# Triangulated Categories and Geometry – Abstracts

Benjamin Antieau (Chicago)

Negative and homotopy K-theory of ring spectra and extensions of the theorem of the heart

Inspired by work of Neeman, Barwick proved that the K-theory of a stable infinity-category with a bounded t-structure agrees with the K-theory of its heart in non-negative degrees. Joint work with David Gepner and Jeremiah Heller extends this to an equivalence of nonconnective K-theory spectra when the heart satisfies certain finiteness conditions such as noetherianity. Applications to negative K-theory and homotopy K-theory of ring spectra are provided, which were the original motivation for our work.

Paul Balmer (Los Angeles)

The module category and tt-geometry

We'll discuss joint work with Henning Krause and Greg Stevenson, on the interplay between a rigidly-compactly generated tensor-triangulated category and the abelian category of modules over its small objects. We shall see how a variety of phenomena about the triangulated category in question can be analyzed in the module category, including Rickard idempotents and the elusive residue fields in tt-geometry.

Andrew Blumberg (Austin)

The Kunneth theorem for topological periodic cyclic homology

Motivated by Deninger's proposal for a cohomological interpretation of the Hasse-Weil zeta function, Hesselholt has advocated the study of a certain cohomology theory for schemes over a finite field, the S^{1}-Tate spectrum of the topological Hochschild homology (THH) of the scheme. In analogy with the description of periodic cyclic homology in terms of Tate cohomology, one might think of this as topological periodic cyclic homology (or in general, since it is not necessarily periodic when the scheme is not over a finite field, a higher de Rham spectrum).

In this talk, I will explain joint work with Mandell that establishes a strong Kunneth theorem for topological periodic cyclic homology (TP) of smooth and proper dg-categories over a finite field. One application of this result is that TP can be thought of as a "noncommutative Weil cohomology" theory, which has interesting consequences in the theory of noncommutative motives.

Ragnar-Olaf Buchweitz (Toronto)

Tilting - The way back

Whenever a triangulated category admits a tilting object T, it identifies with the derived category of E = End(T). How does one get back from that derived category to the original one?

We describe an algorithm for the case that E is an artinian algebra of finite global dimension. As an example, we use this to identify all matrix factorizations of y^{d} - x^{d} for d ≥ 2, thus, answering a question raised several years ago by physicists.

Denis-Charles Cisinski (Regensburg)

Motivic generic base change formula

The generic base change formula was first proved by Deligne in the context of torsion etale sheaves. We shall explain how to extend this formula to constructible motivic sheaves in a suitable sense. This will involve a conceptual approach to finiteness properties in triangulated categories. If time permits, we shall see how to use this kind of methods to approximate geometrically the conjectural motivic t-structure.

Joachim Cuntz (Münster)

Witt vectors and cyclic homology

We first describe briefly a construction of the classical Witt ring. This can then be used to give a reasonable definition of cyclic homology for algebras over a finite field of positive characteristic. Our version of cyclic homology recovers Berthelot's rigid cohomology in the commutative case. The talk is based on joint work with C. Deninger and with Cortinas, Meyer, Tamme.

Ivo Dell'Ambrogio (Lille)

On Mackey 2-motives

Many abelian or triangulated categories in algebra, geometry and topology are of the equivariant persuasion: they come in group-indexed families, and are connected by change-of-group functors such as restriction, induction and conjugation. Here we are thinking of objects like group representations, equivariant sheaves, G-spectra, G-C*-algebras, and so on. Once decategorified – e.g. by applying K_{0} throughout – the resulting algebraic structure is quite well understood, at least in the case of finite groups, and has been axiomatised via Dress's Mackey functors or Bouc's biset functors. On the other hand, it is still unclear how to best capture the underlying richer (2-)categorical information available in the examples.

In joint work with Paul Balmer, we attempt the latter by categorifying the notion of a Mackey functor as an additive pseudo-functor defined on a certain bicategory of spans (i.e. correspondences) of finite groupoids and taking values in additive categories. This bicategory is, roughly speaking, the initial receptacle for a pseudofunctor for which the restriction along the inclusion of any subgroup has an ambidextrous (i.e. left-and-right) adjoint satisfying base-change. One can strictify this bicategory of "Mackey 2-motives" and work with a convenient calculus of string diagrams. For instance, a string computation shows that the separable monadicity of restriction functors always holds true, which explains previous observations by Paul Balmer, myself and Beren Sanders.

Vincent Gelinas (Toronto)

Tilting theory for stable categories of graded Maximal Cohen-Macaulay modules

We discuss the problem of finding tilting objects for stable categories of graded MCM modules over sufficiently nice Gorenstein rings. We will tour examples where they reveal interesting structure, and discuss what to do when they don't exist. Along the way we will classify standard graded complete intersections with isolated singularities which admit a tilting MCM module.

Jack Hall (Tucson)

Mayer-Vietoris squares in algebraic geometry

I will discuss various notions of Mayer-Vietoris squares in algebraic geometry. These are used to generalize a number of gluing and pushout results of Moret-Bailly, Ferrand-Raynaud, Joyet, and Bhatt. An important step is Gabber's rigidity theorem for henselian pairs, which our methods give a new proof of. This is joint work with David Rydh (KTH).

