The Stacks project

20.39 Inverse systems and cohomology, III

This section continues the discussion in Section 20.36 using derived limits.

Lemma 20.39.1. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $A \to \Gamma (X, \mathcal{O}_ X)$ be a ring map and let $f \in A$. Let $E$ be an object of $D(\mathcal{O}_ X)$. Denote

\[ E_ n = E \otimes _{\mathcal{O}_ X} (\mathcal{O}_ X \xrightarrow {f^ n} \mathcal{O}_ X) \]

and set $E^\wedge = R\mathop{\mathrm{lim}}\nolimits E_ n$. For $p \in \mathbf{Z}$ is a canonical commutative diagram

\[ \xymatrix{ & 0 & 0 \\ 0 \ar[r] & \widehat{H^ p(X, E)} \ar[r] \ar[u] & \mathop{\mathrm{lim}}\nolimits H^ p(X, E_ n) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, E)) \ar[r] & 0 \\ 0 \ar[r] & H^0(H^ p(X, E)^\wedge ) \ar[r] \ar[u] & H^ p(X, E^\wedge ) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, E)) \ar[r] \ar@{=}[u] & 0 \\ & R^1\mathop{\mathrm{lim}}\nolimits H^ p(X, E)[f^ n] \ar[u] \ar[r]^\cong & R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, E_ n) \ar[u] \\ & 0 \ar[u] & 0 \ar[u] } \]

with exact rows and columns where $\widehat{H^ p(X, E)} = \mathop{\mathrm{lim}}\nolimits H^ p(X, E)/f^ n H^ p(X, E)$ is the usual $f$-adic completion, $H^ p(X, E)^\wedge $ is the derived $f$-adic completion, and $T_ f(H^{p + 1}(X, E))$ is the $f$-adic Tate module, see More on Algebra, Example 15.93.5. Finally, we have $H^ p(X, E^\wedge ) = H^ p(R\Gamma (X, E)^\wedge )$.

Proof. Observe that $R\Gamma (X, E^\wedge ) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, E_ n)$ by Lemma 20.37.2. On the other hand, we have

\[ R\Gamma (X, E_ n) = R\Gamma (X, E) \otimes _ A^\mathbf {L} (A \xrightarrow {f^ n} A) \]

(details omitted). We find that $R\Gamma (X, E^\wedge )$ is the derived $f$-adic completion $R\Gamma (X, E)^\wedge $. Whence the diagram by More on Algebra, Lemma 15.93.6. $\square$

Lemma 20.39.2. Let $\mathcal{A}$ be an abelian category. Let $f : M \to M$ be a morphism of $\mathcal{A}$. If $M[f^ n] = \mathop{\mathrm{Ker}}(f^ n : M \to M)$ stabilizes, then the inverse systems

\[ (M \xrightarrow {f^ n} M) \quad \text{and}\quad \mathop{\mathrm{Coker}}(f^ n : M \to M) \]

are pro-isomorphic in $D(\mathcal{A})$.

Proof. There is clearly a map from the first inverse system to the second. Suppose that $M[f^ c] = M[f^{c + 1}] = M[f^{c + 2}] = \ldots $. Then we can define an arrow of inverse systems in $D(\mathcal{A})$ in the other direction by the diagrams

\[ \xymatrix{ M/M[f^ c] \ar[r]_-{f^{n + c}} \ar[d]_{f^ c} & M \ar[d]^1 \\ M \ar[r]^{f^ n} & M } \]

Since the top horizontal arrow is injective the complex in the top row is quasi-isomorphic to $\mathop{\mathrm{Coker}}(f^{n + c} : M \to M)$. Some details omitted. $\square$

Example 20.39.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $A \to \Gamma (X, \mathcal{O}_ X)$ be a ring map and let $f \in A$. Let $\mathcal{F}$ be an $\mathcal{O}_ X$-module. Assume there is a $c$ such that $\mathcal{F}[f^ c] = \mathcal{F}[f^ n]$ for all $n \geq c$. We are going to apply Lemma 20.39.1 with $E = \mathcal{F}$. By Lemma 20.39.2 we see that the inverse system $(E_ n)$ is pro-isomorphic to the inverse system $(\mathcal{F}/f^ n\mathcal{F})$. We conclude that for $p \in \mathbf{Z}$ we obtain a commutative diagram

\[ \xymatrix{ & 0 & 0 \\ 0 \ar[r] & \widehat{H^ p(X, \mathcal{F})} \ar[r] \ar[u] & \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/f^ n\mathcal{F}) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, \mathcal{F})) \ar[r] & 0 \\ 0 \ar[r] & H^0(H^ p(X, \mathcal{F})^\wedge ) \ar[r] \ar[u] & H^ p(R\Gamma (X, \mathcal{F})^\wedge ) \ar[r] \ar[u] & T_ f(H^{p + 1}(X, \mathcal{F})) \ar[r] \ar@{=}[u] & 0 \\ & R^1\mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F})[f^ n] \ar[u] \ar[r]^\cong & R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/f^ n\mathcal{F}) \ar[u] \\ & 0 \ar[u] & 0 \ar[u] } \]

with exact rows and columns where $\widehat{H^ p(X, \mathcal{F})} = \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F})/f^ n H^ p(X, \mathcal{F})$ is the usual $f$-adic completion and $M^\wedge $ denotes derived $f$-adic completion for $M$ in $D(A)$.

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 0H3B. Beware of the difference between the letter 'O' and the digit '0'.