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 . Then:

Definition 1TheGrothendieck groupof is the abelian group defined via generators and relations as follows: is generated by symbols for each , and by relations for each exact sequence

Note here that if are isomorphic, then in by considering the exact sequence

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