The Stacks project

Lemma 32.4.13. In Situation 32.4.5 if $S$ is affine, then for some $i_0 \in I$ the schemes $S_ i$ for $i \geq i_0$ are affine.

Proof. By Lemma 32.4.12 we may assume that $S_0$ is quasi-affine for some $0 \in I$. Set $R_0 = \Gamma (S_0, \mathcal{O}_{S_0})$. Then $S_0$ is a quasi-compact open of $T_0 = \mathop{\mathrm{Spec}}(R_0)$. Denote $j_0 : S_0 \to T_0$ the corresponding quasi-compact open immersion. For $i \geq 0$ set $\mathcal{A}_ i = f_{i0, *}\mathcal{O}_{S_ i}$. Since $f_{i0}$ is affine we see that $S_ i = \underline{\mathop{\mathrm{Spec}}}_{S_0}(\mathcal{A}_ i)$. Set $T_ i = \underline{\mathop{\mathrm{Spec}}}_{T_0}(j_{0, *}\mathcal{A}_ i)$. Then $T_ i \to T_0$ is affine, hence $T_ i$ is affine. Thus $T_ i$ is the spectrum of

\[ R_ i = \Gamma (T_0, j_{0, *}\mathcal{A}_ i) = \Gamma (S_0, \mathcal{A}_ i) = \Gamma (S_ i, \mathcal{O}_{S_ i}). \]

Write $S = \mathop{\mathrm{Spec}}(R)$. We have $R = \mathop{\mathrm{colim}}\nolimits _ i R_ i$ by Lemma 32.4.7. Hence also $S = \mathop{\mathrm{lim}}\nolimits _ i T_ i$. As formation of the relative spectrum commutes with base change, the inverse image of the open $S_0 \subset T_0$ in $T_ i$ is $S_ i$. Let $Z_0 = T_0 \setminus S_0$ and let $Z_ i \subset T_ i$ be the inverse image of $Z_0$. As $S_ i = T_ i \setminus Z_ i$, it suffices to show that $Z_ i$ is empty for some $i$. Assume $Z_ i$ is nonempty for all $i$ to get a contradiction. By Lemma 32.4.8 there exists a point $s$ of $S = \mathop{\mathrm{lim}}\nolimits T_ i$ which maps to a point of $Z_ i$ for every $i$. But $S = \mathop{\mathrm{lim}}\nolimits _ i S_ i$, and hence we arrive at a contradiction by Lemma 32.4.6. $\square$

Comments (6)

Comment #4051 by Laurent Moret-Bailly on

Line 3 of proof: should be (twice).

Comment #4052 by Laurent Moret-Bailly on

As an interesting application of Lemma 01Z6, we get the following generalization of Chevalley's theorem (EGA II, (6.7.1)). (I could not fint it in the text, but maybe I did not look in the right place.)

Theorem. Let be a morphism of schemes. Assume integral and surjective, noetherian, and affine. Then is affine.

Chevalley's theorem is the case where is finite. We reduce to this case by writing as the colimit of its finitely generated (hence finite) -algebras . By the lemma, is affine for some . Since it is finite over , we may apply EGA II.

Comment #4055 by on

OK, this theorem is Proposition 32.11.2 where we can even remove the assumption of Noetherianness.

Comment #6665 by Jonas Ehrhard on

Using that the relative spectrum commutes with base change, we see that where is the pull-back of the push-forward of . Is it clear that this is the same as ?

Comment #6882 by on

@#6665 yes because is the identity functor on all modules

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