# The Stacks Project

## Tag 0BLD

Lemma 48.13.4. Let $A$ be a ring and $f \in A$. Let $X$ be a scheme over $A$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. Assume that $\mathcal{F}[f^n] = \mathop{\mathrm{Ker}}(f^n : \mathcal{F} \to \mathcal{F})$ stabilizes. Then $$R\Gamma(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, \mathcal{F})^\wedge$$ where the right hand side indicates the derived completion with respect to the ideal $(f) \subset A$. Let $H^p$ be the $p$th cohomology group of this complex. Then there are short exact sequences $$0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p \to \mathop{\mathrm{lim}}\nolimits H^p(X, \mathcal{F}/f^n\mathcal{F}) \to 0$$ and $$0 \to H^0(H^p(X, \mathcal{F})^\wedge) \to H^p \to T_f(H^{p + 1}(X, \mathcal{F})) \to 0$$ where $T_f(-)$ denote the $f$-adic Tate module as in More on Algebra, Example 15.81.4.

Proof. We start with the canonical identifications \begin{align*} R\Gamma(X, \mathcal{F})^\wedge & = R\mathop{\mathrm{lim}}\nolimits R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} (A \xrightarrow{f^n} A) \\ & = R\mathop{\mathrm{lim}}\nolimits R\Gamma(X, \mathcal{F} \xrightarrow{f^n} \mathcal{F}) \\ & = R\Gamma(X, R\mathop{\mathrm{lim}}\nolimits (\mathcal{F} \xrightarrow{f^n} \mathcal{F})) \end{align*} The first equality holds by More on Algebra, Lemma 15.80.17. The second by the projection formula, see Cohomology, Lemma 20.45.3. The third by Cohomology, Lemma 20.32.2. Note that by Derived Categories of Schemes, Lemma 35.3.2 we have $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^n \mathcal{F}$. Thus to finish the proof of the first statement of the lemma it suffices to show that the pro-objects $(f^n : \mathcal{F} \to \mathcal{F})$ and $(\mathcal{F}/f^n \mathcal{F})$ are isomorphic. There is clearly a map from the first system to the second. Suppose that $\mathcal{F}[f^c] = \mathcal{F}[f^{c + 1}] = \mathcal{F}[f^{c + 2}] = \ldots$. Then we can define an arrow of systems in $D(\mathcal{O}_X)$ in the other direction by the diagrams $$\xymatrix{ \mathcal{F}/\mathcal{F}[f^c] \ar[r]_-{f^{n + c}} \ar[d]_{f^c} & \mathcal{F} \ar[d]^1 \\ \mathcal{F} \ar[r]^{f^n} & \mathcal{F} }$$ Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $\mathcal{F}/f^{n + c}\mathcal{F}$. Some details omitted.

Since $R\Gamma(X, -)$ commutes with derived limits (Injectives, Lemma 19.13.6) we see that $$R\Gamma(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/f^n\mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits R\Gamma(X, \mathcal{F}/f^n\mathcal{F})$$ (for first equality see first paragraph of proof). By More on Algebra, Remark 15.75.9 we obtain exact sequences $$0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0$$ of $A$-modules. The second set of short exact sequences follow immediately from the discussion in More on Algebra, Example 15.81.4. $\square$

The code snippet corresponding to this tag is a part of the file local-cohomology.tex and is located in lines 3173–3202 (see updates for more information).

\begin{lemma}
\label{lemma-formal-functions-principal}
\begin{reference}
\cite[Lemma 1.6]{Bhatt-local}
\end{reference}
Let $A$ be a ring and $f \in A$. Let $X$ be a scheme over $A$.
Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module.
Assume that $\mathcal{F}[f^n] = \Ker(f^n : \mathcal{F} \to \mathcal{F})$
stabilizes. Then
$$R\Gamma(X, \lim \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, \mathcal{F})^\wedge$$
where the right hand side indicates the derived completion
with respect to the ideal $(f) \subset A$. Let $H^p$ be the
$p$th cohomology group of this complex. Then there are short
exact sequences
$$0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p \to \lim H^p(X, \mathcal{F}/f^n\mathcal{F}) \to 0$$
and
$$0 \to H^0(H^p(X, \mathcal{F})^\wedge) \to H^p \to T_f(H^{p + 1}(X, \mathcal{F})) \to 0$$
where $T_f(-)$ denote the $f$-adic Tate module as in
More on Algebra, Example
\ref{more-algebra-example-spectral-sequence-principal}.
\end{lemma}

