In
mathematics, a Henselian ring (or Hensel ring) is a
local ring In abstract algebra, more specifically ring theory, local rings are certain rings that are comparatively simple, and serve to describe what is called "local behaviour", in the sense of functions defined on varieties or manifolds, or of algebraic n ...
in which
Hensel's lemma In mathematics, Hensel's lemma, also known as Hensel's lifting lemma, named after Kurt Hensel, is a result in modular arithmetic, stating that if a univariate polynomial has a simple root modulo a prime number , then this root can be ''lifted'' to ...
holds. They were introduced by , who named them after
Kurt Hensel
Kurt Wilhelm Sebastian Hensel (29 December 1861 – 1 June 1941) was a German mathematician born in Königsberg.
Life and career
Hensel was born in Königsberg, East Prussia (today Kaliningrad, Russia), the son of Julia (née von Adelson) and lan ...
. Azumaya originally allowed Henselian rings to be
non-commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
, but most authors now restrict them to be
commutative
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
.
Some standard references for Hensel rings are , , and .
Definitions
In this article
rings
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.
* A local ring ''R'' with
maximal ideal
In mathematics, more specifically in ring theory, a maximal ideal is an ideal that is maximal (with respect to set inclusion) amongst all ''proper'' ideals. In other words, ''I'' is a maximal ideal of a ring ''R'' if there are no other ideals c ...
''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a
monic polynomial
In algebra, a monic polynomial is a single-variable polynomial (that is, a univariate polynomial) in which the leading coefficient (the nonzero coefficient of highest degree) is equal to 1. Therefore, a monic polynomial has the form:
:x^n+c_x^+\c ...
in ''R''
'x'' then any factorization of its image ''P'' in (''R''/''m'')
'x''into a product of
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
monic polynomials can be lifted to a factorization in ''R''
'x''
* A local ring is Henselian if and only if every finite ring extension is a
product
Product may refer to:
Business
* Product (business), an item that serves as a solution to a specific consumer problem.
* Product (project management), a deliverable or set of deliverables that contribute to a business solution
Mathematics
* Produ ...
of local rings.
* A Henselian local ring is called strictly Henselian if its
residue field In mathematics, the residue field is a basic construction in commutative algebra. If ''R'' is a commutative ring and ''m'' is a maximal ideal, then the residue field is the quotient ring ''k'' = ''R''/''m'', which is a field. Frequently, ''R'' is a ...
is
separably closed.
* By
abuse of terminology, a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
with
valuation is said to be Henselian if its
valuation ring In abstract algebra, a valuation ring is an integral domain ''D'' such that for every element ''x'' of its field of fractions ''F'', at least one of ''x'' or ''x''−1 belongs to ''D''.
Given a field ''F'', if ''D'' is a subring of ''F'' suc ...
is Henselian. That is the case if and only if
extends uniquely to every finite extension of
(resp. to every finite separable extension of
, resp. to
, resp. to
).
* A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.
Properties
* Assume that
is an Henselian field. Then every algebraic extension of
is henselian (by the fourth definition above).
* If
is a Henselian field and
is algebraic over
, then for every
conjugate of
over
,
. This follows from the fourth definition, and from the fact that for every K-automorphism
of
,
is an extension of
. The converse of this assertion also holds, because for a normal field extension
, the extensions of
to
are known to be conjugated.
[A. J. Engler, A. Prestel, ''Valued fields'', Springer monographs of mathematics, 2005, thm. 3.2.15, p. 69.]
Henselian rings in algebraic geometry
Henselian rings are the local rings of "points" with respect to the
Nisnevich topology In algebraic geometry, the Nisnevich topology, sometimes called the completely decomposed topology, is a Grothendieck topology on the category of schemes which has been used in algebraic K-theory, A¹ homotopy theory, and the theory of motives. ...
