The Stacks project

Remark 52.6.13 (Comparison with completion). Let $(\mathcal{C}, \mathcal{O})$ be a ringed site. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $K \mapsto K^\wedge $ be the derived completion functor of Proposition 52.6.12. For any $n \geq 1$ the object $K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$ is derived complete as it is annihilated by powers of local sections of $\mathcal{I}$. Hence there is a canonical factorization

\[ K \to K^\wedge \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n \]

of the canonical map $K \to K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n$. These maps are compatible for varying $n$ and we obtain a comparison map

\[ K^\wedge \longrightarrow R\mathop{\mathrm{lim}}\nolimits \left(K \otimes _\mathcal {O}^\mathbf {L} \mathcal{O}/\mathcal{I}^ n\right) \]

The right hand side is more recognizable as a kind of completion. In general this comparison map is not an isomorphism.


Comments (0)

There are also:

  • 2 comment(s) on Section 52.6: Derived completion on a ringed site

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Your email address will not be published.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