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If you're interested in the $K$-theory of rings, a useful feature of the exact category of finitely generated projective (or free) modules is that it is split exact, i.e. every short exact sequence is isomorphic to a canonical direct sum short exact sequence $$P' \stackrel{i_{1}}{\rightarrowtail} P' \oplus P'' \stackrel{\pi_{2}}{\twoheadrightarrow} P''.$$ Such split exact categories are often simpler to work with than general exact categories. For example, if $\mathcal{P}$ is a split exact category then $K_{1}\mathcal{P}$ agrees with the Bass $K$-theory group $K_{1}^{\text{Bass}}\mathcal{P}$.

When working with split exact categories I usually have categories of f.g. projective (or free) modules in mind. This can cause problems, or make it hard to look for counterexamples to certain statements, as categories of projective modules have stronger properties that do not hold in a general split exact category. I want to broaden my intuition with some more examples.

One can give numerous examples by cheating, since every additive category can be considered as a split exact category with the minimal exact structure (the one comprised of all sequences isomorphic to the direct sum sequences), but this is not usually the natural choice of exact structure.

Question: What are some examples of split exact categories arising naturally, and which are not equivalent to a category of projective or free modules over some ring?

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  • $\begingroup$ Is there any significance to the subscripts in the short exact sequence? $\endgroup$
    – Tait
    Apr 2, 2014 at 15:11
  • $\begingroup$ @Tait the $i_{1}$ is inclusion of the first summand and the $\pi_{2}$ is projection of the second summand. $\endgroup$
    – Tom Harris
    Apr 2, 2014 at 15:14
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    $\begingroup$ The title of this question is misleading, pehaps on purpose: "arising naturally" is not the same as "appearing in nature". If designed to attract more attention, it works! $\endgroup$
    – Wlodek Kuperberg
    Apr 2, 2014 at 16:00
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    $\begingroup$ @WlodekKuperberg I was using them interchangeably, but you're right, they don't mean the same thing. I've changed to title to prevent anyone else being misled :) $\endgroup$
    – Tom Harris
    Apr 2, 2014 at 16:03
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    $\begingroup$ Isn't it true that any karoubian split exact category is equivalent to the category of projective modules over a ring? $\endgroup$
    – Fernando Muro
    Apr 2, 2014 at 21:53

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