, so the
spectra of these rings do not admit non-trivial connected coverings with respect to the Nisnevich topology. Likewise strict Henselian rings are the local rings of geometric points in the
étale topology In algebraic geometry, the étale topology is a Grothendieck topology on the category of schemes which has properties similar to the Euclidean topology, but unlike the Euclidean topology, it is also defined in positive characteristic. The étale ...
.
Henselization
For any local ring ''A'' there is a universal Henselian ring ''B'' generated by ''A'', called the Henselization of ''A'', introduced by , such that any
local homomorphism from ''A'' to a Henselian ring can be extended uniquely to ''B''. The Henselization of ''A'' is unique
up to unique
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
. The Henselization of ''A'' is an algebraic substitute for the
completion of ''A''. The Henselization of ''A'' has the same completion and residue field as ''A'' and is a
flat module
In algebra, a flat module over a ring ''R'' is an ''R''-module ''M'' such that taking the tensor product over ''R'' with ''M'' preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact se ...
over ''A''. If ''A'' is
Noetherian In mathematics, the adjective Noetherian is used to describe objects that satisfy an ascending or descending chain condition on certain kinds of subobjects, meaning that certain ascending or descending sequences of subobjects must have finite lengt ...
,
reduced, normal, regular, or
excellent then so is its Henselization. For example, the Henselization of the
ring of polynomials
In mathematics, especially in the field of algebra, a polynomial ring or polynomial algebra is a ring (which is also a commutative algebra) formed from the set of polynomials in one or more indeterminates (traditionally also called variables ...
''k''
'x'',''y'',... localized at the point (0,0,...) is the ring of algebraic
formal power series (the formal power series satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.
Similarly there is a strictly Henselian ring generated by ''A'', called the strict Henselization of ''A''. The strict Henselization is not quite universal: it is unique, but only up to ''non-unique'' isomorphism. More precisely it depends on the choice of a separable algebraic closure of the residue field of ''A'', and
automorphisms of this separable algebraic closure correspond to automorphisms of the corresponding strict Henselization. For example, a strict Henselization of the field of
''p''-adic numbers is given by the maximal unramified extension, generated by all
roots of unity
In mathematics, a root of unity, occasionally called a de Moivre number, is any complex number that yields 1 when raised to some positive integer power . Roots of unity are used in many branches of mathematics, and are especially important in ...
of order prime to ''p''. It is not "universal" as it has
non-trivial In mathematics, the adjective trivial is often used to refer to a claim or a case which can be readily obtained from context, or an object which possesses a simple structure (e.g., groups, topological spaces). The noun triviality usually refers to a ...
automorphisms.
Examples
*Every field is a Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.)
*Complete
Hausdorff local rings, such as the ring of
''p''-adic integers and rings of formal power series over a field, are Henselian.
*The rings of convergent power series over the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
or
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the fo ...
s are Henselian.
*Rings of algebraic power series over a field are Henselian.
*A local ring that is
integral
In mathematics, an integral assigns numbers to functions in a way that describes displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding integrals is called integration. Along wit ...
over a Henselian ring is Henselian.
*The Henselization of a local ring is a Henselian local ring.
*Every
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of a Henselian ring is Henselian.
*A ring ''A'' is Henselian if and only if the associated
reduced ring In ring theory, a branch of mathematics, a ring is called a reduced ring if it has no non-zero nilpotent elements. Equivalently, a ring is reduced if it has no non-zero elements with square zero, that is, ''x''2 = 0 implies ''x'' = ...
''A''
red is Henselian (this is the quotient of ''A'' by the
ideal of nilpotent elements).
*If ''A'' has only one
prime ideal then it is Henselian since ''A''
red is a field.
References
*
*
*
*
*
*
*
*
*{{citation, last= Raynaud, first= Michel, title= Anneaux locaux henséliens, series= Lecture Notes in Mathematics, volume= 169 , publisher=Springer-Verlag, publication-place= Berlin-New York, year= 1970 , pages=v+129, doi=10.1007/BFb0069571
, isbn =978-3-540-05283-8, mr= 0277519
Commutative algebra