# Local freeness of $\pi_*F(r)$ from flatness of $F$

In 'Fundamental Algebraic Geometry' by Fantechi there is a lemma in section 5.3.2, page 119:

LEMMA 5.5 Let $$S$$ be a noetherian scheme and let $$F$$ be a coherent sheaf on $$\mathbb{P}^n_S$$. Suppose there exist some integer $$N$$ such that for all $$r \ge N$$ the direct image $$\pi_*F(r)= \pi_*(F \otimes O_{\mathbb{P}^n_S}(r))$$ is locally free. ($$\pi$$ is the structure morphism $$\mathbb{P}^n_S \to S$$). Claim: Then $$F$$ is flat over $$S$$.

Now the author leaves as exercise to show that the converse(!) of the above lemma holds as well: if $$F$$ is flat over $$S$$ then $$\pi_*F(r)$$ is locally free for all sufficiently large $$r$$.

Does anybody know how the proof of the converse works or where I can find it?

About the choice of suitable $$N$$ I conjecture that Serre's vanishing theorem may play an important role in the proof:

Let $$A$$ a ring and $$F$$ coherent on $$\mathbb{P}^n_A$$. Then for all $$i \ge 1$$ there exist $$m_0$$ with $$H^i(\mathbb{P}^n_A, F(m))$$ for all $$m \ge m_0$$.

But that's just my suspucion, not more since I otherwise not know how a candidate for $$N$$ can be found.

• See for example Hartshorne, Theorem III.9.9. The idea is to use a Čech resolution and Serre vanishing. Jun 30, 2020 at 21:43
• @R.vanDobbendeBruyn: You mean that one which starts with "Let $T$ be an integral noetherian scheme. Let $X \subset \mathbb{P}^n_T$ be a closed subscheme... And the claim is that $X$ is flat over $T$ iff Hilbert polynomial $P_t$ is independent of $t$. Here is required that $T$ is integral, especially reduced, the book seems to work without it but thats subtile. Jun 30, 2020 at 22:16
• This can be deduced from semicontinuity theorem . For sufficiently large $N$, higher direct images of $F(N)$ vanish. Then the direct image is locally free. Jun 30, 2020 at 23:03
• You're right, the reference doesn't cover the precise statement, but as @Mohan points out Thm III.12.11 does (together with Thm III.8.8). Jul 1, 2020 at 0:09
• I also think the (very short) proof of Thm III.9.9 (i)$\Rightarrow$(ii) applies verbatim in this context, because that direction uses neither that $\mathscr F$ is the structure sheaf nor that $T$ is integral. In fact the first is not even stated as an assumption, and the second is only used for (iii)$\Rightarrow$(ii). Jul 1, 2020 at 0:37