# inverse limit realized as global section

Does the construction (realization) below have some standard name? (Something I could search for?) Does it have any use?

Suppose $I$ a directed set, and $(A_\alpha,\pi_{\alpha,\beta} \text{ for } \alpha > \beta \in I$) an inverse system of modules (or groups, or whatever).

Put a topology on $I$ by declaring sets $D$ which are "downwards closed" (if $\alpha > \beta$ and $\alpha\in D$ then $\beta\in D$) to be open, so $\alpha\downarrow = \{\beta\in I \mid \beta \leq \alpha\}$ is a basis.

Define a sheaf $S$ over $I$ with stalks $S_\alpha = A_\alpha$ ($\alpha\in I$). Toplogy on the stalk space: If $a\in S_\alpha$, an open nhbd is $\{\pi_{\alpha,\beta}(a)\mid \beta \leq \alpha\}$. Then $p:\Gamma S \rightarrow I$ defined by $p(a) = \alpha$ if $a\in S_\alpha$ is a local homemorphism. Addition on the stalk space should be continuous (haven't checked)

Then the inverse limit of the original system is the same thing as the module of global sections. So any completion of a linear topology on a module could be realized in this way as well.

Is this been used for anything I could read about somewhere?

-