Lemma 61.3.8. Let $A$ be a ring. Set $X = \mathop{\mathrm{Spec}}(A)$. The functor

from the category of $A$-algebras $B$ such that $A \to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful.

Lemma 61.3.8. Let $A$ be a ring. Set $X = \mathop{\mathrm{Spec}}(A)$. The functor

\[ B \longmapsto \mathop{\mathrm{Spec}}(B) \]

from the category of $A$-algebras $B$ such that $A \to B$ identifies local rings to the category of topological spaces over $X$ is fully faithful.

**Proof.**
This follows from Lemma 61.3.7 and the fact that if $A \to B$ identifies local rings, then the pullback of the structure sheaf of $\mathop{\mathrm{Spec}}(A)$ via $p : \mathop{\mathrm{Spec}}(B) \to \mathop{\mathrm{Spec}}(A)$ is equal to the structure sheaf of $\mathop{\mathrm{Spec}}(B)$.
$\square$

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