# Locally freeness of twisted direct images $\pi_* \mathcal{F}(i)$

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): 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): 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 $$i$$?

• This is only true for $i \gg 0$, but that's also all you need. (The Hilbert polynomial is determined by its eventual values.) Jul 26 '20 at 19:13
• ...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? Jul 26 '20 at 19:25