Documentation

Mathlib.RingTheory.Henselian

Henselian rings #

In this file we set up the basic theory of Henselian (local) rings. A ring R is Henselian at an ideal I if the following conditions hold:

(Here, saying that a root b of a polynomial g is simple means that g.derivative.eval b is a unit. Warning: if R/I is not a field then it is not enough to assume that g has a factorization into monic linear factors in which X - b shows up only once; for example 1 is not a simple root of X^2-1 over ℤ/4ℤ.)

A local ring R is Henselian if it is Henselian at its maximal ideal. In this case the first condition is automatic, and in the second condition we may ask for f.derivative.eval a ≠ 0, since the quotient ring R/I is a field in this case.

Main declarations #

References #

https://stacks.math.columbia.edu/tag/04GE

TODO #

After a good API for etale ring homomorphisms has been developed, we can give more equivalent characterization of Henselian rings.

In particular, this can give a proof that factorizations into coprime polynomials can be lifted from the residue field to the Henselian ring.

The following gist contains some code sketches in that direction. https://gist.github.com/jcommelin/47d94e4af092641017a97f7f02bf9598

class HenselianRing (R : Type u_1) [CommRing R] (I : Ideal R) :

A ring R is Henselian at an ideal I if the following condition holds: for every polynomial f over R, with a simple root a₀ over the quotient ring R/I, there exists a lift a : R of a₀ that is a root of f.

(Here, saying that a root b of a polynomial g is simple means that g.derivative.eval b is a unit. Warning: if R/I is not a field then it is not enough to assume that g has a factorization into monic linear factors in which X - b shows up only once; for example 1 is not a simple root of X^2-1 over ℤ/4ℤ.)

Instances
    theorem HenselianRing.jac {R : Type u_1} [CommRing R] {I : Ideal R} [self : HenselianRing R I] :
    I .jacobson
    theorem HenselianRing.is_henselian {R : Type u_1} [CommRing R] {I : Ideal R} [self : HenselianRing R I] (f : Polynomial R) :
    f.Monic∀ (a₀ : R), Polynomial.eval a₀ f IIsUnit ((Ideal.Quotient.mk I) (Polynomial.eval a₀ (Polynomial.derivative f)))∃ (a : R), f.IsRoot a a - a₀ I
    class HenselianLocalRing (R : Type u_1) [CommRing R] extends LocalRing :

    A local ring R is Henselian if the following condition holds: for every polynomial f over R, with a simple root a₀ over the residue field, there exists a lift a : R of a₀ that is a root of f. (Recall that a root b of a polynomial g is simple if it is not a double root, so if g.derivative.eval b ≠ 0.)

    In other words, R is local Henselian if it is Henselian at the ideal I, in the sense of HenselianRing.

    Instances
      theorem HenselianLocalRing.is_henselian {R : Type u_1} [CommRing R] [self : HenselianLocalRing R] (f : Polynomial R) :
      f.Monic∀ (a₀ : R), Polynomial.eval a₀ f LocalRing.maximalIdeal RIsUnit (Polynomial.eval a₀ (Polynomial.derivative f))∃ (a : R), f.IsRoot a a - a₀ LocalRing.maximalIdeal R
      @[instance 100]
      instance Field.henselian (K : Type u_1) [Field K] :
      Equations
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      theorem HenselianLocalRing.TFAE (R : Type u) [CommRing R] [LocalRing R] :
      [HenselianLocalRing R, ∀ (f : Polynomial R), f.Monic∀ (a₀ : LocalRing.ResidueField R), (Polynomial.aeval a₀) f = 0(Polynomial.aeval a₀) (Polynomial.derivative f) 0∃ (a : R), f.IsRoot a (LocalRing.residue R) a = a₀, ∀ {K : Type u} [inst : Field K] (φ : R →+* K), Function.Surjective φ∀ (f : Polynomial R), f.Monic∀ (a₀ : K), Polynomial.eval₂ φ a₀ f = 0Polynomial.eval₂ φ a₀ (Polynomial.derivative f) 0∃ (a : R), f.IsRoot a φ a = a₀].TFAE
      @[instance 100]

      A ring R that is I-adically complete is Henselian at I.

      Equations
      • =