# Henselian ring

A **henselian ring** is a local ring \( (R, \mathfrak m, \kappa) \) in which every simple root of a monic polynomial over the residue field lifts to an actual root in the ring: for every monic \( f \in R[T] \) and every \( a_0 \in \kappa \) with \( \overline{f}(a_0) = 0 \) and \( \overline{f'}(a_0) \neq 0 \), there is an \( a \in R \) with \( f(a) = 0 \) and \( \overline{a} = a_0 \)<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup>. The condition is a form of [Hensel's lemma](https://www.edgepedia.org/hensels-lemma), the lifting principle first observed for the \( p \)-adic integers around 1910<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>. A ring is **strictly henselian** if it is henselian and its residue field is separably algebraically closed<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup>.

| Key fact | Detail |
|---|---|
| Defining property | Simple roots of monic polynomials lift from the residue field \( \kappa \) to \( R \)<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup> |
| Strictly henselian | Henselian with separably algebraically closed residue field<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup> |
| Complete local rings | Every complete local ring is henselian<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup> |
| Finite étale algebras | For henselian \( (R, \mathfrak m, \kappa) \), finite étale \( R \)-algebras correspond to finite étale \( \kappa \)-algebras via \( S \mapsto S/\mathfrak m S \)<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup> |
| Henselization | Every local ring embeds in a smallest henselian local ring \( R^h \) with the same residue field, a filtered colimit of étale neighborhoods<sup>[3](https://stacks.math.columbia.edu/tag/04HW)</sup> |
| Relation to completion | For a locally Noetherian scheme, \( \mathcal{O}_{S,s} \subset \mathcal{O}_{S,s}^h \subset \widehat{\mathcal{O}_{S,s}} \), and the henselization has the same completion as the original local ring<sup>[3](https://stacks.math.columbia.edu/tag/04HW)</sup> |
| Global analog | A henselian pair \( (A, I) \): \( I \) in the Jacobson radical, with lifting of coprime monic factorizations from \( (A/I)[T] \)<sup>[4](https://stacks.math.columbia.edu/tag/09XD)</sup> |

## Equivalent characterizations

The root-lifting definition is one of several equivalent formulations. According to Raynaud's definition, a local ring \( R \) has the Henselian property precisely when each finite \( R \)-algebra \( B \) decomposes as a product, finite in number, of local rings.<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup> In constructive mathematics, Alonso García, Henri Lombardi, and Hervé Perdry take the decomposition property as the definition and show that idempotents in any finite algebra over a Henselian ring lift to idempotents of the algebra; conversely, a local ring in which every finite algebra has the idempotent-lifting property is Henselian<sup>[5](https://arxiv.org/abs/2202.06595)</sup>. Another equivalent form asks that coprime factorizations \( \overline{f} = \overline{g}\,\overline{h} \) of a monic polynomial over the residue field lift to factorizations \( f = gh \) over \( R \)<sup>[6](https://www.esaga.uni-due.de/f/thiago.solovera-e-nery/notes/main_hensel.pdf)</sup>.

## Examples and non-examples

**Complete local rings.** The proof is an inductive [Newton's method](https://www.edgepedia.org/newtons-method): a simple root \( a_0 \) of \( \overline{f} \) is lifted successively through \( R/\mathfrak m^{n+1} \), each step using that \( f' \) is invertible modulo \( \mathfrak m \), and completeness supplies the compatible limit \( a \in \lim R/\mathfrak m^n = R \)<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup>.

**Other classes.** A local ring of dimension 0 is henselian, since every finite algebra over it is a product of local rings<sup>[1](https://stacks.math.columbia.edu/tag/04GE)</sup>. Quotients of henselian rings are henselian<sup>[6](https://www.esaga.uni-due.de/f/thiago.solovera-e-nery/notes/main_hensel.pdf)</sup>. Many familiar Noetherian local rings fail to be henselian<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>.

Henselian does not imply that the ring behaves like its completion. There exists a henselian discrete valuation ring whose completion is a purely inseparable algebraic extension, so polynomial equations solvable over the completion need not reflect solvability over the ring itself<sup>[7](https://www.numdam.org/article/PMIHES_1969__36__23_0.pdf)</sup>.

