construct a canonical family $\left({\widetilde{\mathfrak{E}}_{\vec{k}}},{\widetilde{\Theta}_{\vec{k}}}\right)$ on the product $\Sigma_n\times{P_{\vec{k}}}$.

For any scheme $S$, let $T=\Sigma_n\times S$ and let $t_i$, $i=1,2$ be the projections onto the first and the second factor, respectively. Analogously, we introduce the product scheme $T_\infty={\ell_\infty}\times S$, toghether with the projections $u_i$, $i=1,2$.

Let $\vec{k}=(n,r,a,c)\in{\mathbb{Z}}^4$ with $n\geq1$, $r\geq1$ and $0\leq a \leq r-1$. A coherent sheaf ${\mathfrak{F}}$ on $T$ fulfills condition $\vec{k}$ if and only if it is flat on $S$ and for all closed points $s\in S$

-   the restricted sheaf ${\mathfrak{F}}_s$ is torsion-free and trivial at infinity on $T_s\simeq\Sigma_n$;

-   the Chern character of ${\mathfrak{F}}_s$ is $(r,aE,-c-\frac{1}{2}na^2)$.

Note that by Lemma \[torsionfree\] ${\mathfrak{F}}$ is torsion-free.

\[defFam\] Given a vector $\vec{k}$ and a product scheme $T$ as above, a family of framed sheaves on $\Sigma_n$ is a pair $({\mathfrak{F}},\Theta)$, where:

1.  ${\mathfrak{F}}$ is a sheaf on $T$ fullfilling condition $\vec{k}$;

2.  $\Theta$ is an isomorphism ${\mathfrak{F}}|_{T_\infty}\to  \mathcal O_{T_\infty}^{\oplus r}$.

For any sheaf ${\mathfrak{G}}$ of ${\mathcal{O}}_T$-modules we let $${\mathfrak{G}}(p,q)={\mathfrak{G}}\otimes t_1^*{{\mathcal{O}}_{\Sigma_n}}(p,q)\qquad\text{for all $(p,q)\in{\mathbb{Z}}$}.$$

\[teo1\] A sheaf ${\mathfrak{F}}$ on $T$ that satisfies condition $\vec{k}$ is isomorphic to the cohomology of a canonically associated monad $M({\mathfrak{F}})$ on $T$ $$M({\mathfrak{F}}):\qquad
\xymatrix{
0 \ar[r] & {\mathfrak{U}}\ar[r]^A & {\mathfrak{V}}\ar[r]^B & {\mathfrak{W}}\ar[r] & 0
}\,,
