based on a version of Keller's higher multiplication theorem for the $A_{\infty}$-algebra model structure on the Ext-algebra of the sum of standard modules of a quasi-hereditary algebra, which involves very hard and intensive calculations.

Wednesday, 1 June
Marc Stephan: On Benson-Greenlees's "The singularity and cosingularity categories of $C^{*}(BG)$ for groups with cyclic Sylow $p$-subgroups"
Abstract: The talk provides an introduction to $A_\infty$-modules and an exposition of Benson-Greenlees's classification of indecomposables in singularity and cosingularity categories of certain $A_\infty$-algebras.

Wednesday, 8 June
Vincent Rouven Klinksiek: Balanced modules and constructible submodules
Abstract: In the talk we introduce basic definitions and results to understand balanced modules for f.d. algebras. We then explain how we can understand minimal left add$(M)$-approximations of indecomposable projective modules for string tree algebras. Moreover, we give a combinatorial result describing their cokernels. We finish with the definition of constructible modules and a sketch of the proof that balanced modules are constructible for string tree algebras.

Wednesday, 15 June
Raphael Bennett-Tennenhaus. Auslander-Reiten/Serre duality in exact/triangulated categories, I.
Abstract: In this talk I will present a unification of two results by Krause and Enomoto. The result from Krause characterises when a triangulated category has a Serre duality. The result from Enomoto characterises when an exact category has an Auslander—Reiten duality. Both these characterisations are in terms of the dualizing varieties introduced by Auslander and Reiten. The unification I will present is a result of Iyama, Nakaoka and Palu from 2018, which is written in the language of extriangulated categories. Building up the language to present this unifying theorem shall be the focus of the first of two talks.

Wednesday, 22 June
Raphael Bennett-Tennenhaus. Auslander-Reiten/Serre duality in exact/triangulated categories, II.
Abstract: This talk shall be a continuation of the previous talk I gave, with the title "Auslander—Reiten/Serre duality in exact/triangulated categories, I.". Here I presented a result of Iyama, Nakaoka and Palu, which characterises the existence of a certain type of duality in terms of when the stable or costable categories are dualizing varieties. In this second talk I shall explain some of the main ingredients used in the proof. The context I shall work in shall be slightly more general, to fit with a theorem of Enomoto: instead of working with categories enriched over a field, I shall work with those enriched over a (unital, commutative and) complete, local and noetherian ground ring.

Wednesday, 29 June
William Crawley-Boevey. Semilinear clannish algebras
Abstract: This is joint work with Raphael Bennett-Tennenhaus. I will introduce the algebras in the title, which are a generalization of the well-known class of string algebras, explain the classification of indecomposable modules into asymmetric and symmetric strings and bands, and discuss examples such as unsplit Dedekind-like rings. If there is time I will explain some aspects of the proof.

Wednesday, 6 July
Andrew Hubery: The category $Coh_\zeta\mathbb X$
Abstract: We introduce the category of coherent sheaves on a weighted projective line equipped with a zeta connection, and show that it is an abelian length category. In joint work with Crawley-Boevey we aim to describe the simple objects in this category, and apply this to describe all irreducible solutions of the Deligne-Simson problem.