Summer school
July 30 – August 3, 2019

Lidia Angeleri Hügel Silting and cosilting in triangulated categories
This lecture series is centred around the notion of a silting object in a triangulated category with coproducts, introduced independently by Psaroudakis and Vitória, and by Nicolás, Saorín and Zvonareva. We will see that silting objects correspond bijectively to certain triples formed by a t-structure and an adjacent co-t-structure. Although no compactness is required in our definition of silting, these triples are often of finite type, in the sense that they are determined by a set of compact objects. We will also discuss the dual notion of a cosilting object and the role of purity in this context. We will then focus on the unbounded derived category of a ring and present some constructions of silting and cosilting objects. This will lead us to classification results over commutative noetherian rings and over the Kronecker algebra. The lectures will be based on joint work with Michal Hrbek, Frederik Marks, and Jorge Vitória.
Exercise sheet
Osamu Iyama Tilting theory of preprojective algebras and Cohen-Macaulay modules
In this lecture series, I will explain tilting theory of a preprojective algebra of a quiver as a prototype of tilting/silting/cluster-tilting theory of general algebras. In the non-Dynkin (resp. Dynkin) case, the preprojective algebra has a family of tilting (resp. support tau-tilting) modules Iw parametrized by elements w of the corresponding Coxeter group W, and their mutations correspond to simple reflections in W (Buan–I–Reiten–Scott, Mizuno). In particular, the Hasse quiver has a labelling by bricks (I–Reading–Reiten–Thomas), and in the Dynkin case there are only finitely many bricks and they correspond bijectively with indecomposable tau-rigid modules and with join-irreducible elements. I will explain results on this line for general (tau-tilting-finite) algebras (Demonet–I–Jasso, Asai, DIRRT). The module Iw is given as a 2-sided ideal, and the corresponding factor algebra of the preprojective algebra is a finite dimensional 1-Gorenstein algebra. Its Cohen-Macaulay modules form a 2-Calabi-Yau Frobenius category admitting a cluster-tilting object. They play an important role in the categorification of cluster algebras due to Geiss–Leclerc–Schröer. I will explain tilting/silting/cluster-tilting theory of the stable category of (graded) Cohen-Macaulay modules due to Amiot–Reiten–Todorov and Kimura. These results are typical examples of a general phenomenon that "the stable category of (graded) Cohen-Macaulay modules often has tilting/cluster-tilting objects". If time permits, I will explain results on this line for a higher preprojective algebra (I–Oppermann, Amiot–I–Reiten), which is the zero-th cohomology of Keller's Calabi-Yau completion.
Exercise sheet, part 1
Exercise sheet, part 2
Gustavo Jasso Stable ∞–categories: localisations and recollements
In this lecture series we will give a (naive) introduction to the theory of stable ∞–categories in the sense of Lurie. We will illustrate some features of the theory by focusing on two ubiquitous constructions in homological algebra: localisations and recollements.
Exercise sheet
David Pauksztello Silting: mutation, completion and reduction
In this lecture series I will focus on modern “classical” aspects of silting theory. Mutation was first observed in the context of cluster-tilting theory, where having at most two complements to an almost complete tilting module in representation theory was a key observation in the Buan–Marsh–Reineke–Reiten–Todorov categorification of cluster algebras. This classical cluster-tilting theory can be viewed as a “shadow” of silting theory. In this series of lectures, I will introduce finite-dimensional silting objects and describe their relation with other homological aspects of representation theory via the Koenig–Yang correspondences. I will describe mutation of silting objects and describe how to use mutation to develop completion and reduction techniques and illustrate its compatibility with those in cluster-tilting theory. Along the way we will see applications of silting theory, for example, in providing a combinatorial model of Bridgeland's space of stability conditions.
Exercise sheet
Jeremy Rickard Tilting complexes and derived equivalence
I will give an introduction to equivalences of derived categories of module categories, including some history of how the consideration of such equivalences in representation theory grew out of classical tilting theory and led to the definition of tilting complexes, and I will also touch on the relevance to representations of finite groups. I will concentrate on motivation and explicit examples rather than detailed technical proofs, and will describe various constructions of tilting complexes, some of which generalize to constructions of silting objects that will be relevant to other talks at the summer school and conference. I will assume a basic knowledge of derived categories and the representation theory of finite dimensional algebras, particularly as described by quivers, but will assume no previous knowledge of tilting.
Exercise sheet
Jan Šťovíček Silting theory in commutative algebra
The lectures will focus on silting and cosilting torsion pairs and (infinitely generated) silting and cosilting modules over commutative noetherian rings. After recalling basic results on silting torsion pairs over (possibly non-commutative) left noetherian rings, we will turn our attention to the classification in terms of specialization closed subsets of the Zariski spectrum in the commutative case. In the second half of the course we will discuss the relation of silting theory to flat ring epimorphisms. In this context, every partial silting module over a commutative ring gives rise to a flat ring epimorphism, which can be viewed as a generalized localization of the ring. On the other hand, every flat epimorphism starting from a commutative ring arises in this way.
Exercise sheet

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