Schedule and abstracts

The lattice of torsion pairs tors(A) in the category of finite dimensional modules over a finite dimensional algebra A can be studied in terms of mutation of associated cosilting complexes in the unbounded derived category D(Mod(A)). These cosilting complexes are pure-injective, and they correspond bijectively to maximal rigid sets in the Ziegler spectrum Zg(D(A)) of D(Mod(A)). In this talk, we will describe mutation as an operation that exchanges elements in certain closed subsets of Zg(D(A)). This will allow us to interpret properties of tors(A) in terms of topological conditions on Zg(D(A)). The talk will be based on joint work with Rosanna Laking and Francesco Sentieri.

I will recall the notion of a semiorthogonal decomposition of a triangulated category. I will discuss periodic semiorthogonal decompositions and their relation to spherical functors. I will then give examples of periodic decompositions for flops, effective Cartier divisors and blow-ups in smooth centers of codimension two. This is based on a joint work with A. Bondal and W. Donovan.

I will analyze the transfer of the (co)silting property in the derived category of a module category when we apply the (co)extension of scalars functors induced by morphisms of commutative rings. In particular, we will see that the derived versions of the extension of scalars functors preserve silting objects of finite type. The question when these functors reflect the silting property is more delicate. However, we will see that in many reasonable cases the functors associated to faithfully flat morphisms reflect the silting property. In particular, the property “bounded silting” is Zariski local. The talk is based on a joint work with Michal Hrbek and George-Ciprian Modoi.

A classical theorem by Rickard roughly states that the notion of derived equivalence for (noncommutative) rings is independent of the specific type of derived category: one can consider the unbounded derived category of all (left) modules, or any of its relevant triangulated subcategories (like the category of perfect complexes). It is natural to ask if Rickard’s theorem can be generalized so as to include, for instance, also the geometric case where rings are replaced by (quasi-compact and quasi-separated) schemes. It will be shown that a positive answer can be given using the theory of weakly approximable triangulated categories, in combination with some results about uniqueness of dg enhancements. This is joint work, partly in progress, with Amnon Neeman and Paolo Stellari.

I will explain the concept of lax additivity which is an (∞, 2)-categorical analog of the familiar 1-categorical notion of additivity. In this context, direct sums get replaced by lax sums leading to lax variants of matrices along with rules for how to multiply them. We will illustrate the resulting methods by investigating periodicity phenomena for mutations of semiorthogonal decompositions.

Based on joint work with Christ-Walde and Kapranov-Schechtman.

By considering annihilators of objects in the singularity category of a commutative Noetherian ring, we obtain decompositions and upper bounds on the Rouquier dimension. In this talk, I will consider the Gorenstein case and try to motivate our proofs by recalling some now classical ideas in commutative algebra starting from the Depth Lemma. This is joint work with Ryo Takahashi.

Let X be an algebraic variety. Is X determined by its derived category of “sheaves”? Positive answers to different versions of this question were given notably by Bondal-Orlov, and Balmer following Thomason. I’ll discuss a version that applies to a whole family of examples, namely whenever the sheaves are constructible in a precise sense.

Reflexivity is a about a duality between two kinds of derived categories which appear in algebra and geometry. For a finite dimensional algebra these two categories are the bounded derived category of finite dimensional modules and the homotopy category of finitely generated projectives. For a projective scheme these two categories are the bounded derived category of coherent sheaves and the category of perfect complexes. In both cases the difference between the two categories can detect “smoothness” i.e. finite global dimension and being regular. Even in the non-smooth case there is some common information between these two kinds of derived categories; there is a bijection between their semi-orthogonal decompositions and they have the same Hochschild cohomology. Kuznetsov and Shinder showed that the categories mentioned above are so called reflexive DG-categories. We provide a conceptual justification of this common information using a monoidal characterisation of reflexive DG-categories. This approach gives a proof of the Hochschild cohomology invariance result for (semi-)reflexive DG-categories.

I will discuss a replacement of the notion of homotopy cardinality in the setting of even-dimensional Calabi–Yau categories and their relative generalizations (under appropriate finiteness conditions). This includes cases where the usual definition does not apply, such as Z/2-graded dg categories. As a first application, this allows us to define a version of Hall algebras for odd-dimensional Calabi-Yau categories. If time permits, I will also sketch a proof, using the same ideas, of a conjecture of Ng-Rutherford-Shende-Sivek expressing the ruling polynomial of a Z/2m-graded Legendrian knot (which, for m = 1, is a part of the HOMFLY polynomial) in terms of the homotopy cardinality of its augmentation category. The talk is based on joint work with Fabian Haiden.

