The Stacks project

Remark 15.86.6 (Rlim as cohomology). Consider the category $\mathbf{N}$ whose objects are natural numbers and whose morphisms are unique arrows $i \to j$ if $j \geq i$. Endow $\mathbf{N}$ with the chaotic topology (Sites, Example 7.6.6) so that a sheaf $\mathcal{F}$ is the same thing as an inverse system

\[ \mathcal{F}_1 \leftarrow \mathcal{F}_2 \leftarrow \mathcal{F}_3 \leftarrow \ldots \]

of sets over $\mathbf{N}$. Note that $\Gamma (\mathbf{N}, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n$. For an inverse system of abelian groups $\mathcal{F}_ n$ we have

\[ R^ p\mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = H^ p(\mathbf{N}, \mathcal{F}) \]

because both sides are the higher right derived functors of $\mathcal{F} \mapsto \mathop{\mathrm{lim}}\nolimits \mathcal{F}_ n = H^0(\mathbf{N}, \mathcal{F})$. Thus the existence of $R\mathop{\mathrm{lim}}\nolimits $ also follows from the general material in Cohomology on Sites, Sections 21.2 and 21.19.

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