Srikanth Iyengar (Salt Lake City)

Nilpotence theorem and homological conjectures in local algebra

A few years ago Neeman discovered a connection between the nilpotence theorem for perfect complexes over commutative rings, due to Hopkins, and a family of conjectures (most of which have been settled only recently) in local algebra which go by the name of 'the homological conjectures'. The purpose of this talk, based on a ongoing project with Avramov and Neeman, will be to present this connection and some of its implications.

Wendy Lowen (Antwerp)

On tensor products of Grothendieck and triangulated categories

We will present joint work with Julia Ramos González and Boris Shoikhet on tensor products of large categories.

Since both abelian categories and enhanced triangulated categories are used as models for non-commutative spaces, it is natural to wonder about their tensor product, which should generalise the product of schemes. Using representations as sheaf categories over linear sites, we introduce a construction applicable to arbitrary Grothendieck categories, thus improving upon the Deligne tensor product for small abelian categories which is not always defined. We also discuss the relevance of linear sites to deformation theory. In the algebraic setup, Neeman¹s well-generated triangulated categories are the analogues of Grothendieck abelian categories by work of Porta. Making use of their localization theory and Tabuada and Toën's homotopy theory of dg categories, we extend our construction for Grothendieck categories to the realm of algebraic well-generated triangulated categories.

Akhil Mathew (Cambridge)

Polynomial functors and algebraic K-theory

The Grothendieck group K_{0} of a commutative ring is well-known to be a λ-ring: although the exterior powers are non-additive, they induce maps on K_{0} satisfying various universal identities. The λ-operations yield homomorphisms on higher K-groups. In joint work in progress with with Barwick, Glasman, and Nikolaus, we give a general framework for such operations. Namely, we show that the K-theory space is naturally functorial for polynomial functors, and describe a universal property of the extended K-theory functor. This extends an earlier algebraic result of Dold for K_{0}. In this picture, the λ-operations come from the strict polynomial functors of Friedlander-Suslin.

Daniel Murfet (Melbourne)

How to survive without compact objects

One of Neeman’s influential contributions was his development of the theory of well-generated triangulated categories, which are in a sense the “correct” generality for some of the fundamental theorems we rely on in the context of compactly generated triangulated categories. I will give a brief tour of the subject, starting with the definitions and the major early theorems, through to a sketch of some of the more recent progress. Throughout there will be an emphasis on examples.

Dmitri Orlov (Moscow)

Noncommutative varieies, their properties, and geometric realizations

In this talk we are going to discuss a notion of geometric realizations for noncommutative varieties. We will talk about some basic properties of derived and noncommutative schemes and such an operation as gluing. We consider special examples that are directly related to quivers with relations and we also consider examples that coming from Fano varieties and appear in mirror symmetry.

Alice Rizzardo (Edinburgh)

Triangulated categories with multiple enhancements

Triangulated categories are a widely used notion that arises naturally in mathematics, but they are often unsatisfactory to work with because they involve non-functorial constructions. A standard way around this is to endow a given triangulated category with more structure by putting an enhancement on it (for example a DG enhancement). Some natural questions arise: given a triangulated category over a field, does an enhancement always exist? Is it unique? We will discuss these questions during the talk. This is joint work with Michel Van den Bergh.

Pramathanath Sastry (Chennai)

Neeman's work in Grothendieck Duality

The talk will highlight the advances in the subject the past 30 years with special emphasis on Professor Neeman's contribution to the subject.

Stefan Schwede (Bonn)

A triangulated category in equivariant homotopy theory: the global stable homotopy category

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, ...

Jan Šťovíček (Prague)

Infinite dimensional tilting theory

I will discuss recent progress in infinite dimensional tilting theory. This involves recent results with Positselski on derived equivalences to categories of contramodules, classification results for commutative rings with Hrbek, and results with Hrbek and Trlifaj on the local nature of tilting modules with respect to Grothendieck topologies.

Michel Van den Bergh (Hasselt)

Additive invariants of orbifolds

We discuss joint work with Gonçalo Tabuada. We express the non-commutative motive of a global orbifold as a sum of non-commutative motives of fixed point spaces. This implies corresponding decompositions for many invariants such as K-theory and Hochschild homology and gives a uniform proof of the latter. We also consider the case where the orbifold is twisted by a gerbe.

We will also discuss some examples where the motivic decomposition can be lifted to a semi-orthogonal decomposition. This is joint work with Alexander Polishchuk.

Jorge Vitória (London)

Definable subcategories and t-structures

Torsion pairs (and, in particular, t-structures) are useful tools to study triangulated categories. A large class of torsion pairs can be obtained as Hom-orthogonal pairs generated by sets of compact objects, as proved by Aihara and Iyama. In a compactly generated triangulated category, the Hom-orthogonal class to a set of compact objects is a particular case of a definable subcategory. In this talk, we discuss the relation between definable subcategories and t-structures. Concretely, we show that a cosuspended definable subcategory gives rise to a t-structure and that, often, these t-structures have hearts which are Grothendieck abelian categories. In fact, under some assumptions, every Grothendieck heart arises in this way. This is joint work with Lidia Angeleri Hügel and Frederik Marks.