The Stacks project

46.2 Conventions

In this chapter we fix $\tau \in \{ Zar, {\acute{e}tale}, smooth, syntomic, fppf\} $ and we fix a big $\tau $-site $\mathit{Sch}_\tau $ as in Topologies, Section 34.2. All schemes will be objects of $\mathit{Sch}_\tau $. In particular, given a scheme $S$ we obtain sites $(\textit{Aff}/S)_\tau \subset (\mathit{Sch}/S)_\tau $. The structure sheaf $\mathcal{O}$ on these sites is defined by the rule $\mathcal{O}(T) = \Gamma (T, \mathcal{O}_ T)$.

All rings $A$ will be such that $\mathop{\mathrm{Spec}}(A)$ is (isomorphic to) an object of $\mathit{Sch}_\tau $. Given a ring $A$ we denote $\textit{Alg}_ A$ the category of $A$-algebras whose objects are the $A$-algebras $B$ of the form $B = \Gamma (U, \mathcal{O}_ U)$ where $S$ is an affine object of $\mathit{Sch}_\tau $. Thus given an affine scheme $S = \mathop{\mathrm{Spec}}(A)$ the functor

\[ (\textit{Aff}/S)_\tau \longrightarrow \textit{Alg}_ A, \quad U \longmapsto \mathcal{O}(U) \]

is an equivalence.

Comments (0)

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Your email address will not be published.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 06Z3. Beware of the difference between the letter 'O' and the digit '0'.