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A commutative local ring to which the [[Hensel lemma|Hensel lemma]] applies, or, according to another definition, to which the implicit function theorem applies. For a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469401.png" /> with maximal ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469402.png" /> the former definition means that for any unitary polynomial <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469403.png" /> and a simple solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469404.png" /> of the equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469405.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469406.png" />) (i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469407.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469408.png" />) there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h0469409.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694011.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694012.png" />).
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Examples of Hensel rings include complete local rings, rings of convergent power series (and, in a general sense, analytic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694013.png" /> which are algebraic over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694014.png" />). A local ring that is integral over a Hensel ring is a Hensel ring; in particular, a quotient ring of a Hensel ring is a Hensel ring. For any local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694015.png" /> there exists a general construct — a local Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694017.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694018.png" /> such that for any local Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694019.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694020.png" /> there exists a unique homomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694021.png" />-algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694022.png" />. The algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694023.png" /> of a local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694024.png" /> is a strictly-flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694025.png" />-module, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694026.png" /> will be a maximal ideal of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694027.png" />, the residue fields of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694029.png" /> are canonically isomorphic, and the completions of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694030.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694031.png" /> (in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694032.png" /> is a Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694033.png" />-algebra for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694034.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694035.png" /> is a Noetherian (or, respectively, reduced, normal, regular, excellent) ring, so is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694036.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694037.png" /> is an integral ring, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694038.png" /> need not be integral; more exactly, there exists a bijective correspondence between the maximal ideals of the integral closure of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694039.png" /> and the minimal prime ideals of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694040.png" />.
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A Hensel ring with a separably-closed residue field is called strictly local (or strictly Henselian), owing to the locality of its spectrum in the étale topology of schemes; in a manner similar to the construction of the Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694041.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694042.png" /> there is a strict Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694043.png" />-algebra functor <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694044.png" />. The concept of a Hensel ring may be introduced for a semi-local ring and even, in a more-general sense, for the pair ring–ideal.
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A commutative local ring to which the [[Hensel lemma|Hensel lemma]] applies, or, according to another definition, to which the implicit function theorem applies. For a local ring  $  A $
 +
with maximal ideal  $  \mathfrak m $
 +
the former definition means that for any unitary polynomial  $  P( X) \in A[ X] $
 +
and a simple solution  $  a _ {0} \in A $
 +
of the equation  $  P( X) = 0 $(
 +
$  \mathop{\rm mod}  \mathfrak m $)
 +
(i.e. $  P ( a _ {0} ) \in \mathfrak m $
 +
and  $  P  ^  \prime  ( a _ {0} ) \notin \mathfrak m $)
 +
there exists an  $  a \in A $
 +
such that  $  P( a) = 0 $
 +
and  $  a \equiv a _ {0} $(
 +
$  \mathop{\rm mod}  \mathfrak m $).
  
A Hensel ring may be described as a ring over which any finite algebra is a direct sum of local rings. Hensel rings were introduced in [[#References|[1]]]; the general theory of Hensel rings and the construction of Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694045.png" />-algebras are developed in [[#References|[2]]].
+
Examples of Hensel rings include complete local rings, rings of convergent power series (and, in a general sense, analytic rings, cf. [[Analytic ring|Analytic ring]]), and the ring of algebraic power series (i.e. series from  $  k [[ X _ {1} \dots X _ {n} ]] $
 +
which are algebraic over  $  k[ X _ {1} \dots X _ {n} ] $).
 +
A local ring that is integral over a Hensel ring is a Hensel ring; in particular, a quotient ring of a Hensel ring is a Hensel ring. For any local ring  $  A $
 +
there exists a general construct — a local Hensel  $  A $-
 +
algebra  $  {}  ^ {h} A $
 +
such that for any local Hensel  $  A $-
 +
algebra  $  B $
 +
there exists a unique homomorphism of  $  A $-
 +
algebras  $  {}  ^ {h} A \rightarrow B $.  
 +
The algebra  $  {}  ^ {h} A $
 +
of a local ring  $  A $
 +
is a strictly-flat  $  A $-
 +
module,  $  \mathfrak m {}  ^ {h} A $
 +
will be a maximal ideal of  $  {}  ^ {h} A $,
 +
the residue fields of  $  A $
 +
and  $  {}  ^ {h} A $
 +
are canonically isomorphic, and the completions of  $  A $
 +
and  $  {}  ^ {h} A $(
 +
in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in $  X _ {1} \dots X _ {n} $
 +
is a Hensel  $  A $-
 +
algebra for  $  {k [ X _ {1} \dots X _ {n} ] } _ {( X _ {1}  \dots X _ {n} ) } $.
 +
If  $  A $
 +
is a Noetherian (or, respectively, reduced, normal, regular, excellent) ring, so is  $  {}  ^ {h} A $.
 +
Conversely, if  $  A $
 +
is an integral ring,  $  {}  ^ {h} A $
 +
need not be integral; more exactly, there exists a bijective correspondence between the maximal ideals of the integral closure of  $  A $
 +
and the minimal prime ideals of  $  {}  ^ {h} A $.
  
