---
tag: AFC6JMYWEH
kind: concept
name: Affine morphism
msc: [14A15]
specializes: [JXRKH6P1Z4]
sources:
  - url: https://en.wikipedia.org/wiki/Sheaf_of_algebras
---

## Summary

An affine morphism $X \to Y$ of schemes is one for which the preimage of every open affine subscheme of $Y$ is affine.
Affine morphisms over $Y$ correspond to quasi-coherent sheaves of algebras on $Y$, through the relative spectrum construction.
They include closed immersions, finite morphisms and vector bundles.

## Explanations

### L1 human

An affine morphism is a family of affine schemes parametrized by the points of $Y$, glued in a coherent way.
Over each affine piece $\operatorname{Spec} B$ of $Y$, the part of $X$ above it is the spectrum of a single $B$-algebra.
Every morphism between affine schemes is affine.
The map from the plane minus the origin to a point is not affine, since the plane minus the origin is not an affine scheme.