Homological properties of category algebras.

摘要

Let R be a commutative ring with an identity and C a small category. We introduce and study the notion of a category algebra, denoted by R C . This is a type of associative algebra, which simultaneously generalizes several important constructions in representation theory and combinatorics, notably the path algebra of a quiver, the incidence algebra of a poset, and the group algebra of a group. The purpose of introducing category algebras is to use it to understand the representations and cohomology of small categories, which arise when we consider any diagram of modules, or take inverse or direct limits. We are greatly motivated by the representations and cohomology of certain categories constructed from subgroups of a group, which have been intensively studied in recent years and are currently the subject of active investigation.;The categories we consider in the majority of the dissertation are the EI-categories, in which every endomorphism is an isomorphism. We establish a theory of vertices and sources as a tool to parameterize the indecomposable R C -modules. A significant fact is that we can associate to each indecomposable R C -module a full convex subcategory V M of C such that M is totally determined by its values on V M. This special subcategory V M is called the vertex of M, and the restriction M ↓ CVM is the source for M. As a main application of our theory, we compute the Ext groups Ext*RC (M, N) of R C -modules M and N afterwards, by showing the existence of a certain reduction formula. When M = R&barbelow; is the trivial R C -module, it's well-known that Ext*RC (R&barbelow;, N) ≅ lim← *C N, the higher limits of N over C , and thus our formula provides a way to calculate the limits. We also investigate the cohomology ring Ext*RC (R&barbelow;, R&barbelow;) of a small category C , partially because it acts on the Ext groups we mentioned above. A characterization of the ring will give us useful information about the Ext groups and even C itself. It turns out that usually the cohomology ring of a small category is not finitely generated.