The Stacks project

Remark 15.87.2. This remark is a continuation of Remark 15.86.6. A sheaf of rings on $\mathbf{N}$ is just an inverse system of rings $(A_ n)$. A sheaf of modules over $(A_ n)$ is exactly the same thing as an object of the category $\textit{Mod}(\mathbf{N}, (A_ n))$ defined above. The derived functor $R\mathop{\mathrm{lim}}\nolimits$ of Lemma 15.87.1 is simply $R\Gamma (\mathbf{N}, -)$ from the derived category of modules to the derived category of modules over the global sections of the structure sheaf. It is true in general that cohomology of groups and modules agree, see Cohomology on Sites, Lemma 21.12.4.