Lemma 60.3.8. 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 60.3.8. 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 60.3.7 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 $\mathop{\mathrm{Spec}}(B)$.
$\square$

Your email address will not be published.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 (0)