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Henselian ring

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Idea

A Henselian ring is a local ring for which the conclusion of Hensel's lemma? (classically stated for a complete local ring?) holds.

Definition

Definition

Let RR be a local ring with maximal ideal mm. Let k=R/mk = R/m be the residue class field, and for a polynomial fR[x]f \in R[x], let f¯k[x]\bar{f} \in k[x] obtained by reduction of the coefficients of ff modulo mm. Then RR is Henselian if, for any monic fR[x]f \in R[x] such that f¯\bar{f} factorizes as g 0h 0g_0 h_0 with g 0,h 0g_0, h_0 monic and relatively prime in k[x]k[x], there exist monic g,hR[x]g, h \in R[x] such that f=ghf = g h with g¯=g 0\bar{g} = g_0, h¯=h 0\bar{h} = h_0, and the ideal (g,h)(g, h) is the unit ideal in R[x]R[x].

Often this definition is given just for the case when one of g 0g_0, h 0h_0 is a linear factor xax - a, where the idea is that the (simple) root aa can be lifted to RR.

Definition

A Henselian ring RR with residue class field kk is strictly Henselian if kk is separably closed.

Examples

  • Any field kk is trivially Henselian.

  • A complete local ring, such as p-adic number rings, is Henselian. This includes rings of formal power series over a field, k[[x 1,,x n]]k[ [x_1, \ldots, x_n] ].

  • Rings of convergent power series over a local field are Henselian.

Properties

Proposition

The quotient of a Henselian ring is also Henselian.

Proof

In the first place, a quotient RR' of a local ring RR is local (and has the same residue class field kk). Secondly, RR' is clearly Henselian since we can lift factorizations f¯=g 0h 0\bar{f} = g_0 h_0 in k[x]k[x] to factorizations in R[x]R[x], and then push them down to R[x]R'[x].

Proposition

If RR is local and its reduced ring (i.e., RR modulo its ideal of nilpotent elements) is Henselian, then RR itself is Henselian.

Proposition

If RR is Henselian and SS is a local ring that is integral over RR (meaning that SS is an RR-algebra and each xSx \in S is an integral element over RR), then SS is Henselian.

Henselization

Let LocRingLocRing denote the category of local rings (commutative of course) and local ring homomorphisms (f:RSf: R \to S is local if the pullback along ff of the maximal ideal of SS is the maximal ideal of RR).

Theorem

The full subcategory of LocRingLocRing consisting of Henselian rings is reflective. The left adjoint to the full inclusion is called Henselization.

Example

Let RR be a discrete valuation ring, with KK its field of fractions. Let R^\hat{R} be the mm-adic completion of RR with respect to its maximal ideal. Then the Henselization of RR is isomorphic to the subring of R^\hat{R} whose elements are roots of separable polynomials with coefficients in KK.

A rule of thumb, as suggested by this example, is that the Henselization is the algebraic part of a local ring completion.

References

An original reference is

  • Alexandre Grothendieck, (1967), EGA: IV. Étude locale des schémas et des morphismes de schémas , Quatrième partie“, Publications Mathématiques de l’IHÉS 32: 5–361. (web)

Lecture notes are in

Other sources include

  • Alonso, Lombardi, Perdry, Henselian local rings (pdf)

  • Remarks on Henselian rings (pdf);

  • Ieke Moerdijk, Rings of smooth functions and their localizations (pdf)

Revised on November 22, 2013 03:27:26 by Urs Schreiber (77.251.114.72)