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 χV: R → K to every finite dimensional R-module V: χV(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 = χV.
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.
Read more about this topic: Grothendieck Group
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