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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
A -functor is called a -scheme if it is a sheaf for the Zariski Grothendieck topology on .
To We give will more consider details, recall that the closed moral sets of the this Zariski op-ing topology below. on thespectrum of a -ring is defined by
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
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 a codirected limit of finiteformal group scheme -schemes 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. 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 .