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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 the category 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
and it restricts moreover to an adjoint equivalence
between the categories of -birings and the category of commutative affine -group schemes. To see this be aware that a -biring is a commutative ring object in (where the latte latter denotes the category of affine schemes).
A -functor is called a -scheme if it is a sheaf for the Zariski Grothendieck topology on .
We will consider the moral of this op-ing below.
To give more details, recall that the closed sets of the Zariski topology on the spectrum of a -ring is defined by
We can characterize the the elements of also by
where
where denotes the quotient field (aka. field of fractions) of the integral domain .
This construction generalizes to -functors by defining an open subfunctor of a -functor by
where . By the above alternative characterization, the assigned set consists precisely of those for which for all .
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.
We denote the category of -schemes by .
A -group scheme is a group object in . The category of -group schemes we denote by .
Let be a field. A finite -ring is defined to be a -algebra which is a finite dimensional -vector space. The category of finite -rings we denote by . A finite -functor is defined to be a covariant functor . The category of finite -functors we denote by . A finite -scheme is defined to be -scheme which is a finite -functor. The category of finite -schemes we denote by . Analogously we define the category of finite group schemes.
A formal group scheme is defined to be a codirected colimit of finite -schemes.
Recall that we have a covariant embedding
but we equivalently an embedding
where by we denote the category of -corings. A coring is a comonoid in the category of affine schemes (the latter is the opposite category of ). If we restrict to finite -rings by linear algebra we have a bijection and can write
where is a -coring, a finite -ring, the skalar-extended comultiplication, the skalar-extended counit.
(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 .