I am studying Nitsure's wonderful essay Construction of Hilbert and Quot schemes, and I am stuck on page 21.
Given a Noetherian schemes $S$ and a coherent sheaf $\mathcal{F}$ over $\mathbb{P}^{n}_{S}$, Nitsure constructs a finite set of locally closed and mutually disjoint subschemes $V_{i}$ ($i\in\left\{1,...,t\right\}$) so that $\mathcal{F}|_{V_{i}}$ is flat over $V_{i}$, and he concludes from this that there exists an integer $N_{1}\in\mathbb{N}$ so that $\forall r\geq 1, m\geq N_{1}:R^{r}\pi_{*}\mathcal{F}(m)=0$, where $\pi:\mathbb{P}^{n}_{S}\longrightarrow S$ is the projection, and for every $s\in S$ one has $H^{r}(\mathbb{P}^{n}_{S},\mathsf{F}_{s}(m))=0$.
Furthermore, one can show that there exists an $r_{1}\geq N_{1}$ so that for any $m\geq r_{i}$ the map \begin{equation*} (\pi_{*}\mathcal{F}(m))|_{V_{i}}\longrightarrow\pi_{i*}\mathcal{F}_{V_{i}}(m) \end{equation*} is an isomorphism.
So far, so good. Now the author proceeds to claim
As the higher cohomologies of all fibers, in particular the first cohomology, vanish, it follows by semi-continuity theory for the flat family $\mathcal{F}_{V_{i}}$ over $V_{i}$ that for any $s\in V_{i}$ the base change homomorphism $(\pi_{i}\mathcal{F}_{V_{i}}(m))|_{s}\longrightarrow H^{0}(\mathbb{P}^{n}_{s},\mathcal{F}_{s}(m))$ is an isomorphism for $m\geq r_{i}$.
First of all, I assume what the author really meant to write was semi-continuity theorem. Secondly, I don't understand how that theorem implies Nitsure's claim.
Thanks in advance.