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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeJan 16th 2013
    • (edited Jan 16th 2013)
    • CommentRowNumber2.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)

    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.

    • CommentRowNumber3.
    • CommentAuthorZhen Lin
    • CommentTimeJan 17th 2013

    People in the field do that kind of thing all the time though. A “finite AA-algebra” is not one that has finitely many elements but rather is a AA-algebra that is finitely-generated as a module over AA, and the same for “finite AA-modules” and “finite field extension”.

    • CommentRowNumber4.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2013

    In that case, my edit seems justified, in other words to give a caveat for that entry. There may, however, be some better solution,

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2013

    Your edit makes Remark 1 be pointless.

    • CommentRowNumber6.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)

    Pointless yes… but still true. :-)

    I will adjust the wording (perhaps it should be said that AA 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?

    • CommentRowNumber7.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)

    Hm, I think I might create finite abelian category. According to my source, this is a kk-linear abelian category (kk 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.

    • CommentRowNumber8.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2013

    My quote was from the MIT lecture notes that are referenced in the entry. I like your solution.

    • CommentRowNumber9.
    • CommentAuthorTodd_Trimble
    • CommentTimeJan 17th 2013
    • (edited Jan 17th 2013)

    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.

    • CommentRowNumber10.
    • CommentAuthorTim_Porter
    • CommentTimeJan 17th 2013

    That looks good.

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