Grothendieck Group - Grothendieck Group and Extensions

Grothendieck Group and Extensions

Another construction that carries the name Grothendieck group is the following: Let R be a finite dimensional algebra over some field K or more generally an artinian ring. Then define the Grothendieck group G₀(R) as the group generated by the set of isomorphism classes of finitely generated R-modules and the following relations: For every exact sequence

of R-modules add the relation

The abelian group defined by this generators and this relations is the Grothendieck group G₀(R).

This group satisfies a universal property. We make a preliminary definition: A function χ from the set of isomorphism classes to an abelian group A is called additive if, for each exact sequence 0 → A → B → C → 0, we have . Then, for any additive function χ: R-mod → X, there is a unique group homomorphism f: G₀(R) → X such that χ factors through f and the map that takes each object of to the element representing its isomorphism class in G₀(R). Concretely this means that f satisfies the equation f = χ(V) for every finitely generated R-module V and f is the only group homomorphism that does that.

Examples of additive functions are the character function from representation theory: If R is a finite dimensional K-algebra, then we can associate the character χ_V: R → K to every finite dimensional R-module V: χ_V(x) is defined to be the trace of the K-linear map that is given by multiplication with the element x ∈ R on V.

By choosing suitable basis and write the corresponding matrices in block triangular form one easily sees that character functions are additive in the above sense. By the universal property this gives us a "universal character" such that χ = χ_V.

If K = C and R is the group ring C of a finite group G then this character map even gives a natural isomorphism of G₀(C) and the character ring Ch(G). In the modular representation theory of finite groups K can be a the algebraic closure of the finite field with p elements. In this case the analogously defined map that associates to each K-module its Brauer character is also a natural isomorphism onto the ring of Brauer characters. In this way Grothendieck groups show up in representation theory.

This universal property also makes G₀(R) the 'universal receiver' of generalized Euler characteristics. In particular, for every bounded complex of objects in R-mod

we have a canonical element

In fact the Grothendieck group was originally introduced for the study of Euler characteristics.