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Let be a ring. Let denote the category of -rings. Let denote the category of (contravariant) functors . Let denote the category of representable -functors; we call this category thecategory of affine -schemes and an object of this category we write as
We obtain in this way a functor
This functor has a left adjoint
assigning to a -functor its ring of functions. This adjunction restricts to an adjoint equivalence
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
If is a group is a group scheme.
(see also Grothendieck's Galois theory)
An étale group scheme over a field is defined to be a directed colimit
where denotes some set of finite separable field extensions of .
Let be a commutative -group functor (in cases of interest this is a finite flat commutative group scheme). Then the Cartier dual of is defined by
where denotes the group scheme assigning to a ring its multiplicative group consisting of the invertible elements of .
This definition deserves the name duality since we have
Let be a morphism of rings. Then we have an adjunction
from the category of -modules to that of -modules where
is called scalar extension and is called scalar restriction.
If denotes some scheme over a -ring for being a field of characteristic , we define its -torsion component-wise by .