This question already has an answer here:

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?


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.

  • 6
    $\begingroup$ mathoverflow.net/questions/118246/… $\endgroup$ – Dag Oskar Madsen Oct 22 '13 at 19:08
  • 1
    $\begingroup$ 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 Oct 22 '13 at 19:20
  • 2
    $\begingroup$ 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 Oct 22 '13 at 19:28
  • 1
    $\begingroup$ See my question mathoverflow.net/questions/118246/…, which has a very satisfactory answer. $\endgroup$ – David Roberts Oct 22 '13 at 23:45
  • 1
    $\begingroup$ @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 Oct 22 '13 at 23:59

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.


Not the answer you're looking for? Browse other questions tagged or ask your own question.