# Finite abelian categories

## Definition

###### Definition

Let $k$ be a field, and let $\mathcal{C}$ be a $k$-linear abelian category (i.e. one whose Ab-enrichment is lifted to a Vect-enrichment). Then $\mathcal{C}$ is said to be finite (over $k$) if

###### Theorem (Deligne)

For any finite abelian category $C$, there exists a finite-dimensional $k$-algebra $A$ and an $k$-linear equivalence between $C$ and $A$-$Mod_{fd}$, the category of modules over $A$ that are finite-dimensional as vector spaces over $k$.

## Reference

Last revised on November 4, 2016 at 06:53:32. See the history of this page for a list of all contributions to it.