In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Henselian ring (or Hensel ring) is a
local ring in which
Hensel's lemma holds. They were introduced by , who named them after
Kurt Hensel. Azumaya originally allowed Henselian rings to be
non-commutative, but most authors now restrict them to be
commutative.
Some standard references for Hensel rings are , , and .
Definitions
In this article
rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.
* A local ring ''R'' with
maximal ideal ''m'' is called Henselian if Hensel's lemma holds. This means that if ''P'' is a
monic polynomial in ''R''
'x'' then any factorization of its image ''P'' in (''R''/''m'')
'x''into a product of
coprime 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 of local rings.
* A Henselian local ring is called strictly Henselian if its
residue field is
separably closed In field theory, a branch of algebra, an algebraic field extension E/F is called a separable extension if for every \alpha\in E, the minimal polynomial of \alpha over is a separable polynomial (i.e., its formal derivative is not the zero polynom ...
.
* By
abuse of terminology
In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not entirely formally correct, but which might help simplify the exposition or suggest the correct intuition (while possibly minimizing errors an ...
, a
field with
valuation is said to be Henselian if its
valuation ring 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, 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.
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
Local may refer to:
Geography and transportation
* Local (train), a train serving local traffic demand
* Local, Missouri, a community in the United States
* Local government, a form of public administration, usually the lowest tier of administrat ...
from ''A'' to a Henselian ring can be extended uniquely to ''B''. The Henselization of ''A'' is unique
up to Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R''
* if ''a'' and ''b'' are related by ''R'', that is,
* if ''aRb'' holds, that is,
* if the equivalence classes of ''a'' and ''b'' wi ...
unique
isomorphism. 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 over ''A''. If ''A'' is
Noetherian,
reduced, normal, regular, or
excellent
Excellent may refer to:
* ''Excellent'' (album), by Propaganda, 2012
* "Excellent", a song by Sunday Service Choir from the 2019 album ''Jesus Is Born''
* "Excellent", a catchphrase of Mr. Burns in the cartoon ''The Simpsons''
* "Excellent!," a c ...
then so is its Henselization. For example, the Henselization of the
ring of polynomials ''k''
'x'',''y'',... localized at the point (0,0,...) is the ring of algebraic
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
(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
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
s 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 of order prime to ''p''. It is not "universal" as it has
non-trivial 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 or
complex numbers are Henselian.
*Rings of algebraic power series over a field are Henselian.
*A local ring that is
integral over a Henselian ring is Henselian.
*The Henselization of a local ring is a Henselian local ring.
*Every
quotient of a Henselian ring is Henselian.
*A ring ''A'' is Henselian if and only if the associated
reduced ring ''A''
red is Henselian (this is the quotient of ''A'' by the
ideal of nilpotent elements).
*If ''A'' has only one
prime ideal
In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...
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