Köln Algebra and Representation Theory Seminar

Abstract: Given a family of coproduct-preserving tensor-triangulated (tt) functors between rigidly-compactly generated tt-categories, it is natural to ask when they are jointly conservative. Such joint conservativity is the minimal requirement for attempting to descend tt-geometric information along the family. In this talk, I will present a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories, highlighting its homological flavor. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra. The talk is based on joint work with Natalia Castellana. 

Abstract: Cluster algebras and cluster categories have had a revolutionary impact on tilting theory. In particular, they have inspired various theories of mutation, which have become central research topics since then. Important examples include cluster-tilting mutation,𝜏-tilting mutation and relative rigid mutation. Recently, Gorsky—Nakaoka—Palu showed that these, and other, mutations may be unified via the mutation of maximal rigid objects in 0-Auslander extriangulated categories (with certain finiteness assumptions).

For a finite-dimensional algebra, 𝜏-tilting mutation is encoded by a geometric object, called its g-vector fan. In my talk, I expand this notion and define a polyhedral fan which encodes the mutation theories of the 0-Auslander extriangulated categories above. I will illustrate its properties through many examples. Building on my previous work, I will explain how thick subcategories induce an admissible partition of the fan and introduce the notion of a morphism of partitioned fans. This provides a unifying perspective on various results on $\tau$-cluster morphism categories and picture categories of myself and Erlend Børve. This is joint work-in-progress with Erlend Børve.

Abstract: The crystallographic Cambrian lattices are the lattices of torsion classes of representation finite hereditary algebras. Rognerud has shown that Cambrian lattices of linear type A, better known as Tamari lattices, are fractionally Calabi Yau. That is a power of the Serre functor on the derived category of their incidence algebra agrees with a power of the shift.

The m-cluster categories are generalizations of cluster categories which exhibit very similar combinatorics. In particular there is a family of m-cluster tilting objects connected by a notion of mutation. In this talk we will describe the Serre functors of crystallographic Cambrian lattices using the combinatorics of 2-cluster tilting objects in 2-cluster categories. As a consequence we show that all crystallographic Cambrian lattices are fractionally Calabi-Yau. 

Abstract: A metric on a triangulated category, as developed by Neeman, provides a recipe for constructing a metric completion of the category. These completions are guaranteed to be triangulated categories as well and have recently been used to study, among other things, derived Morita theory, cluster categories, and t‑structures. The aim of this talk is to examine metric completions of bounded derived categories of hereditary rings and their connection to the concept of universal localisation. Notably, we explicitly describe the completions of bounded derived categories of hereditary finite dimensional tame algebras and hereditary commutative noetherian rings with respect to additive good metrics.

Abstract: Joint work with Lleonard Rubio y Degrassi. Relative Hochschild (co)homology was first defined by Hochschild in 1956. It has been used by Gerstenhaber-Schack in the context of deformation, and recently by Cibils, Lanzilotta, Marcos, Schroll and Solotar to describe how Hochschild cohomology of bound quiver changes when adding or deleting an arrow of a bound quiver. Let A be a bound quiver algebra and let B be a subalgebra with the same semisimple part. We give a sufficient condition for the first relative Hochschild cohomology to be solvable as a Lie algebra. We also defined the contracted fundamental group and show, similar to the normal case, that the dual maps into the first relative Hochschild cohomology.  

Abstract: Canonical algebras have been introduced by Ringel in 1984 as algebras whose module categories have a particularly nice shape and share properties with those of tame quivers. In this talk, we look at other derived equivalent algebras such as the squid algebra and the Coxeter-Dynkin algebra of canonical type (sometimes called octopus algebra). We will use a result of Barot and Lenzing about one-point-extensions to obtain derived equivalences. Our approach turns out to be particularly useful when passing to the general case where quivers are replaced by species. This talk is based on arxiv:2509.17887.

Abstract: Reflexive DG-categories were introduced by Kuznetsov and Shinder to abstract the duality between the perfect and bounded derived categories of a projective scheme. I'll explain a new characterisation of this notion from which one can deduce that this duality extends to certain invariants such as Hochschild cohomology and derived Picard groups. I'll also mention joint work with Matt Booth and Sebastian Opper in which we produce new examples of reflexive DG-categories from topology, representation theory and commutative algebra.

Abstract: Cluster categories and cluster algebras can be described via triangulations of surfaces.  In this talk, I will present a geometric model for the rank two tubes of the Auslander-Reiten quiver of the cluster category of the affine type D. This model is given in terms of homotopy classes of unoriented tagged generalized arcs in the twice punctured disk. We will extend the model for the tube of rank n-2 given by Baur, Bittmann, Gunawan, Todorov and Yıldırım to the two tubes of rank 2. In order to do that, we will extend the definition of generalized tagged arcs to some particular arcs between the two punctures. This talk is based on arxiv:2510.23280. 

Abstract: We give an overview of our results on the representation types of cyclotomic quiver Hecke algebras (also known as cyclotomic Khovanov–Lauda–Rouquier algebras), with particular emphasis on their quiver representation theory. From this perspective, we observe that all tame cases are in fact Brauer graph algebras. We also outline the strategy by which tilting theory is applied to study derived equivalence classes in this setting.

Abstract: A well known construction associates with every reduced positive braid word a rigid module over a preprojective algebra of type A. More generally, one can associate rigid dg modules over the derived preprojective algebra of type A with non-reduced positive braid words.

In this talk we will consider iterated extensions between such rigid dg modules arising from a family of increasing braid words decorated with positive numbers. Using a geometric model based on curves in the plane, we will give a criterion for the existence of a unique rigid extension. As an application, we will show that the rigid extension is invariant under braid moves.

Based on ongoing joint work with Roger Casals. 

Abstract: Combining results from Keller and Buchweitz, we describe the 1-periodic derived category of a finite dimensional algebra of finite global dimension as the stable category of maximal Cohen-Macaulay modules over some Gorenstein algebra. In the case of gentle algebras, using the geometric model introduced by Opper, Plamondon and Schroll, we describe indecomposable objects in this category using homotopy classes of curves on a surface. In particular, we associate a family of indecompoable objects to each primitive closed curve. I will then discuss the dependence of this category on the marked graded surface associated to A. 

Abstract: Nakajima quiver varieties can be used to give geometric realizations of modules over quantum affine algebras. In the graded case, Keller and Scherotzke gave an algebraic description of these varieties in terms of modules over the singular Nakajima category S and showed that their stratification is governed by the derived category of a Dynkin quiver. In this talk, we explain how this picture extends to Nakajima’s n-fold tensor product varieties, which allow one to geometrically realize n-fold tensor products of standard modules. We do this by introducing a category of filtrations of S-modules with a splitting, whose objects are parametrized by the tensor product varieties. We show that this category is equivalent to the module category of another category S^n, and we describe the category of Gorenstein projective S^n-modules via derived categories. 

Abstract: Given a family of coproduct-preserving tensor-triangulated (tt) functors between rigidly-compactly generated tt-categories, it is natural to ask when they are jointly conservative. Such joint conservativity is the minimal requirement for attempting to descend tt-geometric information along the family. In this talk, I will present a criterion for determining when a family of geometric functors is jointly conservative through the lens of purity in compactly generated triangulated categories, highlighting its homological flavor. We introduce the notion of pure descendability and we apply it to two particular situations involving sequential limits of ring spectra. The talk is based on joint work with Natalia Castellana.