of isomorphisms of monads of the form acts on ${P_{\vec{k}}}$, and the moduli space ${{\mathcal{M}}^n(r,a,c)}$ of framed sheaves on $\Sigma_n$ with given Chern character is set-theoretically defined as the quotient ${P_{\vec{k}}}/{G_{\vec{k}}}$. Theorem \[proMk\] proves that ${{\mathcal{M}}^n(r,a,c)}$ inherits from ${P_{\vec{k}}}$ a structure of smooth algebraic variety. As a corollary, ${{\mathcal{M}}^n(r,a,c)}$ turns out to be irreducible. Moreover, Lemma \[pro3\] states that two monads of the form are isomorphic if and only it their cohomologies are isomorphic, and this ensures that there is a bijection between ${{\mathcal{M}}^n(r,a,c)}$ and set of isomorphism classes of framed sheaves on $\Sigma_n$ (isomorphisms of framed sheaves are introduced in Definition \[DEf3\]). This enables us to show that the set of these classes is nonempty if and only if $ 2c \ge na(1-a)$ (hence, at least when $a\ne 0$, it turns out that the moduli space can be empty even if it has positive expected dimension).

We prove the fineness of the moduli space by constructing a universal family $\left({\ensuremath{\mathfrak{E}_{\vec{k}}}},{\Theta_{\vec{k}}}\right)$ of framed sheaves on $\Sigma_n$ parametrized by ${{\mathcal{M}}^n(r,a,c)}$ (for the precise notion of family see Definition \[defFam\]).

These results are the basis for further work where we give a detailed description of the moduli spaces when the topological invariants 