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In this MSRI lecture Geometry of Quiver Varieties I, Victor Ginzburg describes all abelian categories of homological dimension $1$ as being either

  • a category of representations $\mathrm{Rep}_\mathbf{k}Q$ of a quiver $Q$, or
  • a category of coherent sheaves on a smooth algebraic curve, or
  • one of fifty isolated exceptions that we don't care about.

What are these isolated exceptions? Or to answer this question more generally, what's a reference for this whole classification?

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    $\begingroup$ There exists a classification of hereditary abelian categories (that are noetherian, and have Serre duality) due to Reiten--Van den Bergh. His list is missing out on stacky curves (seen as hereditary orders in Reiten--Van den Bergh). One could argue that the ones which do not have a tilting object, nor are stacky curves, are "isolated", but it's not a finite list. $\endgroup$
    – pbelmans
    Commented Nov 22, 2019 at 15:31
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    $\begingroup$ What about Dedekind domains? To me the number fifty sounds like he is not too serious about this. $\endgroup$
    – Mare
    Commented Nov 22, 2019 at 15:33
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    $\begingroup$ I don't know what he refers to, but to give some references regarding the first two: There is a classification of Ext-finite abelian categories with a tilting object, see e.g. [Happel: A characterization of hereditary categories with tilting object] and [Happel-Reiten: Hereditary abelian categories with tilting object over arbitrary base fields]. Over an algebraically closed field it consists of representations of quivers and coherent sheaves over weighted projective lines. $\endgroup$
    – Julian Kuelshammer
    Commented Nov 22, 2019 at 15:38

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