The Stacks project

[II, Corollary 1.3.2, EGA]

Lemma 29.11.3. Let $f : X \to S$ be a morphism of schemes. The following are equivalent

  1. The morphism $f$ is affine.

  2. There exists an affine open covering $S = \bigcup W_ j$ such that each $f^{-1}(W_ j)$ is affine.

  3. There exists a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras $\mathcal{A}$ and an isomorphism $X \cong \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A})$ of schemes over $S$. See Constructions, Section 27.4 for notation.

Moreover, in this case $X = \underline{\mathop{\mathrm{Spec}}}_ S(f_*\mathcal{O}_ X)$.

Proof. It is obvious that (1) implies (2).

Assume $S = \bigcup _{j \in J} W_ j$ is an affine open covering such that each $f^{-1}(W_ j)$ is affine. By Schemes, Lemma 26.19.2 we see that $f$ is quasi-compact. By Schemes, Lemma 26.21.6 we see the morphism $f$ is quasi-separated. Hence by Schemes, Lemma 26.24.1 the sheaf $\mathcal{A} = f_*\mathcal{O}_ X$ is a quasi-coherent sheaf of $\mathcal{O}_ S$-algebras. Thus we have the scheme $g : Y = \underline{\mathop{\mathrm{Spec}}}_ S(\mathcal{A}) \to S$ over $S$. The identity map $\text{id} : \mathcal{A} = f_*\mathcal{O}_ X \to f_*\mathcal{O}_ X$ provides, via the definition of the relative spectrum, a morphism $can : X \to Y$ over $S$, see Constructions, Lemma 27.4.7. By assumption and the lemma just cited the restriction $can|_{f^{-1}(W_ j)} : f^{-1}(W_ j) \to g^{-1}(W_ j)$ is an isomorphism. Thus $can$ is an isomorphism. We have shown that (2) implies (3).

Assume (3). By Constructions, Lemma 27.4.6 we see that the inverse image of every affine open is affine, and hence the morphism is affine by definition. $\square$

Comments (4)

Comment #5910 by Simon Schirren on

in the proof third line: shouldn't be a quasicoherent sheaf of -algebras?

There are also:

  • 2 comment(s) on Section 29.11: Affine morphisms

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 01S8. Beware of the difference between the letter 'O' and the digit '0'.