Lemma 2

Let $f : U \to V$ and $g : V \to W$ be two L-linear mappings. Iff and g each have an index, the same is true of $g \circ f$ and we have

Put $h = g \circ f$, and denote by N, N', N'' the kernels off, g, h respectively and by C, C', C'' their cokernels. We have $N \subset N'' \subset U$ and $f(N'') = f(U) \cap N'$; hence there exists a linear mapping $\bar{f} : N'' \to N'$ agreeing with f on N'' and with kernel N. The canonical mapping $\pi$ of V onto $C = V / f(U)$ induces a mapping $\pi'$ of N' into C whose kernel is $f(U) \cap N' = \bar{f}(N'')$. By passage to quotients g defines a mapping $\bar{g}$ of $C = V/f(U)$ into $C'' = W/g(f(U))$ whose kernel is clearly $(N' + f(U))/f(U) = \pi'(N')$. Finally the canonical mapping $\rho$ of $C'' = W/g(f(U))$ onto $C' = W/g(V)$ has the kernel $g(V)/g(f(U)) = \bar{g}(C)$. To sum up, we have established the exactness of the sequence

(where i is the canonical injection of N in N").

!Diagram showing the relationships between N, N', N'', U, V, W, C, C', C'', with arrows indicating mappings and kernels

Fig. 1

By hypothesis N, N', C and C' are of finite dimension; hence the same is true of N'' and C. By Cor. 2 (II, p. 295) we then have

whence $\chi(h) = \chi(f) + \chi(g)$.