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> Let $S$ be a connected scheme, let $\pi : \mathbb{P}\_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a flat and locally finitely presented $\mathcal{O}\_{\mathbb{P}\_{S}^{r}}$-module. Is $\pi\_{\ast}(\mathcal{E}(n))$ (nonzero and) flat and locally finitely presented for $n \gg 0$?
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Remarks: If $S$ is a Noetherian scheme, the answer is "yes" using usual cohomology and base change theorems. I included "connected" because I am assuming we could otherwise construct an example with $S$ being an infinite disjoint union of fields. The case when $S$ is quasi-compact reduces to the case when $S$ is affine. Say $S = \operatorname{Spec} A$; we may write $A$ as a filtered colimit $A = \varinjlim\_{\lambda \in \Lambda} A\_{\lambda}$ where each $A\_{\lambda}$ is a finite type $\mathbb{Z}$-subalgebra of $A$. Set $S\_{\lambda} := \operatorname{Spec} A\_{\lambda}$. Then we can descend $\mathcal{E}$ to vector bundles $\mathcal{E}\_{\lambda}$ on $\mathbb{P}\_{S\_{\lambda}}^{r}$ for large enough $\lambda$. Let's fix some $\lambda$. By Serre vanishing [1, III, Theorem 5.2], we may replace $\mathcal{E}\_{\lambda}$ by $\mathcal{E}\_{\lambda}(n)$ to assume that $\mathrm{H}^{i}(\mathbb{P}\_{A\_{\lambda}}^{r},\mathcal{E}\_{\lambda}) = 0$ for $i > 0$. By [2, Section 4, second Theorem] there is a finite complex (the "Grothendieck complex") $$ K^{\bullet} = \{0 \to K^{0} \to \dotsb \to K^{n} \to 0\} $$ of finitely generated projective $A\_{\lambda}$-modules and, for each $i \ge 0$, a functorial isomorphism $$ \mathrm{H}^{i}(\mathbb{P}\_{B}^{r} , \mathcal{E}\_{\lambda}|\_{\mathbb{P}\_{B}^{r}}) \simeq \mathsf{h}^{i}(K^{\bullet} \otimes\_{A\_{\lambda}} B) $$ for every $A\_{\lambda}$-algebra $B$. Now $\mathsf{h}^{i}(K^{\bullet}) = 0$ for $i > 0$ which means $K^{\bullet}$ is a direct sum of complexes of the form $\{\mathrm{id}\_{P} : P \to P\}$ and a complex consisting of a single finitely generated projective $A\_{\lambda}$-module at degree $0$. Thus we may assume that $K^{i} = 0$ if $i \ne 0$. Then $K^{0} \simeq \Gamma(\mathbb{P}\_{A\_{\lambda}}^{r} , \mathcal{E}\_{\lambda})$, and the above gives $\Gamma(\mathbb{P}\_{A}^{r} , \mathcal{E}) \simeq \Gamma(\mathbb{P}\_{A\_{\lambda}}^{r} , \mathcal{E}\_{\lambda}) \otimes\_{A\_{\lambda}} A$. (I guess this argument should work if we replace $\mathbb{P}\_{S}^{r}$ by a flat projective finitely presented morphism $X \to S$ that descends to some $S\_{\lambda}$.)


I am guessing that the difficulty is "flatness" but I also don't know whether the "locally finitely presented" part is true, thus I will include the following subquestion:



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> Let $S$ be an affine scheme, let $\pi : \mathbb{P}\_{S}^{r} \to S$ be projective $r$-space over $S$, and let $\mathcal{E}$ be a locally finitely presented $\mathcal{O}\_{\mathbb{P}\_{S}^{r}}$-module. Is $\pi\_{\ast}\mathcal{E}$ necessarily finitely presented?
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References:


[1] Hartshorne, *Algebraic Geometry*


[2] Mumford, *Abelian Varieties*, Tata Institute of Fundamental Research Studies in Mathematics (1970)


Keywords: Noetherian approximation, cohomology and base change, higher direct images, not Noetherian



Non-noetherian cohomology and base change
The answer to the question about flatness is no and here is a counterexample. Let $S\_n = \mathbf{A}^1$ with coordinate $t$ for $n\geq 1$ (over some base field). Let 
$$
\mathcal{A}\_n = \mathcal{O}\_{\mathbf{P}^1}(-n)\oplus \mathcal{O}\_{\mathbf{P}^1}(n).
$$
We have a non-split extension 
$$ 0\to \mathcal{O}\_{\mathbf{P}^1}(-n)
\to \mathcal{O}\_{\mathbf{P}^1}(-n+1)\oplus \mathcal{O}\_{\mathbf{P}^1}(n-1) 
\to \mathcal{O}\_{\mathbf{P}^1}(n) \to 0 $$
where the middle term is $\mathcal{A}\_{n-1}$. (For $n=1$, this is the Euler sequence twisted by $1$.)
Degenerating this extension to the split one as $t\to 0$ (i.e. pulling it back to $\mathbf{P}^1\times S\_n$ and pushing out along multiplication by $t$ on the left term) yields a vector bundle of rank two $\mathcal{E}\_n$ on $\mathbf{P}^1\_{S\_n}$ whose fiber as $t=0$ (resp. $t\neq 0$) is $\mathcal{A}\_n$ (resp. $\mathcal{A}\_{n-1}$).


(Taking the associated projective bundles, we obtain the well-known degeneration of the Hirzebruch surface $F\_{2n-2}$ to $F\_{2n}$.)


Now let $S$ be an infinite chain of lines obtained by taking the disjoint union of $S\_n=\mathbf{A}^1$ for all $n\geq 1$ and identifying $0\in S\_n$ with $1\in S\_{n-1}$. The above construction yields a vector bundle of rank two on $\mathbf{P}^1\_S$ such that $\pi\_\*\mathcal{E}(m)$ does not have constant rank for every $m$. 


The bottom line is that the statement you want is false for the stack of vector bundles of rank two and degree zero on $\mathbf{P}^1$; above, we exhibited a suitable scheme over that stack.


Note that $S$ is locally noetherian, so all the push-forwards are of course locally finitely presented.


