Definition

An affine -scheme is a ringed space of the form

(where is the prime spectrum of for a commutative ring with it’s structure sheaf.)

Morphisms

A morphism of affine -schemes is a morphism of ringed spaces.

Let be a ring homomorphism. Then we have

we want .

Let , then is open.

Note that exists uniquely since it is defined by the universal property such that

Then since

and

then we can induce the map

from the distinguished maps .

There is an equivalence of categories between rings and affine schemes

with quasi-inverse