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(k-ring, k-functor,affine k-scheme)
For a ring the category of -rings, denoted by is defined to be the category of commutative associative -algebras with unit which are rings. This is equivalently the category of pairs where is a Ring and is a morphism of -algebras.
The category of -functors, denoted by , is defined to be the category of covariant functors .
The forgetful functor sending a -ring to its underlying set is called affine line.
For the full and faithful contravariant functor
(and every isomorphic functor) is called an affine -scheme. (In most modern texts one uses the notation ‘’’‘ instead of ‘’’’.) restricts to an equivalence between the categories of -rings and the category of affine -schemes. We think of this category as of . The functor commutes with limits and skalar extension (see below). Consequently is closed under limits and base change.
The affine line is an affine -scheme.
A function on a -scheme is defined to be an object . is a -ring by component-wise addition and -multiplication.
There is an adjoint equivalence
of the categories of affine k-schemes and -rings.
The category of -functors has limits.
The terminal object is . Products and pullbacks are computed component-wise.
For the ‘’base change’‘ functor induced by given by postcompositions with is called skalar extension.