Realization/embedding for (weakly) finite linear categories

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:

  1. C is essentially small
  2. Every object of C has finite length
  3. 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.)


Source : Link , Question Author : t3suji , Answer Author : Community

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