The Stacks project

Lemma 52.7.3. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $X$ be a Noetherian scheme over $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that $H^ p(X, \mathcal{F})$ is a finite $A$-module for all $p$. Then there are short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathcal{F})^\wedge \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

of $A$-modules where $H^ p(X, \mathcal{F})^\wedge $ is the usual $I$-adic completion. If $f$ is proper, then the $R^1\mathop{\mathrm{lim}}\nolimits $ term is zero.

Proof. Consider the two spectral sequences of Lemma 52.6.20. The first degenerates by More on Algebra, Lemma 15.94.4. We obtain $H^ p(X, \mathcal{F})^\wedge $ in degree $p$. This is where we use the assumption that $H^ p(X, \mathcal{F})$ is a finite $A$-module. The second degenerates because

\[ \mathcal{F}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} \]

is a sheaf by Lemma 52.7.2. We obtain $H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F})$ in degree $p$. Since $R\Gamma (X, -)$ commutes with derived limits (Injectives, Lemma 19.13.6) we also get

\[ R\Gamma (X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F}/I^ n\mathcal{F}) \]

By More on Algebra, Remark 15.87.6 we obtain exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

of $A$-modules. Combining the above we get the first statement of the lemma. The vanishing of the $R^1\mathop{\mathrm{lim}}\nolimits $ term follows from Cohomology of Schemes, Lemma 30.20.4. $\square$


Comments (0)

There are also:

  • 2 comment(s) on Section 52.7: The theorem on formal functions

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Your email address will not be published.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 0A0M. Beware of the difference between the letter 'O' and the digit '0'.