Grothendieck Group - Grothendieck Groups of Exact Categories

Grothendieck Groups of Exact Categories

A common generalization of these two concepts is given by the Grothendieck group of an exact category . Simplified an exact category is an additive category together with a class of distinguished short sequences A → B → C. The distinguished sequences are called "exact sequences", hence the name. The precise axioms for this distinguished class do not matter for the construction of the Grothendieck group.

It is defined in the same way as before as the abelian group with one generator for each (isomorphism class of) object(s) of the category and one relation

for each exact sequence

.

Alternatively one can define the Grothendieck group using a similar universal property: An abelian group G together with a mapping is called the Grothendieck group of iff every "additive" map from into an abelian group X ("additive" in the above sense, i.e. for every exact sequence we have ) factors uniquely through φ.

Every abelian category is an exact category if we just use the standard interpretation of "exact". This gives the notion of a Grothendieck group in the previous section if we choose -mod the category of finitely generated R-modules as . This is really abelian because R was assumed to be artinian and (hence noetherian) in the previous section.

On the other hand every additive category is also exact if we declare those and only those sequences to be exact that have the form with the canonical inclusion and projection morphisms. This procedure produces the Grothendieck group of the commutative monoid in the first sense (here means the "set" of isomorphism classes in .)