spaces are also shown to be fine, and this implies that they are isomorphic to the moduli spaces constructed in [@BruMar] (and therefore embed as open subschemes into Huybrechts-Lehn’s moduli spaces).

The monads we use generalize to torsion-free sheaves those introduced by Buchdahl [@Bu] for the locally-free case (he was actually interested in $\mu$-stable vector bundles on Hirzebruch surfaces with $c_1=0$, $c_2=2$). Indeed, we show in Corollary \[pro2\] that for any framed torsion-free sheaf $({\mathcal{E}},\theta)$ on $\Sigma_n$, the underlying sheaf ${\mathcal{E}}$ is isomorphic to the cohomology of a monad $$\xymatrix{
{{\mathcal{U}}_{\vec{k}}}\ar[r]^-\alpha & {{\mathcal{V}}_{\vec{k}}}\ar[r]^-\beta & {{\mathcal{W}}_{\vec{k}}}}
\label{eqI1}$$ whose terms depend only on the Chern character $\textrm{ch}({\mathcal{E}})=(r,aE,-c-\frac12na^2)$ (here $E$ is the exceptional curve in $\Sigma_n$, i.e., the unique irreducible curve in $\Sigma_n$ squaring to $-n$, and we put $\vec{k}=(n,r,a,c)$). This provides a map $$({\mathcal{E}},\theta)\longmapsto(\alpha,\beta)\in {\operatorname{Hom}}({{\mathcal{U}}_{\vec{k}}},{{\mathcal{V}}_{\vec{k}}})\oplus{\operatorname{Hom}}({{\mathcal{V}}_{\vec{k}}},{{\mathcal{W}}_{\vec{k}}}).
\label{eqxi}$$ We call ${L_{\vec{k}}}$ the image of this map and prove in Proposition \[6prop3\] and in Lemma \[6lm10\] that ${L_{\vec{k}}}$ is a smooth variety. We construct a principal ${\operatorname{GL}}(r,{\mathbb{C}})$-bundle ${P_{\vec{k}}}$ over ${L_{\vec{k}}}$ whose fibre over a point $(\alpha,\beta)$ is naturally identified with the space of framings for the cohomology of the complex . This implies that the map lifts to a map $
({\mathcal{E}},\theta)\longmapsto\theta\in{P_{\vec{k}}}$.

The group ${G_{\vec{k}}}$ 