\begin{proof}
We start with the canonical identifications
\begin{align*}
R\Gamma(X, \mathcal{F})^\wedge
& =
R\lim R\Gamma(X, \mathcal{F}) \otimes_A^\mathbf{L} (A \xrightarrow{f^n} A) \\
& =
R\lim R\Gamma(X, \mathcal{F} \xrightarrow{f^n} \mathcal{F}) \\
& =
R\Gamma(X, R\lim (\mathcal{F} \xrightarrow{f^n} \mathcal{F}))
\end{align*}
The first equality holds by
More on Algebra, Lemma \ref{more-algebra-lemma-derived-completion-koszul}.
The second by the projection formula, see
Cohomology, Lemma \ref{cohomology-lemma-projection-formula-perfect}.
The third by Cohomology, Lemma
\ref{cohomology-lemma-Rf-commutes-with-Rlim}.
Note that by
Derived Categories of Schemes, Lemma \ref{perfect-lemma-Rlim-quasi-coherent}
we have
$\lim \mathcal{F}/f^n\mathcal{F} = R\lim \mathcal{F}/f^n \mathcal{F}$.
Thus to finish the proof of the first statement of the lemma it suffices to
show that the pro-objects $(f^n : \mathcal{F} \to \mathcal{F})$
and $(\mathcal{F}/f^n \mathcal{F})$ are isomorphic. There is clearly
a map from the first system to the second. Suppose that
$\mathcal{F}[f^c] = \mathcal{F}[f^{c + 1}] = \mathcal{F}[f^{c + 2}] = \ldots$.
Then we can define an arrow of systems in $D(\mathcal{O}_X)$
in the other direction by the diagrams
$$\xymatrix{ \mathcal{F}/\mathcal{F}[f^c] \ar[r]_-{f^{n + c}} \ar[d]_{f^c} & \mathcal{F} \ar[d]^1 \\ \mathcal{F} \ar[r]^{f^n} & \mathcal{F} }$$
Since the top horizontal arrow is injective the complex
in the top row is quasi-isomorphic to $\mathcal{F}/f^{n + c}\mathcal{F}$.
Some details omitted.

\medskip\noindent
Since $R\Gamma(X, -)$ commutes with derived limits
(Injectives, Lemma \ref{injectives-lemma-RF-commutes-with-Rlim})
we see that
$$R\Gamma(X, \lim \mathcal{F}/f^n\mathcal{F}) = R\Gamma(X, R\lim \mathcal{F}/f^n\mathcal{F}) = R\lim R\Gamma(X, \mathcal{F}/f^n\mathcal{F})$$
(for first equality see first paragraph of proof).
By More on Algebra, Remark \ref{more-algebra-remark-compare-derived-limit}
we obtain exact sequences
$$0 \to R^1\lim H^{p - 1}(X, \mathcal{F}/f^n\mathcal{F}) \to H^p(X, \lim \mathcal{F}/I^n\mathcal{F}) \to \lim H^p(X, \mathcal{F}/I^n\mathcal{F}) \to 0$$
of $A$-modules. The second set of short exact sequences follow immediately
from the discussion in More on Algebra, Example
\ref{more-algebra-example-spectral-sequence-principal}.
\end{proof}

## References

[Bhatt-local, Lemma 1.6]

## Comments (0)

There are no comments yet for this tag.

## Add a comment on tag 0BLD

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 lower-right corner).

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 box. So in case this where tag 0321 you just have to write 0321. Beware of the difference between the letter 'O' and the digit 0.

This captcha seems more appropriate than the usual illegible gibberish, right?