Seminar
Friday, 03 December 2021
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13:15, Room T2-149
Tiago Cruz (Stuttgart): Relative dominant dimension and quality of split quasi-hereditary covers
Abstract: The dominant dimension of a finite-dimensional algebra A is a homological invariant measuring the connection between module categories A-mod and B-mod, where B is the endomorphism algebra of a faithful projective-injective A-module.
In this talk, we will discuss generalisations of dominant dimension and how they can be used to measure the quality of split quasi-hereditary covers. The Schur algebra S(d, d) together with its faithful projective-injective module is a classic example of a split quasi-hereditary cover of the group algebra of the symmetric group on d letters.
For a regular email announcement please contact birep.
Future Talks
Wednesday, 08 December 2021
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10:15, Room V4-112
Hernán Giraldo (Medellin): Shapes of Auslander-Reiten triangles in the stable category of modules over repetitive algebras
Abstract: Let k be an algebraically closed field, let A be a finite dimensional k-algebra, and let  be the repetitive algebra of A. For the stable category stmod(Â) of finitely generated left Â-modules, we show that the irreducible morphisms fall into three canonical forms: (i) all the component morphisms are split monomorphisms; (ii) all of them are split epimorphisms; (iii) there is exactly one irreducible component. We next use this fact in order to describe the shape of the Auslander-Reiten triangles in stmod(Â). We use the fact (and prove) that every Auslander-Reiten triangle in stmod(Â) is induced from an Auslander-Reiten sequence of finitely generated left Â-modules. Finally, we will talk about applications of this last result.
Friday, 28 January 2022
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13:15, Zoom (Please contact birep for login information.)
Charley Cummings (Bristol): tba
Friday, 04 February 2022
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13:15, Room T2-149
Jun Maillard (Saint-Etienne): tba
Seminar Archive
Friday, 26 November 2021
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13:15, Room T2-149
Tobias Barthel (Bonn): Stratifying integral representations of finite groups
Abstract: Classifying all integral representations of finite groups up to isomorphism is essentially impossible. In this talk, we will introduce an integral version of the stable module category for a finite group G and then explain how to use it to give a `generic' classification of integral G-representations. Our results globalize the modular case established by Benson, Iyengar, and Krause and relies on the notion of stratification in tensor triangular geometry developed in joint work with Heard and Sanders. Time permitting, I will discuss some further directions.
Friday, 19 November 2021
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13:15, Room T2-149
Adam-Christiaan van Roosmalen (Hasselt): Auslander's formula and correspondence for exact categories
Abstract: Following Auslander's philosophy, one can study a small abelian category A by studying the category of finitely presented functors on A. Auslander's formula provides a way to move back: one can recover A as the quotient of the category of finitely presented functors on A by the Serre subcategory of all effaceable functors.
One impediment to formulating a version of Auslander's formula for an exact category E is that the category of finitely presented functors on E is independent of the chosen exact structure: it only depends on the underlying additive category. To address this, we introduce the category of admissibly presented functors on an exact category. In this talk, I will focus on this category of admissibly presented functors, and use it to formulate a version of Auslander's formula and correspondence for exact categories.
This talk is based on joint work with Ruben Henrard and Sondre Kvamme.
Friday, 12 November 2021
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13:15, Room T2-149
Scott Balchin (Bonn): The smashing spectrum of a tt category
Abstract: In joint work with Greg Stevenson, we prove that the frame of smashing tensor ideals of a big tt-category is always spatial. As such, by Stone duality, we are afforded a space: the smashing spectrum. In this talk, I will report on the construction of this new invariant via lattice theoretic techniques, and its relation to the Balmer spectrum. In particular, we will see that there is a surjective comparison map which detects the failure of the telescope conjecture.
Friday, 29 October 2021
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13:15, Room T2-149
Håvard Terland (Trondheim): Identifying components of mutation quivers
Abstract: Tau-tilting theory completes tilting theory from the perspective of mutation. Letting points be support-tau tilting pairs and arrows indicate (left) mutation, one then obtains a so-called mutation quiver whose underlying graph is regular.
The goal of this talk will be to introduce tau-tilting theory and discuss recent efforts to better understand the connected components of (the underlying graphs of) mutation quivers of support tau-tilting pairs.
Friday, 22 October 2021
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13:15, Zoom
Kaveh Mousavand (Kingston): Orbits of bricks of finite dimensional algebras
Abstract: In 2014, Adachi-Iyama-Reiten introduced the tau-tilting theory of finite dimensional algebras to fix the deficiency of mutation of tilting modules. The subject soon received a lot of attention and developed in various directions. Around the same time, Chindris-Kinser-Weyman studied the moduli spaces of quiver representations, in particular the behaviour of Schur representations (bricks) of finite dimensional algebras. In this talk, we try to relate these two lines of research. More specifically, motivated by a conjectural geometric counterpart for the algebraic notion of tau-tilting finiteness, we treat module varieties of finite dimensional algebras, in particular the orbit of bricks. We show that, unlike arbitrary bricks, those used for the labelling of the lattice of functorially finite torsion classes always admit open orbits. From this, we obtain a conceptual proof of the counterpart of the first Brauer-Thrall conjecture for bricks. We push these results further in the treatment of another Brauer-Thrall type conjecture which is still open in full generality.
