In 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.

Properties

Henselian rings in algebraic geometry

Henselian rings are the local rings with respect to the Nisnevich topology in the sense that if R is a Henselian local ring, and is a Nisnevich covering of X = Spec(R), then one of the U_i \to X is an isomorphism. This should be compared to the fact that for any Zariski open covering of the spectrum X = Spec(R) of a local ring R, one of the U_i \to X is an isomorphism. In fact, this property characterises Henselian rings, resp. local rings. 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 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. 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 (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 of order prime to p. It is not "universal" as it has non-trivial automorphisms.

Examples