## Henselization

Every local ring \( (A, \mathfrak m) \) has a henselization \( A^h \), a local ring homomorphism \( A \to A^h \) where \( A^h \) is henselian with residue field \( \kappa(\mathfrak m) \). It is a filtered colimit of étale \( A \)-algebras, and it is characterized by a universal property: any map from \( A \) to a henselian local ring with the same residue field factors through it, and the henselization is unique up to canonical isomorphism<sup>[8](https://stacks.math.columbia.edu/tag/0A03)</sup>. Concretely, \( \mathcal{O}_{S,s}^h \) is the colimit over the filtered category of étale neighborhoods \( (U, u) \) of \( (S, s) \) with \( \kappa(s) = \kappa(u) \)<sup>[3](https://stacks.math.columbia.edu/tag/04HW)</sup>. Raynaud's construction realizes this colimit over the directed set of pairs \( (f, Q) \) with \( f \in R[x] \) monic and \( Q \) a prime with \( f' \notin Q \)<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>.

The henselization is the algebraic substitute for the completion. For a locally Noetherian scheme \( S \), the henselization \( \mathcal{O}_{S,s}^h \) is Noetherian and has the same completion as \( \mathcal{O}_{S,s} \), so \( \mathcal{O}_{S,s} \subset \mathcal{O}_{S,s}^h \subset \widehat{\mathcal{O}_{S,s}} \). The henselization is in general much smaller than the completion and inherits more properties: if \( \mathcal{O}_{S,s} \) is reduced, so is \( \mathcal{O}_{S,s}^h \), which is not true of the completion in general<sup>[3](https://stacks.math.columbia.edu/tag/04HW)</sup>. For a Noetherian local domain, the completion usually has infinite (uncountable) transcendence degree over the ring, while the henselization is an algebraic extension<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>. Henselization also preserves good ring behavior: \( R \) is Noetherian if and only if \( R^h \) is, and excellence, the Nagata property, and geometric regularity of formal fibers pass between \( R \) and \( R^h \)<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>.

## Henselian pairs

The global analog of a henselian local ring is a **henselian pair** \( (A, I) \): the ideal \( I \) is contained in the [Jacobson radical](https://www.edgepedia.org/jacobson-radical) of \( A \), and every factorization of a monic \( f \in A[T] \) into coprime monic factors over \( A/I \) lifts to a factorization over \( A \)<sup>[4](https://stacks.math.columbia.edu/tag/09XD)</sup>. When \( A \) is local and \( I = \mathfrak m \), this recovers the henselian local ring<sup>[4](https://stacks.math.columbia.edu/tag/09XD)</sup>.

The central consequence is that passage to \( A/I \) loses no étale information. For a henselian pair, the functor \( B \mapsto B/IB \) determines an equivalence between finite étale \( A \)-algebras and finite étale \( A/I \)-algebras; the proof turns on the unique lifting of idempotents and Nakayama's lemma<sup>[9](https://stacks.math.columbia.edu/tag/0D49)</sup>. The same equivalence holds for all étale algebras when \( I \) is locally nilpotent, and such a pair is automatically henselian<sup>[4](https://stacks.math.columbia.edu/tag/09XD)</sup>. Henselian pairs also control integral extensions: for an integral map \( A \to B \), idempotents of \( B/IB \) lift bijectively to \( B \)<sup>[10](https://stacks.math.columbia.edu/download/book.pdf)</sup>.

Geometrically, a henselian pair \( (A, I) \) defines a Henselian scheme \( \mathrm{Sph}(A, I) \), a locally ringed space on \( V(I) \subset \mathrm{Spec}(A) \) with a Henselized structure sheaf; these are models for tubular neighborhoods in algebraic geometry<sup>[11](https://arxiv.org/abs/2212.08644)</sup>.

## Role in the étale topology

Strictly henselian local rings are the local rings of the étale topology. The stalk of the structure sheaf on the small étale site of a scheme \( S \) at a geometric point \( \overline{s} \) is the strict henselization \( \mathcal{O}_{S,\overline{s}}^{sh} \) of the local ring at \( s \)<sup>[3](https://stacks.math.columbia.edu/tag/04HW)</sup>. From the étale point of view, \( \mathrm{Spec} \, \mathcal{O}^{sh}_{X,\xi} \) is the true local ring of a scheme at a geometric point<sup>[6](https://www.esaga.uni-due.de/f/thiago.solovera-e-nery/notes/main_hensel.pdf)</sup>.

The category equivalence for finite étale algebras explains why henselian rings are convenient for fundamental group computations: for a henselian local ring, the category of finite étale extensions is equivalent to that over the residue field, so the étale fundamental group is insensitive to passing to the completion<sup>[12](https://mathoverflow.net/questions/250238/clarifying-the-connection-between-etale-locally-and-formally-locally)</sup>.