This is based on my Joint work with Charles Paquette (Royal Military College, Canada).
Friday, 10 September 2021
Thursday, 09 September 2021
Wednesday, 08 September 2021
Tuesday, 07 September 2021
Monday, 06 September 2021
Friday, 03 September 2021
Thursday, 02 September 2021
Wednesday, 01 September 2021
Friday, 16 July 2021
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13:15, Zoom
Joseph Chuang (London): Rank functions on triangulated categories
Abstract: A rank function is a nonnegative real-valued, additive, translation-invariant function on the objects of a triangulated category satisfying the triangle inequality on distinguished triangles. Rank functions on the perfect derived category of a ring are related to Sylvester rank functions on finitely presented modules, and therefore, via the work of Cohn and Schofield, to representations of the ring over skew fields. In my talk I will focus on examples. This is joint work with Andrey Lazarev.
Friday, 09 July 2021
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13:15, Zoom
Paolo Stellari (Milano): Uniqueness of enhancements: derived and geometric categories
Abstract: In this talk we address several open questions and generalize the existing results about the uniqueness of enhancements for triangulated categories which arise as derived categories of abelian categories or from geometric contexts. If time permits, we will also discuss applications to the description of exact equivalences. This is joint work with A. Canonaco and A. Neeman.
Friday, 25 June 2021
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13:15, Zoom
Martin Kalck: A surface and a threefold with equivalent singularity categories
Abstract: We start with an introduction to singularity categories and equivalences between them. In particular, we recall known results about singular equivalences between commutative rings, which go back to Knörrer, Yang, Kawamata and a joint work with Karmazyn. Then we explain a new singular equivalence between an affine surface and an affine threefold. This seems to be the first (non-trivial) example of a singular equivalence involving rings of even and odd Krull dimension.
Friday, 18 June 2021
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13:15, Zoom
Norihiro Hanihara (Nagoya): Yoneda algebras from additive generators
Abstract: Yoneda algebras form a class of algebras which have widely been studied in ring theory and representation theory. They are defined for a ring A and an A-module M as the direct sum of Ext^i_A(M,M) over all i endowed with the Yoneda product. We discuss these Yoneda algebras in the following setting: A is a finite dimensional algebra of finite representation type, and M is the additive generator for the category of A-modules. We will give some fundamental results on such Yoneda algebras, such as coherence, Gorenstein property, periodicity, and a description of the stable category of Cohen-Macaulay modules.
Friday, 11 June 2021
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13:15, Zoom
Job Rock (Boston): Composition series of arbitrary cardinality
Abstract: We discuss a generalization of the notion of a composition series in an abelian category to one of arbitrary cardinality. Then we discuss sufficient axioms that yield "Jordan—Hölder—Schreier like" theorems. Examples of these settings include pointwise finite-dimensional persistence modules and Prüfer modules. We will conclude with evidence that suggests the axioms are necessary for our "Jordan—Hölder—Schreier like" theorems. This is joint work with Eric J. Hanson.
Friday, 30 April 2021
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13:15, Zoom
Sondre Kvamme (Uppsala): Admissibly presented functors
Abstract: Wanting to extend the functorial approach of Auslander to exact categories, we introduce the category of admissibly presented functors mod_{adm}(E) for an exact category E. Using this category, we extend Auslanders formula from abelian to exact categories. Furthermore, we characterize exact categories equivalent to categories of the form mod_{adm}(E), and we show that they have properties similar to module categories of Auslander algebras. For a fixed idempotent complete category C, we use this construction to show that exact structures on C are in bijection with certain resolving subcategories of mod C, and we compare this with the bijection to certain Serre subcategories of mod C due to Enomoto. This is joint work with Ruben Henrard and Adam-Christiaan Van Roosmalen.
Friday, 23 April 2021
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13:15, Zoom
Rene Marczinzik (Stuttgart): Homological algebra and combinatorics
Abstract: We show that the incidence algebra of a finite lattice L is Auslander regular if and only if L is distributive. As an application we show that the order dimension of L coincides with the global dimension of its incidence algebra when L has at least two elements and we give a categorification of the rowmotion bijection for distributive lattices. At the end we discuss the Auslander regular property for other objects coming from combinatorics. This is joint work with Osamu Iyama.
We also report on recent joint work with Aaron Chan, Erik Darpö and Osamu Iyama on fractionally Calabi-Yau algebras and their trivial extension algebras with relations to combinatorics and lattices.
Friday, 16 April 2021
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14:15, Zoom
Xiao-Wu Chen (Hefei): Skew group categories, algebras associated to Cartan matrices and folding of root lattices
Abstract: The folding of root lattices is fundamental in Lie theory when getting from the simply-laced cases to the non-simply-laced cases. Following Gabriel and Geiss-Leclerc-Schroer, the relevant root lattices are categorified by certain module categories. We obtain a categorification of the folding projection, namely a certain functor between the module categories whose K_0-shadow is the folding projection. The main tools are skew group categories and finite EI categories of Cartan type. This is joint with Ren Wang at USTC.
For information on earlier talks please check the complete seminar archive.
