The Stacks project

Lemma 20.36.2. Suppose $X$, $f$, $(\mathcal{F}_ n)$ satisfy condition (1) of Lemma 20.36.1. Let $p \geq 0$ and set $H^ p = \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}_ n)$. Then $f^ cH^ p$ is the kernel of $H^ p \to H^ p(X, \mathcal{F}_ c)$ for all $c \geq 1$. Thus the limit topology on $H^ p$ is the $f$-adic topology.

Proof. Let $c \geq 1$. It is clear that $f^ c H^ p$ maps to zero in $H^ p(X, \mathcal{F}_ c)$. If $\xi = (\xi _ n) \in H^ p$ is small in the limit topology, then $\xi _ c = 0$, and hence $\xi _ n$ maps to zero in $H^ p(X, \mathcal{F}_ c)$ for $n \geq c$. Consider the inverse system of short exact sequences

\[ 0 \to \mathcal{F}_{n - c} \xrightarrow {f^ c} \mathcal{F}_ n \to \mathcal{F}_ c \to 0 \]

and the corresponding inverse system of long exact cohomology sequences

\[ H^{p - 1}(X, \mathcal{F}_ c) \to H^ p(X, \mathcal{F}_{n - c}) \to H^ p(X, \mathcal{F}_ n) \to H^ p(X, \mathcal{F}_ c) \]

Since the term $H^{p - 1}(X, \mathcal{F}_ c)$ is independent of $n$ we can choose a compatible sequence of elements $\xi '_ n \in H^ p(X, \mathcal{F}_{n - c})$ lifting $\xi _ n$. Setting $\xi ' = (\xi '_ n)$ we see that $\xi = f^ c \xi '$ as desired. $\square$

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