Remark 20.37.3 (0BKQ)—The Stacks project

Remark 20.37.3. Let $(X, \mathcal{O}_ X)$ be a ringed space. Let $(K_ n)$ be an inverse system in $D(\mathcal{O}_ X)$ . Set $K = R\mathop{\mathrm{lim}}\nolimits K_ n$ . For each $n$ and $m$ let $\mathcal{H}^ m_ n = H^ m(K_ n)$ be the $m$ th cohomology sheaf of $K_ n$ and similarly set $\mathcal{H}^ m = H^ m(K)$ . Let us denote $\underline{\mathcal{H}}^ m_ n$ the presheaf

$U \longmapsto \underline{\mathcal{H}}^ m_ n(U) = H^ m(U, K_ n)$

Similarly we set $\underline{\mathcal{H}}^ m(U) = H^ m(U, K)$ . By Lemma 20.32.3 we see that $\mathcal{H}^ m_ n$ is the sheafification of $\underline{\mathcal{H}}^ m_ n$ and $\mathcal{H}^ m$ is the sheafification of $\underline{\mathcal{H}}^ m$ . Here is a diagram

$\xymatrix{ K \ar@{=}[d] & \underline{\mathcal{H}}^ m \ar[d] \ar[r] & \mathcal{H}^ m \ar[d] \\ R\mathop{\mathrm{lim}}\nolimits K_ n & \mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n \ar[r] & \mathop{\mathrm{lim}}\nolimits \mathcal{H}^ m_ n }$

In general it may not be the case that $\mathop{\mathrm{lim}}\nolimits \mathcal{H}^ m_ n$ is the sheafification of $\mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n$ . If $U \subset X$ is an open, then we have short exact sequences

20.37.3.1

$\begin{equation} \label{cohomology-equation-ses-Rlim-over-U} 0 \to R^1\mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^{m - 1}_ n(U) \to \underline{\mathcal{H}}^ m(U) \to \mathop{\mathrm{lim}}\nolimits \underline{\mathcal{H}}^ m_ n(U) \to 0 \end{equation}$

by Lemma 20.37.1.

The Stacks project

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