\[lm3\]). This ends the proof.

The following result is the absolute case of Proposition \[teo1\], obtained by letting $S={\operatorname{Spec}}{\mathbb{C}}$, and follows easily from Corollary \[corLocOnS\].

\[pro2\] Any sheaf $\mathcal{E}$ on $\Sigma_n $ that is torsion-free and trivial at infinity is isomorphic to the cohomology of a monad ${M_{\vec{k}}}({\mathcal{E}})$, which is of the form $M(\alpha,\beta)$ for a suitable $(\alpha,\beta)\in\overline{L}_{\vec{k}}$. Note that we do not require $k_1\geq0$ *a priori*. As a consequence however, if $k_1=c+\frac{1}{2}na(a-1)<0$, the set ${\mathcal{M}}^n(r,a,c)$ is empty.

We fix a $\vec{k}$ such that $k_1\geq0$. The functoriality of $M_{\vec{k}}(-)$ implies that ${M_{\vec{k}}}({\mathcal{E}})\simeq{M_{\vec{k}}}({\mathcal{E}}')$ whenever ${\mathcal{E}}\simeq{\mathcal{E}}'$. In particular, ${M_{\vec{k}}}(-)$ provides a map between the set of isomorphism classes of torsion-free sheaves on $\Sigma_n$ that are trivial at infinity and the set of isomorphism classes of monads of the form $M(\alpha,\beta)$. The following two results establish the injectivity of this map, and enable us to characterize its image.

\[pro3\] Let $(\alpha,\beta), \left(\alpha',\beta'\right)$ be any two points in $\overline{L}_{\vec{k}}$ satisfying the conditions (c1) and (c2) introduced in section \[Main result\] (therefore $M=M(\alpha,\beta)$ and $M'=M\left(\alpha',\beta'\right)$ are monads). Then $$M\simeq M'\qquad\text{if and only if}\qquad \mathcal{E}(M)\simeq\mathcal{E}(M').
\label{eq392}$$

The proof of [@Ok Lemma 4.1.3] holds true for non-locally free sheaves as well.

\[6prop3\] For any point $(\alpha,\beta)\in\overline{L}_{\vec{k}}$ satisfying 