nLab
Henselian ring

Idea
Definition
Examples
Properties
Henselization
References

Idea

A Henselian ring is a local ring for which the conclusion of Hensel's lemma? (classically stated for a complete local ring?) holds.

Definition

Let $R$ be a local ring with maximal ideal $m$ . Let $k = R / m$ be the residue class field, and for a polynomial $f \in R [x]$ , let $\bar{f} \in k [x]$ obtained by reduction of the coefficients of $f$ modulo $m$ . Then $R$ is Henselian if, for any monic $f \in R [x]$ such that $\bar{f}$ factorizes as $g_{0} h_{0}$ with $g_{0}, h_{0}$ monic and relatively prime in $k [x]$ , there exist monic $g, h \in R [x]$ such that $f = g h$ with $\bar{g} = g_{0}$ , $\bar{h} = h_{0}$ , and the ideal $(g, h)$ is the unit ideal in $R [x]$ .

Often this definition is given just for the case when one of $g_{0}$ , $h_{0}$ is a linear factor $x - a$ , where the idea is that the (simple) root $a$ can be lifted to $R$ .

Definition

A Henselian ring $R$ with residue class field $k$ is strictly Henselian if $k$ is separably closed.

Examples

Any field $k$ is trivially Henselian.
A complete local ring, such as p-adic number rings, is Henselian. This includes rings of formal power series over a field, $k [[x_{1}, \dots, x_{n}]]$ .
Rings of convergent power series over a local field are Henselian.

Properties

Proposition

The quotient of a Henselian ring is also Henselian.

Proof

In the first place, a quotient $R'$ of a local ring $R$ is local (and has the same residue class field $k$ ). Secondly, $R'$ is clearly Henselian since we can lift factorizations $\bar{f} = g_{0} h_{0}$ in $k [x]$ to factorizations in $R [x]$ , and then push them down to $R' [x]$ .

Proposition

If $R$ is local and its reduced ring (i.e., $R$ modulo its ideal of nilpotent elements) is Henselian, then $R$ itself is Henselian.

Proposition

If $R$ is Henselian and $S$ is a local ring that is integral over $R$ (meaning that $S$ is an $R$ -algebra and each $x \in S$ is an integral element over $R$ ), then $S$ is Henselian.

Henselization

Let $LocRing$ denote the category of local rings (commutative of course) and local ring homomorphisms ( $f : R \to S$ is local if the pullback along $f$ of the maximal ideal of $S$ is the maximal ideal of $R$ ).

Theorem

The full subcategory of $LocRing$ consisting of Henselian rings is reflective. The left adjoint to the full inclusion is called Henselization.

Example

Let $R$ be a discrete valuation ring, with $K$ its field of fractions. Let $\hat{R}$ be the $m$ -adic completion of $R$ with respect to its maximal ideal. Then the Henselization of $R$ is isomorphic to the subring of $\hat{R}$ whose elements are roots of separable polynomials with coefficients in $K$ .

A rule of thumb, as suggested by this example, is that the Henselization is the algebraic part of a local ring completion.

References

An original reference is

Alexandre Grothendieck, (1967), EGA: IV. Étude locale des schémas et des morphismes de schémas , Quatrième partie”, Publications Mathématiques de l’IHÉS 32: 5–361. (web)

Other sources are

Alonso, Lombardi, Perdry, Henselian local rings (pdf)
Remarks on Henselian rings (pdf);
Ieke Moerdijk, Rings of smooth functionss and their localizations (pdf)

Revised on February 4, 2013 22:58:43 by Todd Trimble (67.81.93.26)

nLab Henselian ring

Contents

Idea

Definition

Definition

Definition

Examples

Properties

Proposition

Proof

Proposition

Proposition

Henselization

Theorem

Example

References

nLab
Henselian ring