Abstracts

Abstract

In this talk, by using functor rings and functor categories, we study finiteness and purity of subcategories of the module categories. We give a characterisation of contravariantly finite resolving subcategories of the module category of finite representation type in terms of their functor rings. We also characterize contravariantly finite resolving subcategories of the module category A-mod that contain the Jacobson radical of A of finite type, by their functor categories. We study the pure semisimplicity conjecture for a locally finitely presented category lim X when X is a covariantly finite subcategory of A-mod and every simple object in Mod(X^{op}) is finitely presented. Moreover we give a characterization of covariantly finite subcategories of finite representation type in terms of decomposition properties of their closure under filter colimits. These subcategories include important examples such as n-cluster-tilting subcategories and Gorenstein projective modules. In this talk, we recover and unify known characterizations of finiteness and purity for n-cluster-tilting subcategories and subcategories of finitely generated Gorenstein projective modules.

This talk is based on a joint work with Alireza Nasr-Isfahani.

Abstract

Let A be an associative ring with identity and mod A the category of finitely generated left A-modules. The morphism category Mor(A) of A has as objects the maps in mod A and whose morphisms are given by commutative squares. The monomorphism category Mon(A) of A is the full subcategory of Mor(A) consisting of all monomorphisms, which is also known as the submodule category. The study of monomorphism categories goes back to Birkhoff in 1934. This subject has been attracted more attention and studied in a systematic and deep work by Ringel and Schmidmeier. In this research, we assume that (S; n) is a commutative noetherian local ring with dim S equal or greater than 2 and w in n is non-zerodivisor. Assume that Mon(w, proj S) is the full subcategory of Mon(S) consisting of all S-monomorphisms (f: P to Q) such that P and Q are finitely generated projective S-modules and Coker(f) is annihilated by w. We defined the homotopy category HMon(w, proj S) and we have shown that the category HMon(w, proj S) is a triangulated category. Also a tie connection between category HMon(w, proj S) and the singularity category of the factor ring R = S/(w), will be discovered.

Abstract

In this talk, I will explain how the Auslander-Reiten theory of the morphism category over a dualizing variety is related to that of the category of finitely presented functors over the given dualizing variety. Then, I will examine this observation for some special cases of dualizing varieties. In particular, I will give some information about the Auslander-Reiten translation of simple modules over an Auslander algebra as well as the Auslander-Reiten components over an Auslander algebra containing a simple module.

This talk is based on my joint works with Hossein Eshraghi.

Abstract

The interaction of algebra and (noncommutative) geometry has always been a major theme in Lenzing’s work. I will present some of the highlights, including his fundamental work on weighted projective lines.

Abstract

In this talk, we will consider a class of Nakayama algebras given by the path algebras of the equioriented quiver of type A subject to the nilpotency degree r for each sequence of r consecutive arrows. We classify all the Nakayama algebras having Fuchsian type, that is, derived equivalent to the extended canonical algebras, or equivalently, derived equivalent to a kind of stable category of vector bundles overweighted projective lines. This is achieved by constructing certain tilting complexes in the bounded derived category of coherent sheaves and also in the stable category of vector bundles for weighted projective lines, and the strategy of one-point extension and the perpendicular approach will be used. As a byproduct, we reprove the classication result of Nakayama algebras of piecewise hereditary type due to Happel-Seidel.

This is joint work with Helmut Lenzing and Hagen Meltzer.

Abstract

We study characterization of sincere modules, sincere silting modules and tilting modules in terms of vanishing conditions.

Abstract

In 1958 Nakayama conjectured that a finite-dimensional algebra is self-injective whenever its dominant dimension is infinite. This conjecture is still open and becomes nowadays one of the main problems in the representation theory of Artin algebras. To understand this conjecture, Tachikawa decomposed the conjecture into two related conjectures in 1970's, one of them says that an orthogonal module over a self-injective algebra should be projective. This suggests to study orthogonal generators over self-injective algebras. In this series of lectures we will deal with this topic from the view point of stable module categories. In particular, we establish recollements of relative stable module categories related to orthogonal generators. Consequently, we show that the Nakayama conjecture holds true for Gorenstein-Morita algebras.

The lectures consist of three sections:

I: Introduction: Nakayama conjecture and Tachikawa's conjecture

II: Main results and ideas of proofs

III: Gorenstein-Morita algebras and problems

Abstract

In this talk I will explain the connection between t-structures in the bounded derived category of finitely presented modules and the unbounded derived category of all modules over a coherent ring A using homotopy colimits. I will provide necessary and sufficient conditions for a compactly generated t-structure in the unbounded derived category to restrict to an intermediate t-structure in the bounded derived category, thus describing which t-structures can be obtained via lifting. Based on joint work with Frederik Marks.