Origin of notion of "split Grothendieck group"? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T20:30:16Z http://mathoverflow.net/feeds/question/118275 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/118275/origin-of-notion-of-split-grothendieck-group Origin of notion of "split Grothendieck group"? Jim Humphreys 2013-01-07T13:14:09Z 2013-01-07T15:12:15Z <p>In the construction of Soergel's bimodules in representtion theory , it's essential for him to work with <em>split</em> Grothendieck groups. Here he starts with a certain small additive category <code>$\mathcal{A}$</code> and writes <code>$\langle \mathcal{A} \rangle$</code> for its split Grothendieck group: the free abelian group on objects <code>$\langle A \rangle$</code> corresponding as usual to isomorphism classes, modulo sums<code>$\langle C \rangle = \langle A \rangle + \langle B \rangle$</code> corresponding only to the situation <code>$C \cong A \oplus B$</code>.</p> <p>This is a less familiar situation than the usual Grothendieck group with sums corresponding to short exact sequences which may or may not split. </p> <blockquote> <p>Where does the notion of split Grothendieck group originate, and why?</p> </blockquote> <p>This is mostly asked out of curiosity, but I'm also looking for further interesting examples.</p> http://mathoverflow.net/questions/118275/origin-of-notion-of-split-grothendieck-group/118278#118278 Answer by Angelo for Origin of notion of "split Grothendieck group"? Angelo 2013-01-07T13:33:21Z 2013-01-07T13:33:21Z <p>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.</p> http://mathoverflow.net/questions/118275/origin-of-notion-of-split-grothendieck-group/118285#118285 Answer by Stephan Müller for Origin of notion of "split Grothendieck group"? Stephan Müller 2013-01-07T15:12:15Z 2013-01-07T15:12:15Z <p>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.</p> <p>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.</p>