geometry ag.algebraica – Sections of pulleys in boundary spaces

Leave $ {U _ { nu} } _ { nu in I} $ To be an inverse system of topological spaces on a set of filtered indexes. $ I $ With continuous transition maps.

Leave $ A $ To be a sheaf of abelian groups in the topological space. $ U = varprojlim U _ { nu} $ and call $ U _ { nu} subset U $ the preimage of $ U _ { nu} $ in $ U $ through the map $ U a U _ { nu} $ from the inverse limit.

When we have
$$ A (U) = varinjlim A (U _ { nu}) $$
for each sheaf of abelian groups $ A $? We always have this when $ A $ It is the constant sheaf with values ​​in the Abelian group. $ A $?

Yes $ U $ It is compact Hausdorff this is true.

It is true when each $ U _ { nu} $ it's even a smooth variety, the transition maps are local diffeomorphisms, but $ U $ Is not it necessarily compact?