---
tag: 5A35TYZD02
kind: concept
name: Scheme
msc: [14A15]
specializes: [B7FPT8JTXR]
sources:
  - url: https://en.wikipedia.org/wiki/Scheme_(mathematics)
same_as:
  wikidata: Q1155772
  wikipedia: Scheme (mathematics)
---

## Summary

A scheme is a [[locally ringed space|B7FPT8JTXR]] that looks, near each point, like an [[affine scheme|B3V3XJ2W8G]].
Schemes were introduced by Alexander Grothendieck as the basic objects of algebraic geometry.
They include algebraic varieties over any field, and also objects such as $\operatorname{Spec} \Z$ that come from number theory.
They also allow nilpotent functions, which record multiplicities and infinitesimal information.

## Explanations

### L1 human

A manifold is built by gluing open pieces of $\R^n$; a scheme is built by gluing affine schemes, the spectra of rings.
Each piece carries its own ring of functions, and the pieces agree where they overlap.
The projective line is the main small example.
It is glued from two copies of the affine line, with coordinates $t$ and $s$, along the rule $s = 1/t$ where both are nonzero.
The result is not an affine scheme, but every point has an affine neighbourhood.