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:

Proposition 2 To give a function ${\chi: \mathbf{A} \rightarrow X}$ for an abelian group ${X}$ satisfying ${\chi(M) = \chi(M')+\chi(M'')}$ for each exact sequence as in (1) (i.e. an Euler-Poincaré map, is equivalent to giving a group-homomorphism ${K(\mathbf{A}) \rightarrow X}$.

This property, which follows from the definition, determines the Grothendieck group up to isomorphism as the unique group making the above result valid.

The Grothendieck Group of Representations

Let ${G}$ be a finite group. Then consider the category ${Rep(G)}$ defined as follows: the objects of ${G}$ are the finite-dimensional representations of ${G}$, and morphisms are ${{\mathbb C}[G]}$-module homomorphisms (also called intertwining operators).

Then:

Proposition 3 The Grothendieck group ${K(Rep(G))}$ is the abelian group ${F}$ generated by the irreducible characters.

Suppose the irreducible representations are ${V_1, \dots, V_t}$, corresponding to characters ${\chi_1, \dots, \chi_t}$. We have a group-homomorphism ${F \rightarrow K(Rep(G))}$ sending a sum ${n_1 \chi_1 + \dots + n_t \chi_t}$ to ${\bigoplus_i n_i V_i}$ (this being well-defined as the characters are linearly independent). This is surjective since we can decompose a representation as a direct sum of irreducibles. We define the inverse ${K(Rep(G)) \rightarrow F}$ by using the above universal property. First, define the Euler-Poincaré map ${Rep(G) \rightarrow F}$ by sending ${V= \bigoplus_i m_i V_i \rightarrow \sum m_i \chi_i}$; this is valid by the previous post, since each object can be decomposed uniquely into irreducibles. One then gets a map ${K(Rep(G)) \rightarrow F}$. These two maps between ${K(Rep(G))}$ and ${F}$ (and vice versa) are checked to be inverse to each other.

This result can be generalized to semisimple abelian categories.

The Eilenberg Swindle

Suppose an abelian category ${\mathbf{A}}$ admits infinite direct sums. Then I claim:

Theorem 4 ${K(\mathbf{A})=0}$.

This is proved using the Eilenberg swindle. Given ${X \in \mathbf{A}}$, we show that ${[X] = 0}$. But $\displaystyle X \oplus \bigoplus_{i=1}^\infty X = \bigoplus_{i=1}^\infty X;$

we thus have an exact sequence $\displaystyle 0 \rightarrow X \rightarrow \bigoplus_{i=1}^\infty X \rightarrow \bigoplus_{i=1}^\infty X \rightarrow 0,$

which gives us the relation ${[X]=0}$.

The Eilenberg swindle can also be stated in a slightly different form.

The notion of Grothendieck group is useful in modular representation theory—which works in fields other than ${{\mathbb C}}$, having nonzero characteristic not relatively prime to ${card \ G}$. Then, it’s possible to say that certain maps between categories may not be surjective, but they are “mostly” surjective in that the induced map on Grothendieck groups is so. We’ll probably use Grothendieck groups more later.