Spectrum as functor

Question

I have just started studying about affine schemes and I am searching for books about Spec as a functor from the category CRings to the category AffineSchemes can you suggest me some? Also I am searching about the geometric results of this equivalence of categories, any ideas?

Answer 1

There is no book about these topics in particular, because there is basically no content to speak of here. The definition of the structure sheaf on $\operatorname{Spec}(A)$ takes some work, but once this is done it follows immediately that $\operatorname{Spec}$ is a functor and in particular an equivalence between $\mathsf{CRing}^{\text{op}}$ and $\mathsf{AffSch}$. The proof should take between 1 and 10 lines depending on how detailed you want to be.

The key ingredient for making this stuff seem as "obvious" as I am making it sound is to really deeply understand the definitions of:

(1) The structure sheaf
(2) Morphisms of locally ringed spaces

This will be covered in any introductory book on algebraic geometry, e.g. Vakil's "The Rising Sea".