I would like to ask if the following is true or not: Let $S$ a scheme and $X$ a $S$-scheme which is proper and flat. Let $\mathcal{F}$ a sheaf of $\mathcal{O}\_{X}$-algebras over $X$. Let's suppose that for every geometric point $p$, the pullback of $\mathcal{F}$ to $X\_{p}$ is a finitely generated sheaf of $\mathcal{O}\_{X\_{p}}$-algebras. Is it true then that $\mathcal{F}$ is a finitely generated sheaf of $\mathcal{O}\_{X}$-algebras?


Thank you for your time.



Finitely generated sheaf of algebras over geometric points
This is false for $X=S=\operatorname{Spec}\mathbb{Z}$. For example, take the sheaf of $\mathcal{O}\_X$-algebras corresponding to the $\mathbb{Z}$-algebra
$$
A=\mathbb{Z}+ \mathbb{Q}\varepsilon\subseteq \mathbb{Q}[\varepsilon\,|\,\varepsilon^2=0].
$$
It is easy to see that $A\otimes\_{\mathbb{Z}}K$ is finitely generated as a $K$-module for any ($\mathbb{Z}$-)field $K$ (it is enough to consider $K=\mathbb{Q}$ and $K=\mathbb{F}\_p$). However, $A$ is not finitely generated as a $\mathbb{Z}$-algebra because the additive group $(\mathbb{Q},+)$ cannot be generated by finitely many elements.


One can construct similar examples for $X=S=F[x]$, with $F$ being any field, by replacing $\mathbb{Z}$ with $F[x]$ and $\mathbb{Q}$ by $F(x)$.


