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) |
Hensel ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Hensel_ring&oldid=47211