Categories of Dimension Zero

Abstract

If D is a category and k is a commutative ring, the functors from D to k-Mod can be thought of as representations of D. By definition, D is dimension zero over k if its finitely generated representations have finite length. We characterize categories of dimension zero in terms of the existence of a "homological modulus" (Definition 1.4) which is combinatorial and linear-algebraic in nature.