## Tag `02L5`

Chapter 34: Descent > Section 34.20: Properties of morphisms local in the fpqc topology on the target

Lemma 34.20.18. The property $\mathcal{P}(f) =$''$f$ is affine'' is fpqc local on the base.

Proof.A base change of an affine morphism is affine, see Morphisms, Lemma 28.11.8. Being affine is Zariski local on the base, see Morphisms, Lemma 28.11.3. Finally, let $g : S' \to S$ be a flat surjective morphism of affine schemes, and let $f : X \to S$ be a morphism. Assume that the base change $f' : X' \to S'$ is affine. In other words, $X'$ is affine, say $X' = \mathop{\rm Spec}(A')$. Also write $S = \mathop{\rm Spec}(R)$ and $S' = \mathop{\rm Spec}(R')$. We have to show that $X$ is affine.By Lemmas 34.20.1 and 34.20.6 we see that $X \to S$ is separated and quasi-compact. Thus $f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras, see Schemes, Lemma 25.24.1. Hence $f_*\mathcal{O}_X = \widetilde{A}$ for some $R$-algebra $A$. In fact $A = \Gamma(X, \mathcal{O}_X)$ of course. Also, by flat base change (see for example Cohomology of Schemes, Lemma 29.5.2) we have $g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}$. In other words, we have $A' = R' \otimes_R A$. Consider the canonical morphism $$ X \longrightarrow \mathop{\rm Spec}(A) $$ over $S$ from Schemes, Lemma 25.6.4. By the above the base change of this morphism to $S'$ is an isomorphism. Hence it is an isomorphism by Lemma 34.20.17. Therefore Lemma 34.19.4 applies and we win. $\square$

The code snippet corresponding to this tag is a part of the file `descent.tex` and is located in lines 4940–4944 (see updates for more information).

```
\begin{lemma}
\label{lemma-descending-property-affine}
The property $\mathcal{P}(f) =$``$f$ is affine''
is fpqc local on the base.
\end{lemma}
\begin{proof}
A base change of an affine morphism is affine, see
Morphisms, Lemma \ref{morphisms-lemma-base-change-affine}.
Being affine is Zariski local on the base, see
Morphisms, Lemma \ref{morphisms-lemma-characterize-affine}.
Finally, let
$g : S' \to S$ be a flat surjective morphism of affine schemes,
and let $f : X \to S$ be a morphism. Assume that the base change
$f' : X' \to S'$ is affine. In other words, $X'$ is affine, say
$X' = \Spec(A')$. Also write $S = \Spec(R)$
and $S' = \Spec(R')$. We have to show that $X$ is affine.
\medskip\noindent
By Lemmas \ref{lemma-descending-property-quasi-compact}
and \ref{lemma-descending-property-separated} we see that
$X \to S$ is separated and quasi-compact. Thus
$f_*\mathcal{O}_X$ is a quasi-coherent sheaf of $\mathcal{O}_S$-algebras,
see Schemes, Lemma \ref{schemes-lemma-push-forward-quasi-coherent}.
Hence $f_*\mathcal{O}_X = \widetilde{A}$ for some $R$-algebra $A$.
In fact $A = \Gamma(X, \mathcal{O}_X)$ of course.
Also, by flat base change
(see for example
Cohomology of Schemes, Lemma \ref{coherent-lemma-flat-base-change-cohomology})
we have $g^*f_*\mathcal{O}_X = f'_*\mathcal{O}_{X'}$.
In other words, we have $A' = R' \otimes_R A$.
Consider the canonical morphism
$$
X \longrightarrow \Spec(A)
$$
over $S$ from Schemes, Lemma \ref{schemes-lemma-morphism-into-affine}.
By the above the base change of this morphism to $S'$ is an isomorphism.
Hence it is an isomorphism by
Lemma \ref{lemma-descending-property-isomorphism}.
Therefore Lemma \ref{lemma-descending-properties-morphisms} applies and we win.
\end{proof}
```

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