---
tag: P8YTMGEN9R
kind: concept
name: Local homomorphism
msc: [13H99, 13B10]
specializes: [0VTH8KPCG2]
sources:
  - url: https://en.wikipedia.org/wiki/Local_ring
---

## Summary

A local homomorphism is a [[ring homomorphism|0VTH8KPCG2]] between [[local rings|MZ02BKPHJA]] that sends the maximal ideal into the maximal ideal.
These are the natural maps between local rings, and they induce maps of [[residue fields|NJ3APQ8BHH]].
In geometry, the map of local rings induced by a morphism of spaces at a point is local.

## Explanations

### L1 human

The maximal ideal of a local ring consists of the "functions that vanish at the point".
A local homomorphism must send functions vanishing at the point to functions vanishing at the image point.
For example, reducing the ring $\Z_{(p)}$ modulo $p$ to the field $\Z/p\Z$ is local.
But including $\Z_{(p)}$ into $\Q$ is not, since $p$ is not invertible in $\Z_{(p)}$ but becomes invertible in $\Q$.
The [[definition|RM0FMT0X8W]] states the condition.