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Grothendieck Groups and the Eilenberg Swindle July 12, 2009

Posted by Akhil Mathew in algebra, representation theory.
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7 comments

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…)