Henselian ring

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In mathematics, a Henselian ring (or Hensel ring) is a local ring in which Hensel's lemma holds. They were defined by Azumaya (1951), who named them after Kurt Hensel.

Some standard references for Hensel rings are (Nagata 1962, Chapter VII), (Raynaud 1970), and (Grothendieck 1967, Chapter 18).

Contents

[edit] Definitions

In this article Henselian rings will be assumed to be commutative, though there is also a theory of non-commutative Henselian rings.

A commutative local ring R with maximal ideal m is called Henselian if Hensel's lemma holds. This means that if P is a monic polynomial, then any factorization of P in R/m into a product of coprime monic polynomials can be lifted to factorization in R.

A local ring is Henselian if and only if every finite ring extension is a product of local rings.

A Henselian ring is called strict, if its residue field is separably closed.

[edit] 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 "points" of the étale topology.

[edit] Henselization

For any local ring A there is a universal Henselian ring B generated by A, called the Henselization of A, 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. 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 then so is its Henselization.

Similarly there is a strictly Henselian ring generated by A, called the strict Henselization of A (which not quite universal: it is unique, but only up to non-unique isomorphism).

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 (the formal power series satisfying an algebraic equation). This can be thought of as the "algebraic" part of the completion.

Example A strict Henselization of the ring of p-adic integers 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.

[edit] Examples

  • Any field is a Henselian local ring.
  • Complete 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 Ared 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.

[edit] References