The stable module category and model structures for hierarchically defined groups

We make a general study of Quillen model structures on abelian categories. We show that they are closely related to cotorsion pairs, which were introduced by Salce [Sal79] and have been much studied recently by Enochs and coauthors [EJ00]. This gives a method of constructing model structures on abelian categories, which we illustrate by building two model structures on the category of modules over a (possibly noncommutative) Gorenstein ring. The homotopy category of these model structures is a generalization of the stable module category much used in modular representation theory. This stable module category has also been studied by Benson [Ben97].

Precovers and orthogonality in the stable module category

Bousfield localization is a technique that has been used extensively in algebraic topology, specifically in stable homotopy theory, over the last quarter century. Its name derives from the fundamental work of Bousfield [3], although this is based on earlier work of Brown [4] and Adams [1]. In abstract terms, Bousfield localization deals with the inclusion of a thick subcategory (i.e., a triangulated subcategory closed under direct summands) into a triangulated category and the existence of adjoint functors to such an inclusion. In the original topological setting, the triangulated category was typically the stable homotopy category, and the thick subcategory was defined by the vanishing of some homology theory, but the techniques involved work much more generally. In particular, the triangulated category can be one of interest to representation theorists, such as a stable module category or the derived category of a module category: since the techniques rely heavily on limiting procedures, however, one is forced to work with infinite dimensional modules.

The symmetry, period and Calabi–Yau dimension of finite dimensional mesh algebras

Let A be a representation-finite self-injective algebra over an algebraically closed field k. We give a new characterization for an orthogonal system in the stable module category A-\(\underline {\text {mod}}\) to be a simple-minded system. As a by-product, we show that every Nakayama-stable orthogonal system in A-\(\underline {\text {mod}}\) extends to a simple-minded system.

Localization and duality in topology and modular representation theory

We compare several recent approaches to studying right Bousfield localization and algebras over monads. We prove these approaches are equivalent, and we apply this equivalence to obtain several new results regarding right Bousfield localizations (some classical, some new) for spectra, spaces, equivariant spaces, chain complexes, simplicial abelian groups, and the stable module category. En route, we provide conditions so that right Bousfield localization lifts to categories of algebras, so that right Bousfield localization preserves algebras over monads, and so that right Bousfield localization forms a compactly generated model category.

On endotrivial modules for Lie superalgebras

We classify all triangulated orbit categories of path-algebras of Dynkin diagrams that are triangle equivalent to a stable module category of a representation-finite self-injective standard algebra. For each triangulated orbit category T we give an explicit description of a representation-finite self-injective standard algebra with stable module category triangle equivalent to T.

Tilting theory is known as a powerful tool for studying representation theory of artin algebras. Though there are no non-trivial tilting modules over selfinjective algebras, we have a way of using tilting theory for trivial extension selfinjective algebras, starting from a tilting module, which is given by constructing equivalent functors between corresponding stable module categories over trivial extension selfinjective algebras of original algebras.

Krull–Schmidt decompositions for thick subcategories

Let G be a finite p-group with subgroup H and k a field of characteristic p. We study the endomorphism algebra E = EndkG(kH ↑G), showing that it is a split extension of a nilpotent ideal by the group algebra kNG(H)/H. We identify the space of endomorphisms that factor through a projective kG-module and hence the endomorphism ring of kH ↑G in the stable module category, and determine the Loewy structure of E when G has nilpotency class 2 and [G, H] is cyclic.

In a previous paper we constructed rank and support variety theories for quantum elementary abelian groups, that is, tensor products of copies of Taft algebras. In this paper we use both variety theories to classify the thick tensor ideals in the stable module category, and to prove a tensor product property for the support varieties.

For self-injective algebras, Rickard proved that each derived equivalence induces a stable equivalence of Morita type. For general algebras, it is unknown when a derived equivalence implies a stable equivalence of Morita type. In this article, we first show that each derived equivalenceFbetween the derived categories of Artin algebrasAandBarises naturally as a functorbetween their stable module categories, which can be used to compare certain homological dimensions ofAwith that ofB. We then give a sufficient condition for the functorto be an equivalence. Moreover, if we work with finite-dimensional algebras over a field, then the sufficient condition guarantees the existence of a stable equivalence of Morita type. In this way, we extend the classical result of Rickard. Furthermore, we provide several inductive methods for constructing those derived equivalences that induce stable equivalences of Morita type. It turns out that we may produce a lot of (usually not self-injective) finite-dimensional algebras that are both derived-equivalent and stably equivalent of Morita type; thus, they share many common invariants.

The stable module category of a selfinjective algebra is triangulated, but need not have any nontrivial t-structures, and in particular, full abelian subcategories need not arise as hearts of a t-structure. The purpose of this paper is to investigate full abelian subcategories of triangulated categories whose exact structures are related, and more precisely, to explore relations between invariants of finite-dimensional selfinjective algebras and full abelian subcategories of their stable module categories.