Schedule for: 16w5040 - Triangulated Categories and Applications

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building.

Despite their ubiquity and impressive usefulness in many areas of pure mathematics, triangulated categories and monoidal, triangulated categories have certain defects. This includes the non-functoriality of the cone construction as well as the failure of additivity of traces results. In this talk we survey how the existence of such structures and refinements of them are formal consequences of exactness properties of (monoidal) derivators in the background. For suitable derivators we intend to sketch the construction of a) canonical triangulations, b) canonical higher triangulations a la Belinson--Bernstein--Deligne and Maltsiniotis, and c) canonical monoidal triangulations a la May. To also indicate the added flexibility of derivators, we conclude by mentioning fairly general additivity of traces results of de Souza and Ponto--Shulman.

Triangulated derivators are yet another enhancement of triangulated categories introduced by Grothendieck in the early 1990s. In some sense, they are closer to triangulated categories than any other enhancement, yet they have enough structure to define nice K-theories for them. I will give a survey of results about K-theories of triangulated derivators and how they compare to each other and to Quillen's and Waldhausen's K-theory. I will start with Maltsiniotis's conjectures and then discuss a more recent proposal made jointly with Raptis from a higher categorical perspective.

Meet in the Corbett Hall Lounge for a guided tour of The Banff Centre campus.

Meet in foyer of TCPL to participate in the BIRS group photo. The photograph will be taken outdoors, so dress appropriately for the weather. Please don't be late, or you might not be in the official group photo!

I will give a general introduction to A-infinity algebras and their triangulated categories of modules, with a focus on an important class of concrete examples coming from algebraic geometry, specifically, from isolated hypersurface singularities. Since Dyckerhoff identified the DG-endomorphism algebra of the generator of the homotopy category of matrix factorisations in his thesis, it has been possible to (in principle) apply the general method of minimal models to produce the associated A-infinity algebra. I will describe a good choice of homotopy retract which makes such calculations tractable, and sketch the yoga of Feynman diagrams that computes the A-infinity algebra associated to a given singularity.

We give a simultaneous generalization of exact categories and triangulated categories, which is suitable for considering cotorsion pairs, and which we call extriangulated categories. Extension-closed, full subcategories of triangulated categories are examples of extriangulated categories. We give a bijective correspondence between some pairs of cotorsion pairs which we call Hovey twin cotorsion pairs, and admissible model structures. As a consequence, these model structures relate certain localizations with certain ideal quotients, via the homotopy category which can be given a triangulated structure. This gives a natural framework to formulate reduction and mutation of cotorsion pairs, applicable to both exact categories and triangulated categories. These results can be thought of as arguments towards the view that extriangulated categories are a convenient setup for writing down proofs which apply to both exact categories and (extension-closed subcategories of) triangulated categories. This is a joint work with Yann Palu.

A buffet dinner is served daily between 5:30pm and 7:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building.

Consider varieties X and X^+ related by flopping contractions f: X \to Y, f^+: X^+ \to Y with fibers of dimension less than or equal to one. The null category for f is the category of sheaves on X with vanishing derived direct image. I will show that the derived categories of null categories for f and f^+ form a spherical pair in an appropriate quotient of the derived category of the fiber product of X and X^+ over Y. The associated auto-equivalence of the derived category of X is the flop-flop functor. This is a joint work with A. Bondal.

This is a report on joint work with Martin Olsson. I will review the basic results on equivalences of the derived categories of coherent sheaves on smooth projective varieties and discuss some attempts to use them to produce Torelli theorems in positive characteristic. Derived equivalences between two varieties always give correspondences between the two varieties in many cohomology theories. In particular, in characteristic 0, one can link them to Hodge theory and rephrase Torelli theorems in terms of a package made of the derived category and the Chow theory. This leads one to wonder if a similar thing happens in positive characteristic. For K3 surfaces, we have a complete answer to this question (that I will explain). Among other things, this has a strong relationship to the Tate conjecture. I will finish with some open questions.

I will give a survey over recent results on derived categories of algebraic stacks with an emphasis on compact generation. In a loose sense, compact objects in the derived category is a replacement for ample line bundles on projective schemes. They also generalize vector bundles but are more flexible. From a different perspective, compact objects are like the coherent sheaves among the quasi-coherent sheaves. I will focus on three questions: (1) When are perfect complexes compact? (2) When is the derived category compactly generated? (3) When are complexes of quasi-coherent sheaves good enough? For schemes, these questions are well understood by work of Thomason, Neeman, Bokstedt, Bondal, Van den Bergh and Lipman. For stacks, the picture is not complete yet but there are satisfactory partial results by Ben-Zvi, Francis, Nadler, Toen, Antieau, Gepner, Lurie, Drinfeld, Gaitsgory, Hall, Neeman and the speaker.

Let T be a triangulated category. A basic question is the classification of the triangulated subcategories of T. The interest in this question stemmed from the work of Hopkins who connected it to the Telescope conjecture in algebraic topology. Over the last two decades, spurred by work of Neeman, there has been interest in this from the perspective of algebraic geometry. Most recently, this has been considered by Antieau, Stevenson, Balmer-Favi and Dubey-Mallick. I will discuss some recent work with Rydh that allows us to classify subcategories in the equivariant setting and, more generally, for stacks.

In this talk, which will be primarily expository, I'll present some of the highlights of tensor triangular geometry thus far. In particular, I'll try to draw attention to the various points at which the existence of a tensor product makes small miracles occur.

