I am currently studying a proof by M. Kapranov, from this article, prop. 2.2.2, and I have encountered a difficulty.
We have a smooth projective scheme $S$, a codimension 1 closed subscheme $C$ in $S$ defined by the sheaf of ideals $m$, and a fixed $SL_n$-bundle $V^o$ over $S\setminus C$. (Never mind the fact that the article talks about a surface and a curve; we need to work with families.) Let $V_0$ be an $SL_n$-bundle $V$ equipped with an isomorphism $\phi:V|_{S\setminus C}\cong V^o$. Then one can prove that each pair $(V,\psi)$ with the same properties admits a couple of inclusions $m^nV\subset V_0$, $m^nV_0\subset V$, for some $n$.
Now the Author concludes that, when $n$ is fixed, a locally closed subscheme of $Quot (m^{-n}V_0/m^{n}V_0)$ parametrises all $V$'s as above. My difficulties are now:
1) Why do we have this correspondence? I expect that a pair $(V, psi)$ corresponds to the cokernel of the map $V\subset m^{-n}V_0$, but I would like to see a formal proof that this is a bijection, if this is the case.
2) If (1) works, why do we need a locally closed subscheme and not just a closed one? I think that, when the Author speaks of a locally closed subscheme, he is referring to the Grassmannian parametrising all locally free quotients of the coherent sheaf (see Nitsure, Construction of the Hilbert and Quot schemes, page 6), but is this really locally closed? When reading Nitsure's paper, it would seem that this Grassmannian is proper (in fact projective), not quasiprojective; and moreover, that it actually coincides with the entire mentioned Quot.
Surely, imposing the condition on the determinant (we want $SL_n$-bundles) should in fact lead us to consider a smaller scheme, so this could be the reason why Kapranov talks about a "locally closed subscheme" of the mentioned Quot. But is this really a locally closed subscheme, or is it already a closed subscheme?
Thank you in advance.