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quick note on Deligne tensor product of abelian categories
Just a quicky, when is an abelian category said to be finite? I found
A k-linear abelian category C is said to be finite if it is equivalent to the category A − mod of finite dimensional modules over a finite dimensional k-algebra A.
in those MIT notes, but I do not like that use of the term finite. (The red herring principle might be invoked, but for this to be clear we should give this definition in the entry.) I will do that for the moment, but wanted to flag up the red herring difficulty.
People in the field do that kind of thing all the time though. A “finite $A$-algebra” is not one that has finitely many elements but rather is a $A$-algebra that is finitely-generated as a module over $A$, and the same for “finite $A$-modules” and “finite field extension”.
In that case, my edit seems justified, in other words to give a caveat for that entry. There may, however, be some better solution,
Your edit makes Remark 1 be pointless.
Pointless yes… but still true. :-)
I will adjust the wording (perhaps it should be said that $A$ is only determined up to Morita equivalence so have added that to Remark 1.)
I cross checked to Morita equivalence and found that there are several query boxes still there. I do not feel competent to answer the queries, so can someone more up on that have a look?
Hm, I think I might create finite abelian category. According to my source, this is a $k$-linear abelian category ($k$ some field) such that all homs are finite dimensional, every object has finite length, and there are only finitely many simple objects, each of which has a projective cover. Is that the standard definition? And is this in Deligne’s Catégories tannakiennes (which I don’t have)?
Then remark 1 becomes less pointless.
My quote was from the MIT lecture notes that are referenced in the entry. I like your solution.
I wrote up something very quickly at finite abelian category, but it could surely be improved. Particularly in specifying the equivalence in the theorem that I attributed (maybe incorrectly) to Deligne.
Edit: I also made some amendments to tensor product of abelian categories. Please check for accuracy.
That looks good.
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