Tensor triangular geometry as introduced by Balmer is a powerful idea which can be used to extract the ambient geometry from a given tensor triangulated category. In this talk, I will present a general setting for a compactly generated tensor triangulated category which enables one to classify thick tensor ideals and the spectrum Spc. Examples will be given which illustrate the interactions between the algebra and the geometry. For a classical Lie superalgebra g, I will show how to construct a Zariski space from a detecting subalgebra f and demonstrate that this topological space governs the tensor triangular geometry for the category of finite dimensional g-modules which are semisimple over g_0. A complete determination of thick tensor ideals and Spc will be given for the Lie superalgebra gl(m|n). Conjectures will be presented for arbitrary classical Lie superalgebras. These results represent joint work with Brian Boe and Jonathan Kujawa.

(joint with Greg Stevenson) The singularity category of a ring R is defined to be the quotient of the triangulated category of complexes with finitely generated homology modulo the perfect complexes: D_{sg}(R)=D_{fg}(R)/D_{perf}(R). To define the analogue more generally (eg when R is a DGA or a ring spectrum satisfying mild finiteness hypotheses), we need to find analogues of the triangulated categories D_{fg}(R) and D_{perf}(R). The perfect complexes are precisely the small objects, so the denominator is easy. One way to make sense of the finitely generated R-modules is to suppose given a regular ring S and a map S->R making R a small S-module, and take D_{fg}(R) (which may depend on S) to consist of R-modules small over S. This covers interesting cases, and the talk will discuss the use of Morita theory to understand the resulting singularity category.

In a recent paper, joint with Paul Balmer and Ivo Dell'Ambrogio, we made a general study of the existence and properties of adjoints to an arbitrary coproduct-preserving tensor-triangulated functor between rigidly-compactly generated tensor triangulated categories. One of the highlights of this work was the recognition that such a functor has a left adjoint if and only if it satisfies Grothendieck-Neeman duality, in which case there is a Wirthmueller isomorphism between its left and right adjoint (twisted by the relative dualizing object). In particular, this work provided a purely formal canonical construction of the classical Wirthmueller isomorphism in equivariant stable homotopy theory. In this talk, I will review aspects of the above story before explaining how the Adams isomorphism can also be obtained purely formally by an extension of the theory. The main punch-line is that every such functor -- even one which does not have a left adjoint -- gives rise to a "Wirthmueller type" isomorphism (properly understood). This construction generalizes the Wirthmueller isomorphism of the earlier paper, and includes the Adams isomorphism as a special case.

Global homotopy theory is equivariant homotopy theory with simultaneous and compatible actions of all compact Lie groups. In this survey talk I will advertise the global stable homotopy category, a specific tensor triangulated category that is the natural home of all stable global phenomena. The forgetful functor to the non-equivariant stable homotopy category is part of a recollement. A preferred set of compact generators is given by the suspension spectra of the global classifying spaces of compact Lie groups. Further examples of global stable homotopy types are the global sphere spectrum, global equivariant K-theory, different global flavors of bordims theories, Eilenberg-MacLane spectra of global Mackey functors,... Our model is the well-known category of orthogonal spectra, but we use a much finer notion of equivalence, the `global equivalences', then what is traditionally considered. Indeed, every orthogonal spectrum gives rise to an orthogonal G-spectrum for every compact Lie group G, and the fact that all these individual equivariant objects come from one orthogonal spectrum implicitly encodes strong compatibility conditions as the group G varies. An orthogonal spectrum thus has G-equivariant homotopy groups for every compact Lie group, and a global equivalence is a morphism of orthogonal spectra that induces isomorphisms for all equivariant homotopy groups for all compact Lie groups. The structure on the equivariant homotopy groups of an orthogonal spectrum gives an idea of the information contained in a global homotopy type: the equivariant homotopy groups are contravariantly functorial for continuous group homomorphisms (`restriction maps' and `inflation maps'), and they are covariantly functorial for inclusions of closed subgroups (`transfer maps'). The restriction, inflation and transfer maps enjoy various transitivity properties and interact via a double coset formula. This kind of algebraic structure has been studied under different names (e.g., `global Mackey functor', `inflation functor').

There are two types of triangulated categories which arise routinely in modular representation theory - derived categories, and stable module categories. Derived categories determine many fundamental numerical invariants of block algebras, such as the number of isomorphism classes of simple modules and irreducible characters of prescribed heights. Broue's abelian defect conjecture predicts the nature of the derived categories of blocks of finite groups with abelian defect groups. By contrast, it is not known whether stable module categories determine any of the above mentioned numerical invariants in general. This is one of the major obstacles in modular representation theory. Unlike derived categories, stable module categories need not have any t-structures, and hence their stability spaces in the sense of Bridgeland may be empty. Still, a stable module category may have many abelian subcategories whose exact structure is compatible with the triangulated structure, and whose numerical invariants are, in some cases, those of the original algebra. We will describe a simple construction principle for abelian subcategories of stable module categories.

This talk will be about the representations of a finite group scheme G defined over a field k of positive characteristic. Mimicking the classical local to global principle in commutative algebra, one can tackle some questions concerning these representations by reducing them to questions about p-local p-torsion representations, as p varies over the (not necessarily closed) points in the projective variety defined by the cohomology of G. In recent work we discovered an enhancement of this principle, namely, that the method of passage to generic points in classical algebraic geometry has a counterpart in representation theory that allows one to pass from arbitrary points in the projective variety to closed points. This technique is proving to be quite fruitful. It has lead to a classification of the localising (and also the colocalising) subcategories of the stable module category of G, as well as a form of local Serre duality for the category of finite dimensional representations. The aim of the talk will be to present an overview of these developments. Joint work with Dave Benson, Henning Krause and Julia Pevtsova.

5-day workshop participants are welcome to use BIRS facilities (BIRS Coffee Lounge, TCPL and Reading Room) until 3 pm on Friday, although participants are still required to checkout of the guest rooms by 12 noon.