## Grothendieck Groups and the Eilenberg SwindleJuly 12, 2009

Posted by Akhil Mathew in algebra, representation theory.
Tags: , , , ,

The following topic came up in a discussion with my mentor recently. Since the material is somewhat general and well-known, but relevant to my project area, I decided to write this post partially to help myself understand it better.

Definition

Consider an abelian category ${\mathbf{A}}$. Then:

Definition 1 The Grothendieck group of ${\mathbf{A}}$ is the abelian group ${K(\mathbf{A})}$ defined via generators and relations as follows: ${K(\mathbf{A})}$ is generated by symbols ${[M]}$ for each ${M \in \mathbf{A}}$, and by relations ${[M] - [M'] - [M'']}$ for each exact sequence

$\displaystyle 0 \rightarrow M' \rightarrow M \rightarrow M'' \rightarrow 0.\ \ \ \ \ (1)$

Note here that if ${M,N}$ are isomorphic, then ${[M] = [N]}$ in ${K(\mathbf{A})}$ by considering the exact sequence

$\displaystyle 0 \rightarrow M \rightarrow N \rightarrow 0 \rightarrow 0.$

The Grothendieck group has an important universal property: (more…)