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Let $K$ be a field. Does the category of finitely generated $K$-modules have a nice characterization, for example as the unique abelian category satisfying a certain simple condition? For example, we know that:

  1. Every short exact sequence is split.
  2. The Euler characteristic of every bounded exact sequence is zero.

Are either of those enough to characterize the category?

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    $\begingroup$ mathoverflow.net/questions/118246/… $\endgroup$
    – Dag Oskar Madsen
    Commented Oct 22, 2013 at 19:08
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    $\begingroup$ What do you mean by the Euler characteristic of a complex of objects in an abelian category? Do you mean the alternating sum of the images in the Grothendieck group, and if so, isn't this always equal to zero? $\endgroup$
    – Qiaochu Yuan
    Commented Oct 22, 2013 at 19:20
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    $\begingroup$ I'm also not sure how you expect to recover the field $K$. Are we secretly talking about $K$-linear categories or are you happy to characterize these categories as $K$ runs over all fields? $\endgroup$
    – Qiaochu Yuan
    Commented Oct 22, 2013 at 19:28
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    $\begingroup$ See my question mathoverflow.net/questions/118246/…, which has a very satisfactory answer. $\endgroup$
    – David Roberts ♦
    Commented Oct 22, 2013 at 23:45
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    $\begingroup$ @Fernando: that's not what I mean. The wording of the OP's question suggests that he wants simple conditions that pin down $\text{FinVect}_K$ for fixed $K$, and that seems unlikely to me unless we explicitly code in $K$, e.g. by working with $K$-linear categories. $\endgroup$
    – Qiaochu Yuan
    Commented Oct 22, 2013 at 23:59

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The answer is no, and an easy counterexample is provided by the category of finite-dimensional modules over a division algebra such as the quaternions. Of course this is not a very good example because you can easily add small modifications to your question to get rid of it. This category is for instance not symmetric monoidal, unlike vector spaces over a field. As Oskar points out, there is an already answered question in MO which gives a positive answer to your question under somewhat different conditions. You'll like to look at it. I warn you that your Euler characteristic condition may be complicated to state in an abstract setting.

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