Where in EGA is the result of Serre Vanishing located?

Question

We recall Serre's Vanishing Theorem which states the following. Let $X$ be a closed subscheme of $\Bbb{P}^n_A$ with $A$ a Noetherian ring. Then for any $\mathcal{F} \in \operatorname{Coh}(X)$, there is $m_0 \in \Bbb{N}$ for which $$H^i(X,\mathcal{F}(m)) = 0$$ for all $i > 0$ and $m \geq m_0$. Where in EGA is this theorem recorded? Googling "Serre Vanishing EGA" doesn't seem to come up with anything. Also, is there a version for non-Noetherian rings? Thanks.

Answer 1

This is Theorem 2.2.1 in EGA III. If $A$ is not noetherian, it also works when $X$ is defined by an ideal sheaf of finite type and $F$ is a quasi-coherent sheaf of finite presentation (then you can simply reduce to the noetherian case).