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In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with split Grothendieck groups. Here he starts with a certain small additive category $\mathcal{A}$ and writes $\langle \mathcal{A} \rangle$ for its split Grothendieck group: the free abelian group on objects $\langle A \rangle$ corresponding as usual to isomorphism classes, modulo sums$\langle C \rangle = \langle A \rangle + \langle B \rangle$ corresponding only to the situation $C \cong A \oplus B$.

This is a less familiar situation than the usual Grothendieck group with sums corresponding to short exact sequences which may or may not split.

Where does the notion of split Grothendieck group originate, and why?

This is mostly asked out of curiosity, but I'm also looking for further interesting examples.

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I was going to point out that this concept for finite groups is known as the Green ring. Then I discovered that you were the reviewer for: MR0537496 (80e:20012) Alperin, J. L. Projective modules for SL(2,2n). J. Pure Appl. Algebra 15 (1979), no. 3, 219--234. – Bruce Westbury Jan 7 '13 at 13:36
@Bruce: Yes, the idea is basically the same for Green rings though I had the impression that split Grothendieck groups were defined explicitly in a more geometric framework. I still have to do some rechecking about origins and motivation. – Jim Humphreys Jan 7 '13 at 13:49
up vote 7 down vote accepted

These groups are mentioned in [Swan '68 - Algebraic K-Theory, p.69]. He constructs $K_0(\mathcal{A}, S)$ for a class $S$ of exact sequences in $\mathcal{A}$. Take the free abelian group mod the relations from sequences in $S$. For example the class of all exact sequences for the Grothendieck-group $K_0(\mathcal{A})$ or all exact split seqences for the group you mentioned. The name 'split-Grothendieck-group' does not appear.

He generalizes further to $K_0(\mathcal{A}, F)$ for a bifunctor $F:\mathcal{A} \times \mathcal{A} \to \mathcal{A}$ and obtains a generalized Picard-group.

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This is a convincing source, which I'm only marginally familiar with. The helpful detailed MathSciNet review by Hyman Bass brings out clearly the categorical generality in which Swan revisited Grothendieck's construction. Though the term "split Grothendieck group" doesn't seem to occur exactly in that form, the split version is certainly explicit enough. – Jim Humphreys Jan 7 '13 at 19:10

The split Grothendieck group for vector bundles on a complete variety appears in Nori's PhD thesis on the fundamental group scheme; this was published in the Proceedings of the Indian Academy of Science in 1981. It is used to define and study finite vector bundles. Nori does not give any references, so as far I know the construction might be due to him.

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@Angleo: The general idea seems to originate with Swan in his 1968 lecture notes volume, as pointed out by Stephan. But Nori and others were certainly applying the same strategy in specific geometric or algebraic cases of interest, whether or not they were inspired by Swan. – Jim Humphreys Jan 7 '13 at 19:14

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