The lattice of torsion classes torsA in the category of finite dimensional modules over a finite dimensional algebra A can be studied via an operation of mutation. Work of Adachi, Iyama and Reiten [1] shows that minimal inclusions of functorially finite torsion classes can be regarded as mutations of associated silting complexes. In [2] it is shown that the whole lattice torsA admits a similar interpretation by associating a cosilting complex to each torsion class and considering cosilting mutation.

The cosilting complexes associated to torsion classes are not compact in general, but they are pure-injective, and they correspond bijectively to (pure-injective) cosilting modules. In this talk, we will describe mutation of cosilting modules as an operation that exchanges indecomposable pure-injective summands. This will allow us to regard minimal inclusions of torsion classes as an operation on the Ziegler spectrum of A.

The talk will be based on joint work with Ivo Herzog, Rosanna Laking and Francesco Sentieri.

References

• [1] T. Adachi, O. Iyama, I. Reiten, τ-tilting theory, Compos. Math. 150 (2014), 415-452.
• [2] L. Angeleri Hügel, R. Laking, J. Šťovíček, J. Vitória, Mutation and torsion pairs, arxiv:2201.02147

Alternating strand diagrams (as introduced by Postnikov) on the disk have been used in the study of the coordinate ring of the Grassmannian. In particular, they give rise to clusters of the Grassmannian cluster algebras (Scott) or to cluster-tilting objects of the Grassmannian cluster categories of Jensen-King-Su (Baur-King-Marsh). We introduce orbifold diagrams as quotients of Postnikov under a rotational symmetry and associate quivers with potentials to them as a means towards skew group categories. This is joint work with Andrea Pasquali and Diego Velasco.

Varieties for modules, based on Hochschild cohomology algebra, introduced by N. Snashall and Ø. Solberg, have attracted a lot of attention. Such varieties exist provided the algebra satisfies suitable finite generation properties (now known as (Fg)). We give a survey of what is known, and discuss some open problems.

The notion of T-Koszul algebras was introduced by Green–Reiten–Solberg and provides a unified approach to Koszul duality and tilting equivalence. Building on work by Madsen, we generalize this notion to a higher-dimensional setup and show that a version of classical Koszul duality still holds. Our work is motivated by and has applications for n-hereditary algebras. This talk is based on joint work with Mads H. Sandøy.

We consider representations of finite groups over a commutative noetherian ring and explain a classification of thick and localising tensor ideals via group cohomology. Some focus will be on the definition of the appropriate categories of representations such that the existing machinery can be applied. This is a report on joint work with Dave Benson, Srikanth Iyengar, and Julia Pevtsova.

First arising in a combinatorial form in Fomin and Zelevinsky's theory of cluster algebras, g-vectors have two closely related representation-theoretic incarnations. The first of these is the notion of an index (or coindex) in a 2-Calabi–Yau triangulated category, whereas the second involves projective presentations of modules over finite-dimensional algebras. In this talk I will explain some joint work in progress with Xin Fang, Mikhail Gorsky, Yann Palu and Pierre-Guy Plamondon, in which we show how to lift the relationship between these two computations to an equivalence of extriangulated categories, as well as generalise to the case of indices in stably 2-Calabi–Yau Frobenius exact categories. Part of the talk will serve as a brief introduction to well-behaved classes of extriangulated categories and their relationship to relative homological algebra as developed by Auslander and Solberg.

Homological conjectures has been a central theme in representation theory of finite dimensional algebras. In this talk I will give an overview of my joint works with Øyvind Solberg (and collaborators) on the reduction techniques we developed for various homological invariants, such as the finitistic dimension, Gorensteinness and singular equivalences, and the finite generation condition for Hochschild cohomology.

For a ring R, we say that “injectives generate” if the injective modules generate the unbounded derived category as a triangulated category with arbitrary coproducts. I will discuss this property in the context of finite dimensional algebras (where it implies the finitistic dimension conjecture, and where I know of no examples where injectives don’t generate). I will concentrate particularly on some techniques to prove that injectives generate for specific algebras, in the hope that the audience can suggest examples where these techniques might not work.

In the full generality in which Solberg, Snashall, and coauthors introduced support varieties via Hochschild cohomology, the best understood case is perhaps that of Koszul algebras. Work by Briggs and Gelinas suggests why this should have been so: in particular, they show that the Hochschild cohomology of an algebra surjects along a well-known canonical map onto the \(A_\infty\)-center of the Yoneda algebra of the simple modules of the algebra. Looking to Koszul algebras, one finds that they are characterized by the Yoneda algebra of the simples having trivial \(A_\infty\)-structure, allowing one to work with the graded centre instead. Consequently, checking whether a Koszul algebra has a good theory of support varieties, i.e. satisfies the (Fg) conditions, becomes far easier than what is usually the case. In this talk, I will present progress on generalizing this picture to T-Koszul algebra, a notion introduced by Green, Reiten and Solberg, and further developed by Madsen. This is based in part on joint work with Johanne Haugland.

We study a certain family of 2-Calabi-Yau tilted algebras, which we call dimer tree algebras, since they may be realized as quotients of dimer algebras on a disk. These algebras are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra \(B\), we construct a polygon \(\mathcal{S}\) with a checkerboard pattern in its interior that defines a category diag\((\mathcal{S})\). Its indecomposable objects are the 2-diagonals in \(\mathcal{S}\), and its morphisms are generated by pivoting moves between the 2-diagonals. We prove that the category diag\((\mathcal{S})\) is equivalent to the stable syzygy category of the algebra \(B\).

As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type \(\mathbb{A}\). In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon \(\mathcal{S}\) is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed from the quiver.

In this talk we give a description of full exceptional sequences in the derived category of a gentle algebra and determine which gentle algebras admit full exceptional sequences. We describe when an exceptional sequence can be completed to a full exceptional sequence in terms of cuts of the surface. Finally, we examine the (extended) braid group action on the set of full exceptional sequences in the case of gentle algebras. This is joint work with Wen Chang.

Thanks to recent work of Opper-Plamondon-Schroll (and Palu-Pilaud-Plamondon), every gentle algebra can be realized as a surface algebra of an oriented surface with certain decorations, and the bounded derived category can be understood in terms of homotopy classes of suitable graded arcs on that surface. I will explain a classification of semiorthogonal decompositions of both the perfect derived category and the bounded derived category of a gentle algebra (the classification is the same for both) in terms of suitable cuts of the surface. This is a joint work with Jakub Kopriva.