Henselization

Builds on Local ring Henselian ring Local homomorphism

Definition

Let $(R, \mathfrak{m})$ be a local ring. A Henselization of $R$ is a local ring $R^h$ together with a local homomorphism $i \colon R \to R^h$ such that:

$R^h$ is a Henselian local ring.
For every Henselian local ring $S$ and every local homomorphism $f \colon R \to S$ , there is exactly one local homomorphism $g \colon R^h \to S$ with $g \circ i = f$ .

Remarks

A Henselization exists for every local ring, and the universal property makes it unique up to a unique isomorphism. One construction takes the direct limit of the localizations of étale $R$ -algebras at primes over $\mathfrak{m}$ with trivial residue field extension. The map $i$ is faithfully flat, the ideal $\mathfrak{m} R^h$ is the maximal ideal of $R^h$ , and $i$ induces an isomorphism of residue fields. If $R$ is Noetherian, then $R^h$ is Noetherian and has the same completion as $R$ .

Example: if $R$ is already Henselian, then $R$ with the identity map is a Henselization, because $g = f$ is the only choice in clause 2.

Non-example: $\Z_{(5)}$ with the identity map is not its own Henselization, because it fails clause 1. The polynomial $x^2 - 6$ has the simple root $1$ modulo $5$ , but $6$ is not a square in $\Q$ , so the root does not lift to $\Z_{(5)}$ .

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Henselian ring concept
Local homomorphism concept
Local ring concept
Henselization definition

Tag	`J9QYGM8AKF`
URI	`m2://J9QYGM8AKF`
Kind	definition
Writing	W0 stub
MSC	13J15, 13B40
Defines	Henselization
Uses	Local ring, Henselian ring, Local homomorphism
Changed	2 November 2026

Henselization

Definition

Remarks

Reading path

Further reading

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