Let be a field.
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 k-spectra of finite separable field-extensions of .
We give a characterization of étale -schemes and étale formal -schemes in terms of constant schemes?:
The category of -schemes 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.
Let be a -scheme or a formal -scheme. Then the following statements are equivalent:
is constant. where denotes an algebraic closure of , denotes the subextension of consisting of all separable elements of and denotes skalar extension.
is étale iff its skalar extension is étale. And a -scheme is étale iff it is constant.
This statement is an instance of the main theorem of Grothendieck's Galois theory in the classical case of fields.
Since this functor preserves products we have the analogue statement for group schemes:
Then is étale iff the Frobenius morphism is a monomorphism resp. an isomorphism.