Bondal-Orlov and Ballard showed that a Gorenstein (anti-)Fano variety X can be reconstructed from the triangulated category structure of its derived category. On the other hand, Balmer showed that for any variety X, it is possible to reconstruct X by considering the tensor triangulated structure on its derived category using the derived tensor product. In this talk, we will first observe how much of Balmer’s reconstruction can be understood without the monoidal structure, using the recently introduced Matsui spectrum of a triangulated category. From this perspective, we provide a new proof of the reconstruction theorems of Bondal-Orlov and Ballard and offer further insights and generalizations.

Equivariant categories arise naturally as the categorical structure related to a finite group acting on a category. Recollements of categories (abelian or triangulated) were introduced by Beilinson-Bernstein- Deligne and essentially are short exact sequences of categories with extra structure. In this talk, after reviewing these definitions, we will see how to lift recollements of abelian (resp. triangulated) categories to recollements of abelian (resp. triangulated) equivariant categories. We will then discuss equivariant singularity categories and we will examine under which conditions the quotient functor in an equivariant abelian recollement induces an equivalence between the equivariant singularity categories. This is joint work with Aristeides Kontogeorgis and Chrysostomos Psaroudakis.

The importance of ring theoretic constructions such as trivial extensions and tensor rings has long been established in representation theory. The aim of this talk is to provide a systematic treatment of Gorenstein homological algebra for such classes. In particular, we study the relation of the singularity categories of cleft extension algebras. As a byproduct, we compare Gorensteinness and prove singular equivalences, unifying an abundance of known results.

The curvature problem in the deformation theory of dg algebras stems from the following observations:

(1)  According to the Hochschild complex, deformations of a dg algebra include curved cdga’s and in general it is not possible to realise the full cohomology by means of dg Morita deformations (Keller- Lowen, 2009);

(2)  Since the differential of a cdga does not square to zero, there is no conventional derived category and second kind derived categories in the sense of Positselski may vanish for deformations (Keller- Lowen-Nicolás, 2009).

In this talk, we propose an altogether different approach to (2) by introducing the filtered derived category of an n-th order cdg deformation of a given dga A. This turns out to be a compactly generated triangulated category in which Positselski’s semiderived category embeds admissibly. Further, it allows for a semiorthogonal decomposition into n+1 copies of D(A) and for smooth A it can be interpreted as a categorical resolution in the sense of Kuznetsov of the (in general) nonexistent classical derived category of the cdga.

Time permitting, we will discuss work in progress which allows to view the higher results in a novel framework of infinitesimal deformations of pre-triangulated dg categories governed by Hochschild cohomology.

This is joint work with Wendy Lowen.

In recent work, Neeman has worked with triangulated categories with metrics to prove remarkable results in algebraic geometry, derived Morita theory etc. One of the most striking of those results is a new Brown representability theorem. In this talk, we prove a generalisation of this representability result. To this end, we introduce a generalisation of the notion of an approximable triangulated category, and give examples coming from algebraic geometry.

Neeman’s novel method of creating new triangulated categories from old ones involves assigning a metric to a triangulated category and constructing its completion analogously to the completion of a metric space. In this talk, we will explore those metric completions in the context of triangulated categories arising from nice enough algebras. In particular, we provide a concrete description of all completions of bounded derived categories of hereditary finite dimensional algebras of finite representation type. This talk is based on pre-print arXiv:2409.01828.

Cell theory has been a crucial ingredient in the development of the 2-representation theory of finitary 2-categories, which are locally additive and k-linear with strong finiteness conditions. In this talk, I will present some generalisations of the concept of cells and cell 2-representations to 2-categories which are locally differential graded.

We consider the derived category D(A) of modules over a noetherian ring A. In this talk we will present a variant of Brown-Adams theorem which is suitable for the full subcategory of D(A) consisting of those complexes which have coherent cohomology. The key ingredient of the proof is that these complexes are always of pure projective dimension less or equal than 1. In fact, we will deal with a generalization of this result which is obtained by replacing D(A) with a sufficiently nice compactly generated triangulated category.

Since the appearance of Bondal-van den Bergh’s work on the representability of functors, proving existence of strong generators of the bounded derived category of coherent sheaves on a scheme has been a central problem. While for a quasi-excellent, separated scheme the existence of strong generators is established, explicit examples of such generators are not common. In this talk, we show that explicit generators can be produced in prime characteristics using the Frobenius pushforward functor. As a consequence, we will see that for a prime characteristic p domain R with finite Frobenius endomorphism, R^{1/p^n} - for large enough n - generates the bounded derived category of finite R-modules. This recovers Kunz’s characterization of regularity in terms of flatness of Frobenius. We will discuss examples indicating that in contrast to the affine situation, for a smooth projective scheme whether some Frobenius pushforward of the structure sheaf is a generator, depends on the geometry of the underlying scheme. Part of the talk is based on a joint work with Matthew Ballard, Srikanth Iyengar, Patrick Lank and Josh Pollitz.

