http://www.maths.leeds.ac.uk/~marsh/research_articles/pp18.pdf

Tilting theory and cluster combinatorics

Applications of contravariantly finite subcategories

Gorenstein injective and projective modules

We introduce a new category C, which we call the cluster category, obtained as a quotient of the bounded derived category D of the module category of a finite-dimensional hereditary algebra H over a field. We show that, in the simply-laced Dynkin case, C can be regarded as a natural model for the combinatorics of the corresponding Fomin-Zelevinsky cluster algebra. In this model, the tilting modules correspond to the clusters of Fomin-Zelevinsky. Using approximation theory, we investigate the tilting theory of C, showing that it is more regular than that of the module category itself, and demonstrating an interesting link with the classification of self-injective algebras of finite representation type. This investigation also enables us to conjecture a generalisation of APR-tilting.

We introduce the notion of mutation of n-cluster tilting subcategories in a triangulated category with Auslander–Reiten–Serre duality. Using this idea, we are able to obtain the complete classifications of rigid Cohen–Macaulay modules over certain Veronese subrings.

Mutation in triangulated categories and rigid Cohen–Macaulay modules

1. Artin rings 2. Artin algebras 3. Examples of algebras and modules 4. The transpose and the dual 5. Almost split sequences 6. Finite representation type 7. The Auslander-Reiten-quiver 8. Hereditary algebras 9. Short chains and cycles 10. Stable equivalence 11. Modules determining morphisms.

/pdf/representation-theory-of-artin-algebras-29j39wh1ji.pdf

Representation Theory of Artin Algebras

Preprojective modules over artin algebras

Almost split sequences in subcategories

Torsion Theories and Tilting Modules

In a series of papers additive subbifunctors F of the bifunctor ExtΛ( , ) are studied in order to establish a relative homology theory for an artin algebra Λ. On the other hand, one may consider the elements of F (X, Y ) as short exact sequences. We observe that these exact sequences make mod Λ into an exact category if and only if F is closed in the sense of Butler and Horrocks. Concerning the axioms for an exact category we refer to Gabriel and Roiter’s book. In fact, for our general results we work with subbifunctors of the extension functor for arbitrary exact categories. In order to study projective and injective objects for exact categories it turns out to be convenient to consider categories with almost split exact pairs, because many earlier results can easily be adapted to this situation. Exact categories arise in representation theory for example if one studies categories of representations of bimodules. Representations of bimodules gained their importance in studying questions about representation types. They appear as domains of certain reduction functors defined on categories of modules. These reduction functors are often closely related to the functor ExtΛ( , ) and in general do not preserve at all the usual exact structure of mod Λ. By showing the closedness of suitable subbifunctors of ExtΛ( , ) we can equip mod Λ with an exact structure such that some reduction functors actually become ‘exact’. This allows us to derive information about the projective and injective objects in the respective categories of representations of bimodules appearing as domains, and even show that almost split sequences for them. Examples of such domains appearing in practice are the subspace categories of a vector space category with bonds. We provide an example showing that existence of almost split sequences for them is not a general fact but may even fail if the vector space category is finite.

/pdf/exact-categories-and-vector-space-categories-11pb9idf9o.pdf

Sverre O. Smalo

Papers

Representation Theory of Artin Algebras

Preprojective modules over artin algebras

Almost split sequences in subcategories

Torsion Theories and Tilting Modules

Exact categories and vector space categories