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

and the corresponding inverse system of long exact cohomology sequences

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. $\square$

## 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Your email address will not be published.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.

## Comments (0)