Workshop on bricks and endofinite representations – Abstracts

Critical modules
Lidia Angeleri Hügel

To every torsion class t in the category modA of finite dimensional modules over a finite dimensional algebra A we can associate a family of indecomposable pure-injective modules, usually infinite dimensional, that are called critical. We will see that these modules encode information about neighbours of t in the lattice torsA of torsion classes in modA, and we will discuss how they determine the existence of infinite dimensional bricks. This is a report on joint work with Rosie Laking and Francesco Sentieri.

On (co)silting bijections involving the category of large projective presentations
Diego Alberto Barceló Nieves

Based on results by Adachi-Iyama-Reiten, Marks-Šťovíček, Pauksztello-Zvonareva and Adachi-Tsukamoto, García successfully completed a commutative 'triangular prism' of bijections connecting the classes of support τ-tilting modules, functorially-finite torsion pairs and left finite wide subcategories in the category of finitely-generated A-modules—where A is a finite-dimensional algebra over an algebraically closed field—to the classes of 'silting objects', complete cotorsion pairs and thick subcategories with enough injectives in the category of projective presentations of objects in mod(A)—which has many powerful properties. In this talk, we will present some generalizations of these results to the realm of infinite-dimensional modules, as well as their dualizations. It is based on joint work in progress with Lidia Angeleri Hügel.

Generic bricks over tame algebras
Raymundo Bautista

In the following A is a finite dimensional k-algebra with k an algebraically closed field. Following K. Mousavand and Ch. Paquette we say that the algebra A is brick-infinite if it admits an infinite family of non-isomorphic finite-dimensional bricks. It is called brick-continuous if it admits an infinite family of non-isomorphic bricks with the same finite dimension over k.

In this talk we consider the following theorem: If A is a tame algebra then A is brick-continuous if and only if A admits a generic brick module. The main tool used for the proof of this result is the use of a full but not faithful functor from the category of representations of a minimal bocs to the category of A-modules, sending a generic module in the first category to a generic module in the second category; see [W.W. Crawley-Boevey, Tame algebras and generic modules, 1991]. In the talk we will see how the properties of the minimal bocs are used for proving the theorem.

The talk is based on joint work with Efrén Pérez and Leonardo Salmerón; see arXiv:2408.16127.

Radical preservation & algebras of quasi-uniform Loewy length
Odysseas Giatagantzidis

The Finitistic Dimension Conjecture, a central problem in the representation theory of Artin algebras, investigates the finiteness of the little and big finitistic dimensions. This talk focuses on the relationship between the finitistic dimensions of two Artin algebras connected via a homomorphism.

Kirkmann, Kuzmanovich and Small showed that if f is a surjective Artin algebra homomorphism from A to B with kernel contained in the Jacobson radical of A and such that the projective dimension of B as a right module over A is finite, then the finiteness of the little or big finitistic dimension of B implies the finiteness of the respective dimension of A. We extend this classical result by showing that the surjectivity condition can be weakened to the condition of f being radical-preserving.

Furthermore, we introduce the notion of quasi-uniform Loewy length for bound quiver algebras and demonstrate that every such algebra has finite big finitistic dimension. This provides a new class of algebras with this property, which we illustrate through an explicit example.

τ-tilting theory and silting theory of skew group algebra extensions
Yuta Kimura

This talk is based on joint work with Koshio, Kozakai, Minamoto and Mizuno. There are many known results on support τ-tilting modules for various kinds of algebras. One reason for studying these modules is that they correspond to many important objects, including two-term silting complexes, torsion classes and semibricks. In this talk, we focus on such objects over a finite-dimensional algebra A on which a finite group G acts. We show that G-stable support τ-tilting modules are in bijection with (mod G)-stable support τ-tilting modules over the skew group algebra AG, via the restriction-induction functors. A similar bijection also holds for other objects related to support τ-tilting modules. As an application of these results, we obtain consequences regarding the τ-tilting finiteness of both A and AG. Our findings generalize some earlier results by Huang-Zhang, Breaz-Marcus-Modoi and Koshio-Kozakai.

Kleiner-Roiter regular subalgebras vs Frobenius extensions
Julian Külshammer

Typical examples of bricks are standard modules for a quasi-hereditary algebra. They however don't form a semibrick as there are in general many non-trivial homomorphisms between them. Nevertheless, in this situation, one can construct a subalgebra of the quasi-hereditary algebra satisfying a regularity condition due to Kleiner and Roiter. Inducing the simple modules for this subalgebra gives the standard modules for the quasi-hereditary algebra. In this talk, we will explore how Kleiner-Roiter regularity interacts with another common property on algebra extensions, namely Frobenius extensions. This is joint work with Tomasz Brzezinski, Teresa Conde, and Steffen Koenig.

Wide intervals and closed rigid sets
Rosanna Laking

An old result of Ringel states that semibricks (i.e., sets of pairwise Hom-orthogonal bricks) in the category modA of finite-dimensional modules over a finite-dimensional algebra are in bijection with the wide subcategories of modA (i.e. subcategories that are closed under kernels, cokernels and extensions). These subcategories are interesting abelian length subcategories of modA that arise naturally in various contexts throughout the representation theory of A.

In this talk we will be focusing on one such context: when non-trivial wide subcategories arise as intersections of a torsion class T and a torsion-free class V in modA. Such a pair T and V corresponds to an interval in the lattice of torsion pairs in modA that Asai and Pfeifer call wide intervals. We will report on joint work with Lidia Angeleri Hügel and Francesco Sentieri in which we show that wide intervals are parametrised by certain closed sets of the Ziegler spectrum of the unbounded derived module category of A.

