Tag - Tilting theory

Jan Trlifaj: Tree modules, and limits of the approximation theory

19th August, 2016

Classes of modules closed under transfinite extensions often provide for precovers, and hence fit in the machinery of relative homological algebra. However, there are important exceptions: the Whitehead groups, and flat Mittag-Leffler modules over non-perfect rings. The latter class is just the zero dimensional instance (for T = R and n = 0) of non-precovering of the class of all locally T-free modules, where T is any n-tilting module which is not Σ-pure split. The phenomenon occurs even for finite dimensional algebras, when R is hereditary of infinite representation type, and T is the Lukas tilting module. The key tools here are the tree modules, which have recently been generalized in order to solve Auslander's problem on the existence of almost split sequences.

Julian Külshammer: Higher Nakayama algebras

19th August, 2016

Nakayama algebras are among the best understood representation-finite algebras. They are defined as those algebras such that each indecomposable projective and each indecomposable injective module admits a unique composition series. An equivalent characterisation is that τjS is simple (or zero) for all j ∈ ℤ and every simple module S. Here, τ denotes the Auslander–Reiten translation. Nakayama algebras can be classified by the sequence of lengths of their indecomposable projective modules, called the Kupisch series.

In this talk, we introduce a higher analogue of a Nakayama algebra for each Kupisch series 𝓁 in the sense of Iyama's higher Auslander–Reiten theory. More precisely, (in type A) the higher Nakayama algebra A𝓁(d) is a quotient of the higher Auslander algebra An(d) of type A, constructed by Iyama and studied extensively by Oppermann and Thomas. In type ̃A, one has to use an infinite version of An(d). The higher Nakayama algebra has a d-cluster-tilting module, i.e. a module M with

add(M) = {N | Exti(M,N) = 0 ∀i = 1, . . . , d−1 } = {N | Exti(N,M) = 0 ∀i = 1, . . . , d−1 }.

There are n simple modules in add(M) and they satisfy that τdjS is simple for all j ∈ ℤ and every simple module S in add(M), where τd = τΩd−1 is Iyama's higher Auslander–Reiten translation.

William Crawley-Boevey: Quiver Grassmannians and orbit closures for representation-finite algebras

18th August, 2016

We show that Auslander algebras have a unique tilting and cotilting module which is generated and cogenerated by a projective-injective; its endomorphism ring is called the projective quotient algebra. For any representation-finite algebra, we use the projective quotient algebra to construct desingularizations of quiver Grassmannians, orbit closures in representation varieties, and their desingularizations. This generalizes results of Cerulli Irelli, Feigin and Reineke.

Claire Amiot: Cluster categorification and applications to tilting theory

11th August, 2016

This series of talks is based on joint works with Oppermann, Grimeland, Labardini and Plamondon. Cluster categories are triangulated categories where quiver mutation appears as a natural operation. A first class of example is given by cluster categories associated with surfaces with marked points. A second class is constructed using the derived category of finite-dimensional algebras of global dimension 2. Mixing both constructions, one may consider surface cut algebras, that are algebras of global dimension 2 constructed from a surface and show how cluster combinatorics permits to deduce information on their derived category.