nLab Henselian ring

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

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, $k[ [x_1, \ldots, x_n] ]$.

• 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.

Related concepts

• restricted formal power series
• Henselian pair

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)

Lecture notes are in

• James Milne, section 4 of Lectures on Étale Cohomology

Other sources include

• Michel Raynaud, Anneaux locaux henséliens, Lecture Notes in Mathematics Volume 169 1970 doi:10.1007/BFb0069571

• Jacob Lurie, Descent Theorems, section 3.

• Alonso, Lombardi, Perdry, Henselian local rings (pdf)

• Krzysztof Jan Nowak, Remarks on Henselian rings (pdf);

• Ieke Moerdijk, Rings of smooth functions and their localizations (pdf)