---
tag: CDS1VWB7A3
kind: concept
name: Morphism of varieties
msc: [14A10]
sources:
  - url: https://en.wikipedia.org/wiki/Morphism_of_algebraic_varieties
same_as:
  wikidata: Q7309604
  wikipedia: Morphism of algebraic varieties
---

## Summary

A morphism of varieties is a map between algebraic varieties that is given locally by polynomials, or by quotients of polynomials with nonvanishing denominators.
Morphisms are the maps of the category of varieties, and an isomorphism is a morphism with an inverse morphism.
Between affine varieties, morphisms correspond exactly to homomorphisms of the coordinate rings in the opposite direction.

## Explanations

### L1 human

A morphism is a map between solution sets of polynomial equations that is itself written with polynomials.
The map $t \mapsto (t, t^2)$ from the line onto the parabola $y = x^2$ is a morphism, and so is its inverse $(x, y) \mapsto x$.
So the line and the parabola are isomorphic varieties.
The map $t \mapsto (t^2, t^3)$ onto the cusp $y^2 = x^3$ is a morphism and a bijection, but its inverse is not a morphism, so these two are not isomorphic.