BIREP – Representations of finite dimensional algebras at Bielefeld

Location: Bielefeld University. All talks will be given in room V2-210/216 in the main university building (UHG).

The goal of this informal workshop is to bring together experts working on triangulated categories and connections to geometry, homotopy, and representations theory.

Registration: There is no formal registration, but please send an e-mail to Nannette Nopto if you intend to participate.

Local organiser: Henning Krause

Support: DFG CRC 701 "Spectral Structures and Topological Methods in Mathematics"

- Paul Balmer (Los Angeles)
- Ivo Dell'Ambrogio (Lille)
- Amnon Neeman (Canberra)
- Beren Sanders (Copenhagen)
- Olaf Schnürer (Bonn)
- Johan Steen (Trondheim)
- Greg Stevenson (Bielefeld)
- Amnon Yekutieli (Be’er Scheva)

All talks will be given in room V2-210/216 in the main university building (UHG).

10:00 | – | 11:00 | Amnon Neeman (Canberra) Grothendieck duality via Hochschild homology |

Coffee break | |||

11:30 | – | 12:30 | Amnon Yekutieli (Be’er Scheva) Differential Graded Rings and Derived Categories of Bimodules |

Lunch break | |||

14:00 | – | 15:00 | Olaf Schnürer (Bonn) Six operations on dg enhancements of derived categories of sheaves and applications |

Coffee break | |||

15:30 | – | 16:30 | Ivo Dell'Ambrogio (Lille) Modules in triangulated categories and Eilenberg-MacLane ring spectra |

19:00 | – | Workshop Dinner at Restaurant Brauhaus (see also Google Maps) |

10:00 | – | 11:00 | Greg Stevenson (Bielefeld) A new(ish) approach to the local-to-global principle |

Coffee break | |||

11:30 | – | 12:30 | Johan Steen (Trondheim) A triangulated Eilenberg–Watts theorem |

Lunch break | |||

14:00 | – | 15:00 | Beren Sanders (Copenhagen) Reconciling the reconstruction theorems of Bondal-Orlov and Balmer |

Coffee break | |||

15:30 | – | 16:30 | Paul Balmer (Los Angeles) Relative stable categories in tt-geometry |

Relative stable categories in tt-geometry

Following work of Carlson (unpublished) and of Stevenson et al, we discuss relative stable categories through the lense of tt-geometry. (The talk might contain more open problems than theorems.)

Modules in triangulated categories and Eilenberg-MacLane ring spectra

I will report on recent results in a joint project with Paul Balmer and Beren Sanders. Given a monad on a triangulated category, one would like to know under which conditions the modules (a.k.a. algebras) for that monad again form a triangulated category. Balmer (2011) proved that this holds provided the monad is separable, but there are easy examples showing that separability is not a necessary condition. Our new result says that the situation when this can happen is quite rigid: If the modules are triangulated, then essentially every triangulated adjunction realising the same monad must be monadic (modulo the obvious adjustments). As an application, we obtain a characterisation of those rings R for which the category of naive modules and that of highly structured modules over the Eilenberg-MacLane ring spectrum HR coincide. This explains and improves some results of Casacuberta and Gutierrez (2005).

Grothendieck duality via Hochschild homology

Hochschild cohomology was introduced in a 1945 paper by Hochschild, and Grothendieck duality dates back to the early 1960s. The fact that the two have some relation with each other is very new – it came up in papers by Avramov and Iyengar [2008], Avramov, Iyengar, and Lipman [2010] and Avramov, Iyengar, Lipman and Nayak [2011]. We will review this history, and the surprising formulas that come out. We will then discuss more recent progress. The remarkable feature of all this is the role played by Hochschild homology. One example, which we will discuss in some detail, comes about as follows. The new techniques permit us to write formulas giving trace and residue maps in Grothendieck duality in terms of expressions that are very Hochschild-homological – Alonso, Jeremias and Lipman gave such a formula, but couldn't prove that it agrees with the usual formula dating back to Verdier in the 1960s. The proof that these two agree, due to Lipman and the speaker, turns out to hinge on considering the action of ordinary Hochschild homology on the various objects in the formula.

Reconciling the reconstruction theorems of Bondal-Orlov and Balmer

The Balmer reconstruction theorem shows that a quasi-compact quasi-separated scheme can be reconstructed from the tensor triangulated structure of its derived category of perfect complexes. On the other hand, the Bondal-Orlov reconstruction theorem (of smooth projective varieties whose canonical bundle is ample or anti-ample) does not use the tensor structure at all. In this talk, I will discuss the relationship between these two theorems and explain how the lack of a tensor structure in the Bondal-Orlov theorem can be reconciled with the tensor structure used in Balmer’s construction.

Six operations on dg enhancements of derived categories of sheaves and applications

We lift Grothendieck’s six functor formalism for derived categories of sheaves on ringed spaces over a field to differential graded enhancements. Then we explain applications concerning homological smoothness of derived categories of schemes.

A triangulated Eilenberg–Watts theorem

The ordinary Eilenberg–Watts theorem states that any right exact functor between module categories which commutes with coproducts necessarily is given as tensoring with a bimodule.

In this talk, based on joint work with Greg Stevenson, I will describe how (a variant of) this theorem looks in the realm of triangulated categories. More specifically, Paul Balmer has shown that separable monoids in a tensor triangulated category S give rise to triangulated module categories, and we obtain an Eilenberg–Watts theorem for exact functors compatible with the S-structure. I will discuss how this setting naturally leads us to consider functors enriched in S.

A new(ish) approach to the local-to-global principle

The local-to-global principle is a very useful property of a tensor triangulated category which asserts, roughly speaking, that the local pieces of an object determine that object. Traditionally one proves this property by appealing to some noetherian hypothesis. In this talk I'll give some background on the local-to-global principle and outline an approach to proving it that relies on a weaker form of finiteness hypothesis which is phrased in terms of rank functions on topological spaces. To illustrate this I'll give examples of non-noetherian rings, whose spectra are very far from being noetherian spaces, which nonetheless satisfy the local-to-global principle.

Differential Graded Rings and Derived Categories of Bimodules

Homological algebra plays a major role in noncommutative ring theory. One important homological construct related to a noncommutative ring A is the rigid dualizing complex, which is a special kind of complex of A-bimodules. When A is a ring containing a central field, this concept is well-understood now. However, little is known about dualizing complexes when the ring A does not contain a central field (I shall refer to this as the noncommutative arithmetic setting). The main technical issue is finding the correct derived category of A-bimodules.

In this talk I will propose a promising definition of the derived category of A-bimodules in the noncommutative arithmetic setting. The idea is to use DG (differential graded) rings: resolve the ring A by a flat DG ring A'. A recent theorem says that the category of DG A'-bimodules is independent of the resolution A', up to a canonical equivalence.

Working with the category of DG A'-bimodules allows us to define rigid dualizing complexes over A, and to prove their uniqueness.

What is noticeably missing is a result about existence of rigid dualizing complexes. We are now trying to extend Van den Bergh’s method to the noncommutative arithmetic setting. This is work in progress, joint with Rishi Vyas.

In this talk I will explain, in broad strokes, what are DG rings, DG modules, and the associated derived categories and derived functors. Also, I will try to go into the details of a few results and examples, to give the flavor of this material.

Notes: https://www.math.bgu.ac.il/~amyekut/lectures/der-cat-bimodules/notes_compact.pdf