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 with maximal ideal the former definition means that for any unitary polynomial and a simple solution of the equation () (i.e. and ) there exists an such that and ().

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 which are algebraic over ). 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 there exists a general construct — a local Hensel -algebra such that for any local Hensel -algebra there exists a unique homomorphism of -algebras . The algebra of a local ring is a strictly-flat -module, will be a maximal ideal of , the residue fields of and are canonically isomorphic, and the completions of and (in the topologies of the local rings) coincide. Thus, the ring of algebraic power series in is a Hensel -algebra for . If is a Noetherian (or, respectively, reduced, normal, regular, excellent) ring, so is . Conversely, if is an integral ring, need not be integral; more exactly, there exists a bijective correspondence between the maximal ideals of the integral closure of and the minimal prime ideals of .

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 -algebra there is a strict Hensel -algebra functor . 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 -algebras are developed in [2].

In the theory of étale morphisms and étale topology a Hensel -algebra is understood to be the inductive limit of étale extensions of the ring. In a commutative algebra a Hensel -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) is called the Henselization or Hensel closure of the local ring .

The ideal–ring pair formulation of the Hensel property is as follows. Let be a pair consisting of a ring and an ideal . Then if is such that and is a unit in , then there exists an such that .

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)