This entry is about a section of the text
Let be a field. Let denote the category of finite dimensional -rings.
A -scheme is called a -formal scheme if it is is equivalent to a codirected colimit of finite (affine) -schemes.
A -scheme is a -formal scheme if it is presented by a profinite -ring or -equivalently- by a -ring which is the limit of discrete quotients which are finite -rings. If is such a topological -ring denotes the set of continous morphisms from to the topological discrete ring . We have is a contravariant equivalence between the category of profinite -rings and the category of formal -schemes.
Instead of defining as the opposite of define it covariantly on the category of finite dimensional -corings.
A formal -scheme is precisely a left exact (commutig with finite limits) functor .
The inclusion induces a functor
called completion functor.
A -coalgebra is a -vector space equipped with a -linear map .
A -coalgebra is called a -coring if is
coassociative in that
cocommutative in that the image of consists only of symmetric tensors.
has a counit to satisfying .
Let and be two finite -rings, let denote the dual k-coring? of .
Linear maps correspond bijectively to elements of the tensor product . The -linear maps and extends to -linear maps
denoted by and .
A -linear map associated to is a ring morphism iff and .
There is a functorial isomorphism
and the -formal spectrum of the coring is defined by
in particular is a covariant functor from the category of -corings to the category of -formal functors.