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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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marked as duplicate by David Roberts, Daniel Moskovich, Eric Wofsey, Andrey Rekalo, David White Oct 23 '13 at 14:23

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

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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? –  Qiaochu Yuan Oct 22 '13 at 19:20
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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? –  Qiaochu Yuan Oct 22 '13 at 19:28
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See my question mathoverflow.net/questions/118246/…, which has a very satisfactory answer. –  David Roberts Oct 22 '13 at 23:45
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@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. –  Qiaochu Yuan Oct 22 '13 at 23:59

1 Answer 1

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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