Seminars in Triangulated Categories

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Emanuele Macrì: Deformations of t-structures

14th November, 2024

Bridgeland stability conditions were introduced about 20 years ago, with motivations from algebraic geometry, representation theory, and physics. One of the fundamental problems is that we currently lack methods to construct and study such stability conditions in full generality. In this talk, I will present a new technique to construct stability conditions by deformations, based on joint works with Li, Perry, Stellari, and Zhao. As an application, we can construct stability conditions on very general abelian varieties and deformations of Hilbert schemes of points on K3 surfaces, and we prove a conjecture by Kuznetsov and Shinder on quartic double solids.

David Jorgensen: Asymptotic vanishing of cohomology in triangulated categories

7th November, 2024

Given a graded-commutative ring acting centrally on a triangulated category, the main result of this talk shows that if the cohomology of a pair of objects of the triangulated category is finitely generated over the ring acting centrally, then the asymptotic vanishing of the cohomology is well-behaved. In particular, enough consecutive asymptotic vanishing of cohomology implies all eventual vanishing. Several key applications are also given.

Martin Frankland: Toda brackets in n-angulated categories

17th October, 2024

Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.

Janina Letz: Generation time for biexact functors and Koszul objects in triangulated categories

25th July, 2024

One way to study triangulated categories is through finite building. An object X finitely builds an object Y, if Y can be obtained from X by taking cones, suspensions and retracts. The X-level measures the number of cones required in this process; this can be thought of as the generation time. I will explain the behaviour of level with respect to tensor products and other biexact functors for enhanced triangulated categories. I will further present applications to the level of Koszul objects.

Kent Vashaw: A Chinese remainder theorem and Carlson theorem for monoidal triangulated categories

6th June, 2024

Carlson's connectedness theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. In this talk, we will discuss a generalization, where it is proved that the Balmer support for an arbitrary monoidal triangulated category satisfies the analogous property. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.

Dave Benson: The nucleus and the singularity category of cochains on the classifying space

30th May, 2024

The definition of the nucleus was originally formulated in joint work with Carlson and Robinson, to capture the supports of modules with no cohomology. This definition works in various contexts such as finite groups, restricted Lie algebras, and more generally, suitable triangulated categories of modules. In the finite group context it has a characterization in terms of subgroups whose centralizer is not p-nilpotent. In the restricted Lie algebra context, it is described in terms of the Richardson orbit, at least for large primes. Recent work with Greenlees has highlighted a connection with the singularity category of the cochains on the classifying space, in the group theoretic context. My plan is to give an introduction to these ideas.

Kent Vashaw: A Chinese remainder theorem and Carlson’s theorem for monoidal triangulated categories

27th May, 2024

Carlson’s Connectedness Theorem for cohomological support varieties is a fundamental result which states that the support variety for an indecomposable module of a finite group is connected. For monoidal triangulated categories, the Balmer spectrum is an intrinsic geometric space associated to the category which generalizes the notion of cohomological support for finite groups. In this talk, we will discuss a generalization of the Carlson Connectedness Theorem: that the Balmer support of any indecomposable object in a monoidal triangulated category with a thick generator is a connected subset of the Balmer spectrum. This is shown by proving a version of the Chinese remainder theorem in this context, that is, giving a decomposition for a Verdier quotient of a monoidal triangulated category by an intersection of coprime thick tensor ideals.

Milen Yakimov: Non-commutative tensor triangular geometry and finite tensor categories

27th May, 2024

Describing the thick ideals of a monoidal triangulated category is a key component of the analysis of the category. We will show how this can be done by non-commutative tensor triangular geometry (NTTG), thus extending the celebrated Balmer’s theorem from the symmetric case. We will then use NTTG to analyse the stable categories of finite tensor categories, which play an important role in representation theory, mathematical physics and quantum computing. We will present general results linking this approach to the traditional one through cohomological support, based on a notion of categorical centers of cohomology rings of monoidal triangulated categories.

Eloísa Grifo: Searching for modules that are not virtually small

2nd May, 2024

Pollitz gave a characterization of complete intersection rings in terms of the triangulated structure of their derived category, akin to the Auslander-Buchsbaum-Serre characterization of regular rings. In this talk, we will explore how to bring this characterization back to the world of modules, and discuss the role of cohomological support varieties in solving this problem.

Carles Casacuberta: Homotopy reflectivity is equivalent to the weak Vopenka principle

21st March, 2024

We discuss reflectivity of colocalizing subcategories of triangulated categories under suitable set-theoretical assumptions. In earlier joint work with Gutierrez and Rosicky, we proved that if K is any locally presentable category with a stable model category structure, then Vopenka's principle implies that every full subcategory L of the homotopy category of K closed under products and fibres is reflective. Moreover, if L is colocalizing, then the reflection is exact. Using recent progress in large-cardinal theory, we show that the statement that every full subcategory closed under products and fibres is reflective is, in fact, equivalent to the so-called weak Vopenka principle. Hence this statement cannot be proved using only the ZFC axioms.