## Approximation

Henselization also enters the theory of solving polynomial equations. For a local Noetherian G-ring, having solutions to a system of polynomial equations in the completion is often enough to produce solutions in the henselization<sup>[13](https://stacks.math.columbia.edu/tag/07QX)</sup>. This is the setting of Artin's algebraic approximation theorem, which treats structures over complete local rings and shows that, when \( A \) is henselian, the maximal ideal in such approximation statements may be replaced by any proper ideal<sup>[7](https://www.numdam.org/article/PMIHES_1969__36__23_0.pdf)</sup>.

## History

The lifting principle originates in [Kurt Hensel](https://www.edgepedia.org/kurt-hensel)'s creation of the theory of \( p \)-adic numbers, where his lemma reduces the solution of an algebraic equation over a complete local ring to the corresponding equation over the residue field<sup>[14](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)</sup>. The Henselian property was first observed in algebraic number theory around 1910 for the \( p \)-adic integers<sup>[2](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)</sup>. The abstract ring-theoretic notion was introduced by Go Azumaya, and [Masayoshi Nagata](https://www.edgepedia.org/masayoshi-nagata)'s 1953 paper in the Nagoya Mathematical Journal defined (local) Henselian rings and Henselizations of [integrally closed domains](https://www.edgepedia.org/integrally-closed-domain) and proved uniqueness results for Henselizations<sup>[15](https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-theory-of-henselian-rings/D56C072E961F0E8101D75D535F8B9537)</sup>. Grothendieck's theory of étale morphisms later gave the notion its natural home, developing Nagata's theory of Henselian rings together with that of strictly local rings<sup>[16](https://jcreinhold.github.io/ega/iv/31-ch4-18-complements-etale-morphisms.html)</sup>.

## References

1. [Section 10.153: Henselian local rings, The Stacks Project](https://stacks.math.columbia.edu/tag/04GE)
2. [Heinzer, Lantz, and Shah, *Power Lines, Excellent Rings and Formal Fibers*, Section 8.3, Henselian rings](https://www.math.purdue.edu/~heinzer/preprints/power.pdf)
3. [Section 59.33: Stalks of the structure sheaf, The Stacks Project](https://stacks.math.columbia.edu/tag/04HW)
4. [Section 15.11: Henselian pairs, The Stacks Project](https://stacks.math.columbia.edu/tag/09XD)
5. [Alonso García, Lombardi, and Perdry (2022). Elementary Constructive Theory of Henselian Local Rings. arXiv.](https://arxiv.org/abs/2202.06595)
6. [Solovera-E Nery, Henselian local rings, lecture notes](https://www.esaga.uni-due.de/f/thiago.solovera-e-nery/notes/main_hensel.pdf)
7. [Grothendieck (1969). Algebraic approximation of structures over complete local rings. Publications Mathématiques de l'IHÉS 36.](https://www.numdam.org/article/PMIHES_1969__36__23_0.pdf)
8. [Lemma 15.12.3, The Stacks Project](https://stacks.math.columbia.edu/tag/0A03)
9. [Section 15.13: Lifting and henselian pairs, The Stacks Project](https://stacks.math.columbia.edu/tag/0D49)
10. [The Stacks Project, full book PDF, Section 15.11](https://stacks.math.columbia.edu/download/book.pdf)
11. [Henselian schemes in positive characteristic, arXiv](https://arxiv.org/abs/2212.08644)
12. [Clarifying the connection between 'étale locally' and 'formally locally', MathOverflow](https://mathoverflow.net/questions/250238/clarifying-the-connection-between-etale-locally-and-formally-locally)
13. [Section 16.13: The approximation property for G-rings, The Stacks Project](https://stacks.math.columbia.edu/tag/07QX)
14. [Hensel lemma, Encyclopedia of Mathematics](https://encyclopediaofmath.org/index.php?title=Hensel_lemma)
15. [Nagata, M. (1953). On the Theory of Henselian Rings. Nagoya Mathematical Journal 5, 45–57.](https://www.cambridge.org/core/journals/nagoya-mathematical-journal/article/on-the-theory-of-henselian-rings/D56C072E961F0E8101D75D535F8B9537)
16. [EGA IV, §18: Complements on étale morphisms, Henselian local rings (English translation)](https://jcreinhold.github.io/ega/iv/31-ch4-18-complements-etale-morphisms.html)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Algebraic structures › Ring theory › Commutative algebra › Completions, associated graded rings and power series*

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