Representability of Hom implies flatness (Q1431068)

Let \(X\) be a projective scheme over a noetherian base scheme \(S\), and let \(\mathcal{F}\) be a coherent sheaf on \(X\). For any coherent sheaf \(\mathcal{E}\) on \(X\), consider the set-valued contravariant functor \(\mathcal{H}\text{om}_{(\mathcal{E},\mathcal{F})}\) on \(S\)-schemes, defined by \(\mathcal{H}\text{om}_{(\mathcal{E},\mathcal{F})}(T)=\Hom(\mathcal{E}_T,\mathcal{F}_T)\) where \(\mathcal{E}_T,\text{ and }\mathcal{F}_T\) are the pull-backs of \(\mathcal{E}\) and \(\mathcal{F}\) to \(X_T=X\times_S T\). A basic result of Grothendieck says that if \(\mathcal{F}\) is flat over \(S\) then \(\mathcal{H}\text{om}_{(\mathcal{E},\mathcal{F})}\) is representable for all \(\mathcal{E}\). The author proves the converse of this result, that is, if \(\mathcal{L}\) is a relatively ample line bundle over \(X\) over \(S\) such that the functor \(\mathcal{H}\text{om}_{(\mathcal{L}^{-n},\mathcal{F})}\) is representable for infinitely many positive integers \(n\), then \(\mathcal{F}\) is flat over \(S\). Taking \(S=X\) this results in the corollary that if \(\mathcal{F}\) is coherent on \(S\), then the functor \(T\mapsto H^0(T,\mathcal{F}_T)\) on the category of \(S\)-schemes is representable if and only if \(\mathcal{F}\) is locally free. The techniques used by the author involves a reduction to \(S=\text{Spec}(R)\) where \(R\) is local, a flattening stratification of \(\text{Spec}(R)\), a reduction to the Artin local case with principal \(I\) with \(\mathfrak{m}I=0\), decomposition of \(\pi_{\ast}\mathcal{F}{\text{(r)}}\), the structure of a hypothetical representing scheme \(G^{r}\), the kernel of the map \(G^{r}(R)\rightarrow G^{r}(R/I)\) and finally a functorial description of this kernel. The last functorial description, leads to a contradiction proving the main theorem. All in all a very nice and elegant proof, using results related to deformation theory and geometric invariant theory to prove a useful result.