The Stacks project

Generalization of [Lemma 6.5.9 (2), BS]. Compare with [Theorem 6.5, HL-P] in the setting of quasi-coherent modules and morphisms of (derived) algebraic stacks.

Lemma 52.6.19. Let $f : (\mathop{\mathit{Sh}}\nolimits (\mathcal{D}), \mathcal{O}') \to (\mathop{\mathit{Sh}}\nolimits (\mathcal{C}), \mathcal{O})$ be a morphism of ringed topoi. Let $\mathcal{I} \subset \mathcal{O}$ be a finite type sheaf of ideals. Let $\mathcal{I}' \subset \mathcal{O}'$ be the ideal generated by $f^\sharp (f^{-1}\mathcal{I})$. Then $Rf_*$ commutes with derived completion, i.e., $Rf_*(K^\wedge ) = (Rf_*K)^\wedge $.

Proof. By Proposition 52.6.12 the derived completion functors exist. By Lemma 52.6.7 the object $Rf_*(K^\wedge )$ is derived complete, and hence we obtain a canonical map $(Rf_*K)^\wedge \to Rf_*(K^\wedge )$ by the universal property of derived completion. We may check this map is an isomorphism locally on $\mathcal{C}$. Thus, since derived completion commutes with localization (Remark 52.6.14) we may assume that $\mathcal{I}$ is generated by global sections $f_1, \ldots , f_ r$. Then $\mathcal{I}'$ is generated by $g_ i = f^\sharp (f_ i)$. By Lemma 52.6.9 we have to prove that

\[ R\mathop{\mathrm{lim}}\nolimits \left( Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) \right) = Rf_*\left( R\mathop{\mathrm{lim}}\nolimits K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right) \]

Because $Rf_*$ commutes with $R\mathop{\mathrm{lim}}\nolimits $ (Cohomology on Sites, Lemma 21.22.3) it suffices to prove that

\[ Rf_*K \otimes ^\mathbf {L}_\mathcal {O} K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = Rf_*\left( K \otimes ^\mathbf {L}_{\mathcal{O}'} K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n) \right) \]

This follows from the projection formula (Cohomology on Sites, Lemma 21.48.1) and the fact that $Lf^*K(\mathcal{O}, f_1^ n, \ldots , f_ r^ n) = K(\mathcal{O}', g_1^ n, \ldots , g_ r^ n)$. $\square$


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Your email address will not be published.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 0A0G. Beware of the difference between the letter 'O' and the digit '0'.