Henselian ring

In

Some standard references for Hensel rings are (Nagata 1975, Chapter VII), (Raynaud 1970), and (Grothendieck 1967, Chapter 18).

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
  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.
• By abuse of terminology, a field ${\displaystyle K}$ with valuation ${\displaystyle v}$ is said to be Henselian if its valuation ring is Henselian. That is the case if and only if ${\displaystyle v}$ extends uniquely to every finite extension of ${\displaystyle K}$ (resp. to every finite separable extension of ${\displaystyle K}$, resp. to ${\displaystyle K^{alg}}$, resp. to ${\displaystyle K^{sep}}$).
• A ring is called Henselian if it is a direct product of a finite number of Henselian local rings.

Properties

• Assume that ${\displaystyle (K,v)}$ is an Henselian field. Then every algebraic extension of ${\displaystyle K}$ is henselian (by the fourth definition above).
• If ${\displaystyle (K,v)}$ is a Henselian field and ${\displaystyle \alpha }$ is algebraic over ${\displaystyle K}$, then for every conjugate ${\displaystyle \alpha '}$ of ${\displaystyle \alpha }$ over ${\displaystyle K}$, ${\displaystyle v(\alpha ')=v(\alpha )}$. This follows from the fourth definition, and from the fact that for every K-automorphism ${\displaystyle \sigma }$ of ${\displaystyle K^{alg}}$, ${\displaystyle v\circ \sigma }$ is an extension of ${\displaystyle v|_{K}}$. The converse of this assertion also holds, because for a normal field extension ${\displaystyle L/K}$, the extensions of ${\displaystyle v}$ to ${\displaystyle L}$ are known to be conjugated.^[1]

Henselian rings in algebraic geometry

Henselian rings are the local rings with respect to the Nisnevich topology in the sense that if ${\displaystyle R}$ is a Henselian local ring, and ${\displaystyle \{U_{i}\to X\}}$ is a Nisnevich covering of ${\displaystyle X=Spec(R)}$, then one of the ${\displaystyle U_{i}\to X}$ is an isomorphism. This should be compared to the fact that for any Zariski open covering ${\displaystyle \{U_{i}\to X\}}$ of the spectrum ${\displaystyle X=Spec(R)}$ of a local ring ${\displaystyle R}$, one of the ${\displaystyle U_{i}\to X}$ 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

• Every field is a Henselian local ring. (But not every field with valuation is "Henselian" in the sense of the fourth definition above.)
• Complete
  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 then it is Henselian since A_red is a field.

References

• MR 0040287
• Danilov, V. I. (2001) [1994], "Hensel ring", Encyclopedia of Mathematics, EMS Press
• doi:10.1007/BF02732123
• Kurke, H.; Pfister, G.; Roczen, M. (1975), Henselsche Ringe und algebraische Geometrie, Mathematische Monographien, vol. II, Berlin:
  MR 0491694
• MR 0051821
• MR 0067865
• MR 0109835
• Nagata, Masayoshi (1975) [1962], Local rings, Interscience Tracts in Pure and Applied Mathematics, vol. 13 (reprint ed.), New York-London: Interscience Publishers a division of John Wiley & Sons, pp. xiii+234,
  MR 0155856
• Raynaud, Michel (1970), Anneaux locaux henséliens, Lecture Notes in Mathematics, vol. 169, Berlin-New York: Springer-Verlag, pp. v+129,
  MR 0277519