Hensel ring

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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)

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 $f( a) = 0$.