4.6. (拟) 凝聚层
Now we jump to quasi-coherent/coherent sheaves on stack.
定义 22.214.171.124. We define Lisse-etale site on as follows (the topos is denoted by ). The site has objectswith being scheme, and morphismsThe coverings are given by family of diagrams of the formsuch that we have etale surjection .
Then, we define as .
Now let be the following category. The objects are: for all a choice of etale sheaf of sets and for all a choicesuch that:
if is etale then is isomorphism.
For diagramwe have
A morphism between in is a collection of morphisms so the following diagram commutes
We have given by maps to the sheaf given by .
This is a presheaf where transition maps of come from .
It is a sheaf because etale covers in are already coverings in .
Conversely, given by maps to the object .
Therefore, we get .
定义 126.96.36.199. Let be a sheaf of rings on . Let be a sheaf of -module. Then:
We say is Cartesian if for all , we get
We say is quasi-coherent if it is Cartesian -module and for all , we have is quasi-coherent.
We say is coherent if is locally Noetherian (note this implies for anyu we get locally Noetherian), is quasi-coherent and for all , we have .