Minimal brick-infinite algebras
Charles Paquette

Many open problems related to bricks can be reduced to minimal brick-infinite algebras. These are the algebras which admit infinitely many non-isomorphic bricks but every proper quotient of them is brick-finite. In this talk, I will present some algebraic and geometric properties of minimal brick-infinite algebras. I will also explain how, in some cases, one can construct integral limit rays which lie outside of the g-vector fan. The existence of such rays is conjecturally believed to be tied to the existence of infinite families of bricks of the same dimension. If time permits, we will consider particular classes of algebras for which more can be said.

Generic modules for the category of filtered by standard modules
Jesús Efrén Pérez Terrazas

The notion of homological system is related to that of quasi-hereditary algebra, and in a similar way we study the category of filtered by standard modules.

In this talk it will be discussed versions of classical results for this category, like Tame and Wild dichotomy, and the existence of generic modules under suitable conditions.

Stable modules, τ-regular components and valued quivers
Calvin Pfeifer

Moduli spaces of semistable modules and τ-regular components of schemes of modules may be seen as two sides of the same coin. In particular, τ-tilting theory can be used as an approach towards classifying stable modules. We will illustrate the theory with Geiß-Leclerc-Schröer's algebras associated to valued quivers. Their algebras are often of wild representation type, but they can be deformed to hereditary algebras which makes parts of their representation theory still approachable.

The second Brauer-Thrall conjecture for subcategories
Kevin Schlegel

For the module category of a finite dimensional algebra we consider subcategories that can be, in some sense, finitely defined. We conjecture that such a subcategory is either of finite or strongly unbounded type, generalizing the second Brauer-Thrall conjecture. Using generic modules and the Zielger spectrum, the newly formulated conjecture is proven over algebraically closed fields if the Krull-Gabriel dimension of the algebra is defined. Motivated by this, we consider generic modules that arise from parametrizations of finite dimensional modules over affine varieties and relate their existence to the conjecture.

Characterisation of band bricks over certain string algebras
Annoy Sengupta

In a recent work, Dequêne et al. provided a connection between some band bricks over a particular family of gentle algebras and perfectly clustering words over a linearly ordered alphabet using a geometric model of the module category. We use classical combinatorial descriptions of the module category of a string algebra to significantly generalise this result by characterising band bricks for string algebras whose underlying quiver does not contain a directed cycle in terms of weakly perfectly clustering pairs of crowns – a variant of perfectly clustering words. Furthermore, the combination of our result and a result of Mousavand and Paquette provides an algorithm to determine whether such a string algebra is brick-infinite. This is joint work with Amit Kuber.

A study of Sylvester rank functions via functor categories
Simone Virili

Given a ring R, a Sylvester rank function is a suitable invariant of the category of finitely presented right R-modules with values in the non-negative reals. In this talk I will start by observing that any Sylvester rank function can be uniquely extended to a suitable "normalized length function" on the category of functors from finitely presented modules to Abelian groups. This extension of the ground category has many advantages, in fact, the richer structure of functor categories makes it possible to use tools like torsion-theoretic localizations, rings of definable scalars and the Ziegler spectrum in the study of rank functions. To prove this point, I will give several examples of results about rank functions whose original proof is technically challenging, but become almost trivial when extended to functors.

Quantum cluster algebras of weighted projective lines
Fang Yang

Fan Qin used quantum Caldero–Chapoton characters of representations of acyclic quivers to categorify the quantum cluster algebras of corresponding quivers. Bernstein–Rupel showed that quantum cluster characters are homomorphisms from Hall algebras to the specialized quantum cluster algebras of acyclic quivers.

In this talk, we will construct such quantum cluster characters from Hall algebras of weighted projective lines to quantum torus and show that the quantum cluster characters of indecomposable rigid objects are generic. Moreover we prove that the quantum cluster algebra associated to a weighted projective line is a subquotient of the Hall algebra. This talk is based on a joint work with Fan Xu.

The shift homological spectrum and the space of rank functions
Alexandra Zvonareva

In this talk for a compactly generated triangulated category I will introduce two topological spaces that can be viewed as non-monoidal analogues of the Balmer spectrum and the homological spectrum from tensor-triangulated geometry, called the shift spectrum and shift homological spectrum respectively. The aim is to classify certain families of thick subcategories of the compact objects defined analogously to radical thick subcategories. The shift homological spectrum has as points maximal shift-invariant Serre subcategories of the abelianisation of T^c and is equipped with a support topology given by vanishing sets of compact objects. While the shift spectrum is defined in terms of certain thick subcategories of T^c and can be realised as the Kolmogorov quotient of the shift homological spectrum. The Balmer spectrum of a tensor-triangulated categories generated by the unit is a subspace of the shift spectrum, justifying the claim of it being a non-monoidal analogue.

The connection to endofinite objects comes through an intimate relationship between the shift homological spectrum and the set of irreducible integral rank functions which plays a central role in this study. For example, the kernel of each integral rank function is a radical thick subcategory. The main tool used to compute the spaces is an alternative description of the shift (homological) spectrum in terms of the Ziegler spectrum. Which can be used to provide examples of radical thick subcategories, and show in certain cases, for instance for the perfect derived category of a tame hereditary algebra or thick subcategories of a monogenic tensor triangulated category, that every thick subcategory is radical.

This talk is based on joint work with I. Bird and J. Williamson.