Hensel ring

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 géométrie 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)