The Stacks project

Remark 84.16.6 (Ringed variant over an object). Let $\mathcal{C}$ be a site. Let $\mathcal{O}_\mathcal {C}$ be a sheaf of rings. Let $X \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and denote $\mathcal{O}_ X = \mathcal{O}_\mathcal {C}|_{\mathcal{C}/X}$. Then we can combine the constructions given in Remarks 84.16.4 and 84.16.5 to get

  1. a ringed site $((\mathcal{C}/K)_{total}, \mathcal{O})$ for a simplicial $K$ object of $\text{SR}(\mathcal{C}, X)$,

  2. a morphism of ringed topoi $a : (\mathop{\mathit{Sh}}\nolimits ((\mathcal{C}/K)_{total}), \mathcal{O}) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$,

  3. morphisms of ringed topoi $a_ n : (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/K_ n), \mathcal{O}_ n) \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}/X), \mathcal{O}_ X)$,

  4. a functor $a_! : \textit{Mod}(\mathcal{O}) \to \textit{Mod}(\mathcal{O}_ X)$ left adjoint to $a^*$.

Of course, all the results mentioned in Remark 84.16.5 hold in this setting as well.

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Your email address will not be published.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 0D9X. Beware of the difference between the letter 'O' and the digit '0'.



View Remark 84.16.6 as pdf