In the theory of étale morphisms and étale topology a Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694046.png" />-algebra is understood to be the inductive limit of étale extensions of the ring. In a commutative algebra a Hensel <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694047.png" />-algebra often replaces the operation of completion, which plays an important role in local studies of objects.
+
A Hensel ring with a separably-closed residue field is called strictly local (or strictly Henselian), owing to the locality of its spectrum in the étale topology of schemes; in a manner similar to the construction of the Hensel  $  A $-
 +
algebra  $  {}  ^ {h} A $
 +
there is a strict Hensel  $  A $-
 +
algebra functor  $  {}  ^ {sh} A $.
 +
The concept of a Hensel ring may be introduced for a semi-local ring and even, in a more-general sense, for the pair ring–ideal.
 +
 
 +
A Hensel ring may be described as a ring over which any finite algebra is a direct sum of local rings. Hensel rings were introduced in [[#References|[1]]]; the general theory of Hensel rings and the construction of Hensel  $  A $-
 +
algebras are developed in [[#References|[2]]].
 +
 
 +
In the theory of étale morphisms and étale topology a Hensel $  A $-
 +
algebra is understood to be the inductive limit of étale extensions of the ring. In a commutative algebra a Hensel $  A $-
 +
algebra often replaces the operation of completion, which plays an important role in local studies of objects.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Azumaya,  "On maximally central algebras"  ''Nagoya Math. J.'' , '''2'''  (1951)  pp. 119–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géometrie algébrique. IV"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  G. Azumaya,  "On maximally central algebras"  ''Nagoya Math. J.'' , '''2'''  (1951)  pp. 119–150</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  M. Nagata,  "Local rings" , Interscience  (1962)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A. Grothendieck,  "Eléments de géometrie algébrique. IV"  ''Publ. Math. IHES'' :  32  (1967)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
The ring (algebra) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694048.png" /> is called the Henselization or Hensel closure of the local ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694049.png" />.
+
The ring (algebra) $  {}  ^ {sh} A $
 +
is called the Henselization or Hensel closure of the local ring $  A $.
  
The ideal–ring pair formulation of the Hensel property is as follows. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694050.png" /> be a pair consisting of a ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694051.png" /> and an ideal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694052.png" />. Then if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694053.png" /> is such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694055.png" /> is a unit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694056.png" />, then there exists an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694057.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046940/h04694058.png" />.
+
The ideal–ring pair formulation of the Hensel property is as follows. Let $  ( A , I) $
 +
be a pair consisting of a ring $  A $
 +
and an ideal $  I $.  
 +
Then if $  f \in A[ I] $
 +
is such that $  f( 0) \in I $
 +
and $  f ^ { \prime } ( 0) $
 +
is a unit in $  A / I $,  
 +
then there exists an $  a \in I $
 +
such that $  f( a) = 0 $.
  
 
For a discussion of the solution of systems of polynomial equations and implicit-function type statements in the context of Hensel rings cf., for instance, [[#References|[a2]]], Chapt. 2.
 
For a discussion of the solution of systems of polynomial equations and implicit-function type statements in the context of Hensel rings cf., for instance, [[#References|[a2]]], Chapt. 2.

Latest revision as of 22:10, 5 June 2020


A commutative local ring to which the Hensel lemma applies, or, according to another definition, to which the implicit function theorem applies. For a local ring $ A $ with maximal ideal $ \mathfrak m $ the former definition means that for any unitary polynomial $ P( X) \in A[ X] $ and a simple solution $ a _ {0} \in A $ of the equation $ P( X) = 0 $( $ \mathop{\rm mod} \mathfrak m $) (i.e. $ P ( a _ {0} ) \in \mathfrak m $ and $ P ^ \prime ( a _ {0} ) \notin \mathfrak m $) there exists an $ a \in A $ such that $ P( a) = 0 $ and $ a \equiv a _ {0} $( $ \mathop{\rm mod} \mathfrak m $).