Gregory Arone: The tensor triangular geometry of functor categories

14th March, 2024

We consider the (infinity) category of excisive (aka polynomial) functors from Spectra to Spectra. Understanding this category is a basic problem in functor calculus. We will approach it from the perspective of tensor triangular geometry. Day convolution equips the category of excisive functors with the structure of a rigid monoidal triangulated category. We describe completely the Balmer spectrum of this category, i.e., its spectrum of prime tensor ideals. This leads to a Thick Subcategory Theorem for excisive functors. A key ingredient in the proof is a blueshift theorem for the generalized Tate construction associated with the family of non-transitive subgroups of products of symmetric groups. If there is time, I will say something about work in progress to extend these results to more general functor categories.

Leonid Positselski: Semi-infinite algebraic geometry of quasi-coherent torsion sheaves

8th February, 2024

This talk is based on the book Semi-infinite algebraic geometry of quasi-coherent sheaves on ind-schemes—quasi-coherent torsion sheaves, the semiderived category, and the semitensor product. I will start with some examples serving as special cases of the general theory, such as the tensor structure on the category of unbounded complexes of injective quasi-coherent sheaves on a Noetherian scheme with a dualizing complex. Then I will proceed to explain the setting of a flat affine morphism of ind-schemes into an ind-Noetherian ind-scheme with a dualizing complex, and the main ingredient concepts of quasi-coherent torsion sheaves, pro-quasi-coherent pro-sheaves, and the semiderived category. In the end, I will spell out the construction of the semi-tensor product operation on the semi-derived category of quasi-coherent torsion sheaves, making it a tensor triangulated category.

Scott Balchin: A jaunt through the tensor-triangular geometry of rational G-spectra for G profinite or compact Lie

26th October, 2023

In this talk, I will report on joint work with Barnes-Barthel and Barthel-Greenlees which analyses the category of rational G-equivariant spectra for G a profinite group or compact Lie group respectively. In particular, I will focus on a series of results regarding the Balmer spectra of these categories, and how the topology of these topological spaces informs structural results regarding the category.

Yann Palu: 0-Auslander extriangulated categories

13th October, 2023

Categorification of cluster algebras has instilled the idea of mutation in representation theory. Nice theories of mutation, for some forms of rigid objects, have thus been developed in various settings. In a collaboration with Mikhail Gorsky and Hiroyuki Nakaoka, we axiomatized the similarities between most of those settings under the name of 0-Auslander extriangulated categories. The prototypical example of a 0-Auslander extriangulated category is the category of two-term complexes of projectives over a finite-dimensional algebra. In this talk, we will give several examples of 0-Auslander categories, and explain how they relate to two-term complexes.

Matthew Hamil: Stratifying rings for the stable category of modules over detecting Lie subalgebras

12th May, 2023

Tensor triangulated categories arise across many areas in mathematics. Examples of tensor triangulated categories include the derived category of perfect complexes of a suitably nice scheme X, the stable category of kG-modules where G is a finite group and k is a field of characteristic dividing the order of G, and many more. In his 2005 paper titled The spectrum of prime ideals in tensor triangulated categories, Paul Balmer associates a topological space, now called the Balmer spectrum, to tensor triangulated categories in a manner analogous to the spectrum of a commutative ring.

In the early 2000s Boe, Kujawa, and Nakano published a series of papers studying Lie superalgebras. They show that for classical Lie superalgebras 𝔤 admitting stable and polar actions of the algebraic group G0, there exist interesting subalgebras 𝔣 which 'detect' the cohomology of 𝔤 relative to its even part. They also consider the stable category of 𝔤 modules which are completely reducible over 𝔤0. This is a TTC, and in special cases, including for the detecting subalgebras and the Lie superalgebra 𝔤𝔩(m|n), they give a concrete description of the Balmer spectrum using methods from geometric invariant theory. In our talk we discuss how to recover this result for the detecting subalgebras using a different approach. Namely, we will discuss some circumstances in which we have a stratifying ring.

José Vélez Marulanda: Exact weights for triangulated categories

30th April, 2023

Inspired by the work of P. Bubenik et al., we define exact weights on objects in a triangulated category. In particular, we extend the concept of path metrics for abelian categories to triangulated categories and discuss some of their properties. We are particularly interested in triangulated categories induced by Frobenius categories.

Alexey Elagin: Dimension for triangulated categories

25th April, 2023

I will talk about two notions of dimension of a triangulated category. The first one is the classical Rouquier dimension, based on generation time with respect to a generator, while the second one is the more recent concept of Serre dimension, based on behavior of iterations of the Serre functor. I will propose 'ideal' properties of dimension that one would like to have, and compare them to properties of Rouquier and Serre dimension, both known and conjectural. Various examples of categories where dimension is known will be given and discussed.

Francesca Fedele: Ext-projectives in subcategories of triangulated categories

21st February, 2023

Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b→c in C, then the triangle x→b→c→ extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C' of T and see how this operation is related to Iyama-Yoshino mutation of C with respect to a rigid subcategory. Time permitting, we will see the application of the result to cluster categories of type A.

Henning Krause: Central support for triangulated categories

7th February, 2023

Various notions of support have been studied in representation theory (by Carlson, Snashall-Solberg, Balmer, Benson-Iyengar-Krause, Friedlander-Pevtsova, Nakano-Vashaw-Yakimov, to name only few). My talk offers some new and unifying perspective: For any essentially small triangulated category the centre of its lattice of thick subcategories is introduced; it is a spatial frame and yields a notion of central support. A relative version of this centre recovers the support theory for tensor triangulated categories and provides a universal notion of cohomological support. Along the way we establish Mayer-Vietoris sequences for pairs of central subcategories.