Lemma 20.36.2. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(\mathcal{F}_ n^\bullet )$ be an inverse system of complexes of $\mathcal{O}_ X$-modules. Let $m \in \mathbf{Z}$. Assume there exist a set $\mathcal{B}$ of open subsets of $X$ and an integer $n_0$ such that

1. every open in $X$ has a covering whose members are elements of $\mathcal{B}$,

2. for every $U \in \mathcal{B}$

1. the systems of abelian groups $\mathcal{F}_ n^{m - 2}(U)$ and $\mathcal{F}_ n^{m - 1}(U)$ have vanishing $R^1\mathop{\mathrm{lim}}\nolimits$ (for example these have the Mittag-Leffler condition),

2. the system of abelian groups $H^{m - 1}(\mathcal{F}_ n^\bullet (U))$ has vanishing $R^1\mathop{\mathrm{lim}}\nolimits$ (for example it has the Mittag-Leffler condition), and

3. we have $H^ m(\mathcal{F}_ n^\bullet (U)) = H^ m(\mathcal{F}_{n_0}^\bullet (U))$ for all $n \geq n_0$.

Then the maps $H^ m(\mathcal{F}^\bullet ) \to \mathop{\mathrm{lim}}\nolimits H^ m(\mathcal{F}_ n^\bullet ) \to H^ m(\mathcal{F}_{n_0}^\bullet )$ are isomorphisms of sheaves where $\mathcal{F}^\bullet = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n^\bullet$ is the termwise inverse limit.

Proof. Let $U \in \mathcal{B}$. Note that $H^ m(\mathcal{F}^\bullet (U))$ is the cohomology of

$\mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m - 2}(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m - 1}(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^ m(U) \to \mathop{\mathrm{lim}}\nolimits _ n \mathcal{F}_ n^{m + 1}(U)$

in the third spot from the left. By assumptions (2)(a) and (2)(b) we may apply More on Algebra, Lemma 15.85.2 to conclude that

$H^ m(\mathcal{F}^\bullet (U)) = \mathop{\mathrm{lim}}\nolimits H^ m(\mathcal{F}_ n^\bullet (U))$

By assumption (2)(c) we conclude

$H^ m(\mathcal{F}^\bullet (U)) = H^ m(\mathcal{F}_ n^\bullet (U))$

for all $n \geq n_0$. By assumption (1) we conclude that the sheafification of $U \mapsto H^ m(\mathcal{F}^\bullet (U))$ is equal to the sheafification of $U \mapsto H^ m(\mathcal{F}_ n^\bullet (U))$ for all $n \geq n_0$. Thus the inverse system of sheaves $H^ m(\mathcal{F}_ n^\bullet )$ is constant for $n \geq n_0$ with value $H^ m(\mathcal{F}^\bullet )$ which proves the lemma. $\square$

## 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Your email address will not be published.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.

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 08BY. Beware of the difference between the letter 'O' and the digit '0'.