is injective, and $\mathcal{E}^{\ast\ast}$ and $\mathcal{E}'^{\ast\ast}$ are locally free.

This result generalizes to the relative situation. Let $S$ be a scheme, and let ${\mathfrak{F}}$ and ${\mathfrak{F}}'$ be two sheaves on $T=\Sigma_n\times S$ satisfying condition $\vec{k}$.

\[proInjInfty\] The restriction morphism $$\xymatrix{
{\operatorname{Hom}}\left({\mathfrak{F}},{\mathfrak{F}}'\right) \ar[r]^-R & {\operatorname{Hom}}\left({\mathfrak{F}}|_{T_\infty},{\mathfrak{F}}'|_{T_\infty}\right)
}$$ is injective.

Let ${\mathscr{H}}={\operatorname{\mathcal{H}\!\textit{om}}}\left({\mathfrak{F}},{\mathfrak{F}}'\right)$. Since both ${\mathfrak{F}}$ and ${\mathfrak{F}}'$ are locally free along $T_\infty$ one gets ${\operatorname{\mathcal{T}\!\textit{or}}}_i\left({\mathscr{H}},{\mathcal{O}}_{T_\infty}\right)=0$ for $i>0$ (see for example [@GH p. 700]). Thus, if we twist the structure sequence of the divisor $T_\infty$ by ${\mathscr{H}}$, we get $$\xymatrix{
0 \ar[r] & {\mathscr{H}}(-T_\infty) \ar[r] & {\mathscr{H}}\ar[r] & {\mathscr{H}}|_{T_\infty} \ar[r] & 0\,,
}$$ where ${\mathscr{H}}|_{T_\infty}\simeq{\operatorname{\mathcal{H}\!\textit{om}}}\left({\mathfrak{F}}|_{T_\infty},{\mathfrak{F}}'|_{T_\infty}\right)$. It follows that $$\ker R=H^0\left({\mathscr{H}}(-T_\infty)\right)=H^0\left(t_{2*}\left({\mathscr{H}}(-T_\infty)\right)\right).$$ By Propositions \[teo1\] and \[teo1’\], the sheaf ${\mathscr{H}}(-T_\infty)$ is flat on $S$. Moreover Lemmas \[pro1\] and \[lm2\] yield the following vanishing result for all closed points $s\in S$: $$\begin{split}
H^0\left({\mathscr{H}}(-T_\infty)_s\right)&=H^0\left({\operatorname{Hom}}\left({\mathfrak{F}}_s,{\mathfrak{F}}'_s\right)(-{\ell_\infty})\right)=0\,.
\end{split}$$ The Semicontinuity Theorem entails the vanishing of the sheaf $t_{2*}\left({\mathscr{H}}(-T_\infty)\right)$. This ends the proof.

\[cor1\] The action of ${G_{\vec{k}}}$ on ${P_{\vec{k}}}$ is free.

Let $\left(\alpha,\beta;{\bar{\psi}}\right)\in{L_{\vec{k}}}\times{G_{\vec{k}}}$, and put $\left(\alpha',\beta'\right)={\bar{\psi}}\cdot(\alpha,\beta)$. It follows from Lemma \[lm2\] that a morphism $\Lambda\in{\operatorname{Hom}}\left({\mathcal{E}}_{\alpha,\beta},{\mathcal{E}}_{\alpha',\beta'}\right) $ is fully determined by its restriction $\Lambda_\infty$ to $\ell_{\infty}$. It is not difficult to see [@Ok Lemma 4.1.3] that $\Lambda$ is induced by a unique 