I am trying to determine the status of the following claim. I know how to prove this (unless I made a stupid mistake), so the question is mostly
- Is it in the literature?
- If not, is there something similar that could be useful?
Any comments would be very helpful. Thanks!
Let C be a linear category over a field k (that is, an abelian category enriched over k-spaces). Suppose that it satisfies the following finiteness relations:
- C is essentially small
- Every object of C has finite length
- All hom spaces are finite-dimensional.
Claim. C is equivalent to the category of finite-dimensional modules over a pro-finite-dimensional k-algebra.
Here a pro-finite-dimensional k-algebra is a filtered projective limit of finite-dimensional k-algebras, or in more classical terms, a topological k-algebra that is complete and whose topology has a basis given by two-sided ideals of finite codimension. And of course, the modules must agree with this structure (i.e., they are topological).
Background. If we assume in addition that C has finitely many simple objects (up to isomorphism) and enough projectives, this becomes a known statement. Such categories are called finite (abelian) categories, and the claim is that C is equivalent to the category of finite-dimensional representations of a finite-dimensional algebra. (By the way, the statement is attributed to Deligne on nLab; do you know the source of this attribution? The reference on nLab is to a paper by Etingof and Ostrik, which as far as I see simply gives the statement.)