Examples of Hensel rings include complete local rings, rings of convergent power series (and, in a general sense, analytic rings, cf. Analytic ring), and the ring of algebraic power series (i.e. series from $ k [[ X _ {1} \dots X _ {n} ]] $ which are algebraic over $ k[ X _ {1} \dots X _ {n} ] $). A local ring that is integral over a Hensel ring is a Hensel ring; in particular, a quotient ring of a Hensel ring is a Hensel ring. For any local ring $ A $ there exists a general construct — a local Hensel $ A $- algebra $ {} ^ {h} A $ such that for any local Hensel $ A $- algebra $ B $ there exists a unique homomorphism of $ A $- algebras $ {} ^ {h} A \rightarrow B $. The algebra $ {} ^ {h} A $ of a local ring $ A $ is a strictly-flat $ A $- module, $ \mathfrak m {} ^ {h} A $ will be a maximal ideal of $ {} ^ {h} A $, the residue fields of $ A $ and $ {} ^ {h} A $ are canonically isomorphic, and the completions of $ A $ and $ {} ^ {h} A $( in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in $ X _ {1} \dots X _ {n} $ is a Hensel $ A $- algebra for $ {k [ X _ {1} \dots X _ {n} ] } _ {( X _ {1} \dots X _ {n} ) } $. If $ A $ is a Noetherian (or, respectively, reduced, normal, regular, excellent) ring, so is $ {} ^ {h} A $. Conversely, if $ A $ is an integral ring, $ {} ^ {h} A $ need not be integral; more exactly, there exists a bijective correspondence between the maximal ideals of the integral closure of $ A $ and the minimal prime ideals of $ {} ^ {h} A $.

A Hensel ring with a separably-closed residue field is called strictly local (or strictly Henselian), owing to the locality of its spectrum in the étale topology of schemes; in a manner similar to the construction of the Hensel $ A $- algebra $ {} ^ {h} A $ there is a strict Hensel $ A $- algebra functor $ {} ^ {sh} A $. The concept of a Hensel ring may be introduced for a semi-local ring and even, in a more-general sense, for the pair ring–ideal.

A Hensel ring may be described as a ring over which any finite algebra is a direct sum of local rings. Hensel rings were introduced in [1]; the general theory of Hensel rings and the construction of Hensel $ A $- algebras are developed in [2].

In the theory of étale morphisms and étale topology a Hensel $ A $- algebra is understood to be the inductive limit of étale extensions of the ring. In a commutative algebra a Hensel $ A $- algebra often replaces the operation of completion, which plays an important role in local studies of objects.

References

[1] G. Azumaya, "On maximally central algebras" Nagoya Math. J. , 2 (1951) pp. 119–150
[2] M. Nagata, "Local rings" , Interscience (1962)
[3] A. Grothendieck, "Eléments de géometrie algébrique. IV" Publ. Math. IHES : 32 (1967)

Comments

The ring (algebra) $ {} ^ {sh} A $ is called the Henselization or Hensel closure of the local ring $ A $.

The ideal–ring pair formulation of the Hensel property is as follows. Let $ ( A , I) $ be a pair consisting of a ring $ A $ and an ideal $ I $. Then if $ f \in A[ I] $ is such that $ f( 0) \in I $ and $ f ^ { \prime } ( 0) $ is a unit in $ A / I $, then there exists an $ a \in I $ such that $ f( a) = 0 $.

For a discussion of the solution of systems of polynomial equations and implicit-function type statements in the context of Hensel rings cf., for instance, [a2], Chapt. 2.

References

[a1] M. Raynaud, "Anneaux locaux Henséliens" , Lect. notes in math. , 169 , Springer (1970)
[a2] H. Kurke, G. Pfister, M. Roczen, "Henselsche Ringe" , Deutsch. Verlag Wissenschaft. (1975)
How to Cite This Entry:
Hensel ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hensel_ring&oldid=17423
This article was adapted from an original article by V.I. Danilov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article