Lemma 29.11.5. Let $S$ be a scheme. There is an anti-equivalence of categories

which associates to $f : X \to S$ the sheaf $f_*\mathcal{O}_ X$. Moreover, this equivalence is compatible with arbitrary base change.

Lemma 29.11.5. Let $S$ be a scheme. There is an anti-equivalence of categories

\[ \begin{matrix} \text{Schemes affine}
\\ \text{over }S
\end{matrix} \longleftrightarrow \begin{matrix} \text{quasi-coherent sheaves}
\\ \text{of }\mathcal{O}_ S\text{-algebras}
\end{matrix} \]

which associates to $f : X \to S$ the sheaf $f_*\mathcal{O}_ X$. Moreover, this equivalence is compatible with arbitrary base change.

**Proof.**
The functor from right to left is given by $\underline{\mathop{\mathrm{Spec}}}_ S$. The two functors are mutually inverse by Lemma 29.11.3 and Constructions, Lemma 27.4.6 part (3). The final statement is Constructions, Lemma 27.4.6 part (2).
$\square$

Your email address will not be published.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.

## Comments (2)

Comment #8440 by ElĂas Guisado on

Comment #9063 by Stacks project on

There are also: