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 $D(A)$.
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