Proposition 1.6.6, Etale Cohomology theory, Lei Fu

I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.

$$\mathrm{Proposition} 1.6.6$$ Let $$g:S'\rightarrow S$$ be a quasi-compact faithfully flat morphism, and let $$\mathcal F$$ be a quasi-coherent $$\mathcal O_S$$-module. Then $$g^* \mathcal F$$ is locally of finite type(resp. locally of finite presentation,resp. locally free of finite rank) if and only if $$\mathcal F$$ is so.

The author reduces the proof to the case where both S' and S are affine, but the detail is left to the reader.

How can I confirm this reduction? S' and S seem to be schemes but not necessarily Noetherian.

• I would cover $S$ with open affines (which are in particular quasi-compact). Their preimages are open and quasi-compact, since $g$ is quasi-compact. Then we cover these preimages with open affines which can be done by finitely many (quasi-compactness). Doesn't this work? – Paul K Jan 24 at 18:12
• @PaulK I was thinking about the same thing, but I have no idea about how to use faithfully flatness. It seems for me that there's no morphism like $\mathrm{Spec}A'\rightarrow \mathrm{Spec} A$ where $A'$ is faithfully flat over $A$. – Hardy Jan 24 at 18:50
• What exactly do you mean? – Paul K Jan 24 at 18:55
• @PaulK We may suppose $S=\mathrm{Spec}A$, $S'$ can be covered by finite affine schemes $\mathrm{Spec} A'_i$ and g is faithfully flat. Then we need to reduce the case $S'=\mathrm{Spec} A' \rightarrow \mathrm{Spec} A$ is faithfully flat, but $A'_i$ is not necessarily faithfully flat over A. So, where can I use the case where S' and S are affine? – Hardy Jan 24 at 19:09
• Now I got it. How about looking at the disjoint union of $A_i'$ which is an affine scheme by finiteness? Maybe that works? – Paul K Jan 24 at 20:30