Exact categories and vector space categories

Abstract:

In a series of papers additive subbifunctors $F$ of the bifunctor $\operatorname {Ext}_{\Lambda } ( , )$ are studied in order to establish a relative homology theory for an artin algebra $\Lambda$. On the other hand, one may consider the elements of $F(X,Y)$ as short exact sequences. We observe that these exact sequences make $\operatorname {mod}\Lambda$ 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 $\operatorname {Ext}_{\Lambda }( , )$ and in general do not preserve at all the usual exact structure of $\operatorname {mod}\Lambda$. By showing the closedness of suitable subbifunctors of $\operatorname {Ext}_{\Lambda }( , )$ we can equip $\operatorname {mod}\Lambda$ 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.