as a consequence of [@Eis Corollary 2.9]. At the same time for all ${\mathcal{O}}_T$-modules $\mathcal N$ and all closed points $(x,s)\in T$ there are isomorphisms $$\label{trick}
\mathcal N_{(x,s)} \simeq (\mathcal N_s)_x \otimes_{{\mathcal{O}}_{(T_s)_x}} {\mathcal{O}}_{T,(x,s)}$$ and this is a flat base change. This implies that the germs $\varphi_{(x,s)}$ are surjective (resp. injective) for all closed points $(x,s)\in T$. By appling [@Eis Corollary 2.9] again the thesis follows.

\[lmFlat\] Let $M$ be a monad over the product scheme $T=X\times S$ (see eq. ). The cohomology ${\mathcal{E}}(M)$ is flat over $S$ if and only if the restricted morphisms $a_s$ are injective for all closed points $s\in S$.

By Lemma 2.1.4 in [@HL-book], what we need to prove is that if $a_s$ is injective for all [*closed*]{} then it is so for all $s$. Note that $\mathcal B$ is flat over $S$ as it is locally free and $S$ is a factor of $T=X\times S$. Since flatness over $S$ is local over $S$, we can assume $S={\operatorname{Spec}}R$. If $s$ is not closed, it is the generic point of a closed irreducible subscheme $Z\subseteq S$. We know that $a_{\vert Z}$ is injective for all closed points $s\in Z$. By Lemma \[lmInjSurj\], $a_{\vert Z}$ is injective. Since 