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is copowered (= tensored) over . We define the constant -scheme on a set by
For a scheme we compute and see that there is an adjunction
A constant formal scheme is defined to be a completion of constant scheme. The completion functor induces an equivalence between the category of constant schemes and the category of constant formal schemes.
An étale -scheme is defined to be a directed colimit of -spectra of finite separable field-extensions of .
An étale formal -scheme is defined to be a directed colimit of formal -spectra of finite separable field-extensions of .
Let be a -scheme or a formal -scheme. Then the following statements are equivalent:
where denotes an algebraic closure of , denotes the subextension of consisting of all separable elements of and denotes skalar extension.
The completion functor
is an equivalence of categories.