Lemma 3

Let G be an abelian group written additively, and $\varphi :\mathscr{F}_{\mathfrak{a}}\rightarrow G$ a map; by abuse of notation, we denote by $\varphi (F)$ the image under $\varphi$ of the class of any finite dimensional $\mathfrak{a}$-module F. Assume that, for any exact sequence

of finite dimensional $\mathfrak{a}$-modules, we have $\varphi (F) =\varphi (F') +\varphi (F'')$. Then, there exists a unique homomorphism $\theta :\mathscr{R}(\mathfrak{a})\rightarrow G$ such that $\theta ([F]) =\varphi (F)$ for every finite dimensional $\mathfrak{a}$-module F.

There exists a unique homomorphism $\theta$ from $\mathscr{R}(\mathfrak{a})$ to G such that $\theta ([E]) =\varphi (E)$ for every finite dimensional simple $\mathfrak{a}$-module E. Let F be a finite dimensional $\mathfrak{a}$-module, and $(F_n,F_{n-1}, . . . ,F_0)$ a Jordan-Hölder series of F; if $n >0$, we have, by induction on $n$,

If $n= 0$ then [F] = 0 so $\theta ([F]) = 0$; on the other hand, by considering the exact sequence $0\longrightarrow 0\longrightarrow 0\longrightarrow 0\longrightarrow 0$ we see that $\varphi (0) = 0$.