Remark 1

Let $f$ : S $\rightarrow S'$ be a morphism of tori. The snake diagram (Algebra, Chap. X, §1, no. 2) associated to (1) gives an exact sequence

0 $\longrightarrow$ Ker$\Gamma (f)\longrightarrow$ Ker $L(f)\longrightarrow$ Ker $f\longrightarrow^d$ (4)

$\longrightarrow^d$ Coker $\Gamma (f)\longrightarrow$ Coker $L(f)\longrightarrow$ Coker $f\longrightarrow 0$.

In particular, assume that $f$ is surjective, with finite kernel N, so that we have an exact sequence

0 $\longrightarrow$ N $\longrightarrow^i$ S $\longrightarrow^fS'\longrightarrow 0$,

where $i$ is the canonical injection. Then, $L(f)$ is bijective, and (4) gives an isomorphism $N\rightarrow$ Coker $\Gamma (f)$, hence an exact sequence

0 $\longrightarrow \Gamma (S)\overset{\Gamma(f)}{\longrightarrow}\Gamma (S')\longrightarrow$ N $\longrightarrow 0$. (5)

Moreover, by Spectral Theories, Chap. II, §1, no. 7, Th. 4, the sequence

0 $\longrightarrow X(S')\overset{X(f)}{\longrightarrow}$ X(S) $\longrightarrow^{X(i)}$ X(N) $\longrightarrow$ 0 (6)

is exact.