Editing Commutative ring (section)

=== Affine schemes ===
The notion of a spectrum is the common basis of commutative algebra and [[algebraic geometry]]. Algebraic geometry proceeds by endowing Spec ''R'' with a [[sheaf (mathematics)|sheaf]] <math>\mathcal O</math> (an entity that collects functions defined locally, i.e. on varying open subsets). The datum of the space and the sheaf is called an [[affine scheme]]. Given an affine scheme, the underlying ring ''R'' can be recovered as the [[global section]]s of <math>\mathcal O</math>. Moreover, this one-to-one correspondence between rings and affine schemes is also compatible with ring homomorphisms: any ''f'' : ''R'' → ''S'' gives rise to a [[continuous map]] in the opposite direction
{{block indent|1= Spec ''S'' → Spec ''R'', ''q'' ↦ ''f''<sup>−1</sup>(''q''), i.e. any prime ideal of ''S'' is mapped to its [[preimage]] under ''f'', which is a prime ideal of ''R''. }}
The resulting [[equivalence of categories|equivalence]] of the two said categories aptly reflects algebraic properties of rings in a geometrical manner.

Similar to the fact that [[manifold (mathematics)|manifolds]] are locally given by open subsets of '''R'''<sup>''n''</sup>, affine schemes are local models for [[scheme (mathematics)|schemes]], which are the object of study in algebraic geometry. Therefore, several notions concerning commutative rings stem from geometric intuition.