A differential graded algebra (DGA) is cluster tilting if the rank 1 free module is a cluster tilting object in its derived category. In this talk we will concentrate in the periodic case. In this case, we will classify cluster tilting DGAs. We will also characterise the bimodule Calabi-Yau property for these DGAs. We will stress the importance of Hochschild cohomology and its rich algebraic structure in the development of this theory.

Contraherent cosheaves are the dual concept to quasi-coherent sheaves. They are constructed from contraadjusted or cotorsion modules over commutative rings by gluing together over a nonaffine scheme using the colocalization functors. The semiderived category is constructed in a relative context (for a fibration) by mixing the co/contraderived category along the base and the conventional derived category along the fibers. The semico-semicontra correspondence theorem is an equivalence between the semiderived categories of quasi-coherent sheaves and contraherent cosheaves for a flat, not necessarily affine morphism from a quasi-compact semi-separated scheme to a semi-separated Noetherian scheme with a dualizing complex.

In recent joint work with Tiago Cruz we introduced relative Auslander-Gorenstein pairs which provides a further homological feature of Gorenstein algebras in terms of relative dominant dimension with respect to a self-orthogonal module. We further proved an Auslander-Iyama-Solberg correspondence for these pairs which led to a new notion that generalises precluster tilting modules in the sense of Iyama and Solberg. A key ingredient for these new developmenets is the relative homological algebra based on the self-orthogonal module. In this talk, after summarising the above concepts and results, we will discuss how we can use this theory in order to describe the stable category of Cohen-Macaulay modules of a relative Auslander-Gorenstein pair. This is joint work with Tiago Cruz.

In module categories the inclusion Prod(M) ⊆ Add(M) holds if and only if M is Σ-pure-injective and product-rigid, see Angeleri Hügel [1, Proposition 4.8]. Also, according to [2] and [3], under the assumption (V = L), we have Add(M) ⊆ Prod(M) if and only if M is a Σ-pure-injective module.

In this talk we will give characterizations for these inclusions in compactly generated triangulated category.

[1]  L. Angeleri Hügel, L. Angeleri Hügel, On some precovers and preenvelopes, Habilitationsschrift, 2000.

[2]  S. Breaz, Σ-pure injectivity and Brown Representability, Proc. Amer. Math. Soc. 143 (2015), No. 7,
2789–2794.

[3]  J. Šaroch, Σ-algebraically compact modules and Lω1ω-compact cardinals, Math. Log. Quart. 61 (2015),
No. 3, 196–201.

A theorem by Keller states that the full subcategory F(M) of M-filtered modules for a finite-dimensional algebra can be described, up to Morita equivalence, via the Ext-infinity algebra of M. In this talk, I would like to speak about applications of this theorem, in particular in the case where M is the direct sum of the standard modules for a quasi-hereditary algebra. More concretely, I would like to explain how it relates to existence and uniqueness results for exact Borel subalgebras, and, time permitting, how it can be used in order to give a simple criterion for uniseriality of the category of standardly filtered modules.

Proxy-smallness, introduced by Dwyer, Greenlees, and Iyengar, is a rather mysterious condition on objects of a compactly generated triangulated category. Every compact object is proxy-small, but there are many more examples. For instance, the residue field of any commutative noetherian local ring is proxy-small in the derived category. I’ll try to convince you that proxy-smallness is actually a very natural generalization of compactness, both from the point of view of some functor preserving coproducts, and from the point of view of presenting triangulated categories via derived categories. The results in this talk are based on joint work with Benjamin Briggs and Srikanth Iyengar.

Building on results of Bazzoni–Šťovíček we give an explicit construction of an infinite family of commutative rings such that the telescope conjecture fails and which generalise an example of Keller. Generalising this construction further, we give a complete classification of the frame of smashing ideals for the derived category of a finite dimensional valuation domain. As a consequence, we deduce that the Krull dimension of the Balmer spectrum and the smashing spectrum can differ arbitrarily for rigidly-compactly generated tensor-triangulated categories. The talk will be based on the preprint https://arxiv.org/pdf/2407.11791, cover the basic idea and provide plenty of examples.

Global equivariant homotopy theory captures equivariant structures that appear uniformly among a family of groups. In particular, there are tensor-triangulated categories of global spectra which globalize stable equivariant homotopy theory for a fixed group. In this talk, we study these categories from the perspective of tensor-triangular geometry. Remarkably, even the case of rational global spectra exhibits rich phenomena. This is ongoing joint work with Barrero, Barthel, Pol, and Strickland.