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).
$\square$

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