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I have a question about a step in the proof of the Existence of Flattening Stratification I found in Nitsure's paper here: https://arxiv.org/abs/math/0504590 This question is closely related to Local freeness of $\pi_*F(r)$ from flatness of $F$ where the proof of Lemma 3.2 was discussed which we are probably going to apply here.

That's the converse implication of part (ii) (page 22):

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If $\phi : T \to S$ is a morphism such that $\mathcal{F}_T$ is flat, then the Hilbert polynomial is locally constant over $T$. Let $T_f$ (this notation is introduced in part (i) of the proof) be the open and closed subscheme of $T$ where the Hilbert polynomial is $f$. Clearly, the set map $|T_f | \to |S|$ factors via $|S_f |$.

Now the step I not understand:

But as the direct images $(\pi_T)_*\mathcal{F}_T(i)$ are locally free of rank $f(i)$ on $T_t$, it follows in fact that the schematic morphism $T_f \to S$ factors via $S_f$ ...

Question: Why flatness of $\mathcal{F}_T$ implies that all twisted direct images $(\pi_T)_*\mathcal{F}_T(i)$ are locally free?

Seemingly, Nitsure used implicitely here Lemma 3.2 + a certain exercise (p 15):

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Lemma + Exercise imply that $\mathcal{F}$ is flat if and only if there exist some integer $N$ such that for all $r ≥ N$ the direct image $\pi_*\mathcal{F}(r)$ is locally free.

But, it says only that for flat $\mathcal{F}_T$ the twisted direct images $(\pi_T)_*\mathcal{F}_T(r)$ are locally free IF the $r$ twist is big enough! On the other hand in the proof is clamed that $(\pi_T)_*\mathcal{F}_T(i)$ are locally free for ALL $i$'s. This aspect confuses me. Does anybody know why assuming $\mathcal{F}_T$ flat, implies that all twisted direct images $(\pi_T)_*\mathcal{F}_T(i)$ are locally free independend of $i$?

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    $\begingroup$ This is only true for $i \gg 0$, but that's also all you need. (The Hilbert polynomial is determined by its eventual values.) $\endgroup$ Jul 26 '20 at 19:13
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    $\begingroup$ ...you mean in the sense if $f$ Hilbert has degree $n$ then it is uniquely determined by $f(N), f(1+N),... f(i+N),...,f(n-1+N)$ for arbitrary $N$, ie can be choosen big, that's the point, right? And in the quoted text the statement on locally freeness of $(\pi_T)_*\mathcal{F}_T(i)$ is just formulated awkwardly? $\endgroup$ Jul 26 '20 at 19:25

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