---
tag: B3V3XJ2W8G
kind: concept
name: Affine scheme
msc: [14A15]
specializes: [5A35TYZD02]
sources:
  - url: https://en.wikipedia.org/wiki/Spectrum_of_a_ring
---

## Summary

An affine scheme is the [[spectrum|FB45BHHA2J]] $\operatorname{Spec} A$ of a [[commutative ring|0310REJJY8]] $A$, made into a [[locally ringed space|B7FPT8JTXR]] by its structure sheaf.
Every commutative ring gives one, and the ring can be read back as the ring of global functions.
Affine schemes are the local building blocks of all [[schemes|5A35TYZD02]], just as open balls are the local pieces of manifolds.

## Explanations

### L1 human

An affine scheme turns a ring into a geometric object.
Its points are the prime ideals of the ring, and the elements of the ring act as functions on these points.
For the polynomial ring $\C[x, y]$, the maximal ideals $(x - a, y - b)$ are the points $(a, b)$ of the complex plane, and other primes, such as $(y - x^2)$, stand for whole curves.
The structure sheaf records which fractions are allowed as functions on each open set.