The Stacks project

Lemma 36.30.6. Let $R$ be a ring. Let $X$ be a scheme and let $f : X \to \mathop{\mathrm{Spec}}(R)$ be proper, flat, and of finite presentation. Let $(M_ n)$ be an inverse system of $R$-modules with surjective transition maps. Then the canonical map

\[ \mathcal{O}_ X \otimes _ R (\mathop{\mathrm{lim}}\nolimits M_ n) \longrightarrow \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n \]

induces an isomorphism from the source to $DQ_ X$ applied to the target.

Proof. The statement means that for any object $E$ of $D_\mathit{QCoh}(\mathcal{O}_ X)$ the induced map

\[ \mathop{\mathrm{Hom}}\nolimits (E, \mathcal{O}_ X \otimes _ R (\mathop{\mathrm{lim}}\nolimits M_ n)) \longrightarrow \mathop{\mathrm{Hom}}\nolimits (E, \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n) \]

is an isomorphism. Since $D_\mathit{QCoh}(\mathcal{O}_ X)$ has a perfect generator (Theorem 36.15.3) it suffices to check this for perfect $E$. By Lemma 36.3.2 we have $\mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n = R\mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n$. The exact functor $R\mathop{\mathrm{Hom}}\nolimits _ X(E, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(R)$ of Cohomology, Section 20.44 commutes with products and hence with derived limits, whence

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathop{\mathrm{lim}}\nolimits \mathcal{O}_ X \otimes _ R M_ n) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathcal{O}_ X \otimes _ R M_ n) \]

Let $E^\vee $ be the dual perfect complex, see Cohomology, Lemma 20.50.5. We have

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathcal{O}_ X \otimes _ R M_ n) = R\Gamma (X, E^\vee \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*M_ n) = R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} M_ n \]

by Lemma 36.22.1. From Lemma 36.30.4 we see $R\Gamma (X, E^\vee )$ is a perfect complex of $R$-modules. In particular it is a pseudo-coherent complex and by More on Algebra, Lemma 15.102.3 we obtain

\[ R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} M_ n = R\Gamma (X, E^\vee ) \otimes _ R^\mathbf {L} \mathop{\mathrm{lim}}\nolimits M_ n \]

as desired. $\square$

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Your email address will not be published.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 0D2Q. Beware of the difference between the letter 'O' and the digit '0'.