The Teichmüller lift of a Henselian local ring with finite residue field #
Let R be a Henselian local ring whose residue field k is finite, of cardinality q. Reduction
Rˣ → kˣ identifies the (q - 1)-st roots of unity on both sides, because q - 1 is a
unit in R. Since every unit of k is a (q - 1)-st root of unity, the inverse equivalence gives
the Teichmüller lift teichmuller R : kˣ →* Rˣ.
Thus this construction reuses the general Henselian roots-of-unity equivalence. Its public
characterization says that the lift of x is the unique (q - 1)-st root of unity reducing to
x; in particular its image is exactly μ_{q-1}(R), and it is the only multiplicative section
of reduction.
Since the lift is a section of reduction, Rˣ is the internal direct product of μ_{q-1}(R)
and the kernel 1 + 𝔪 of reduction on units, the principal units: this is the Teichmüller
splitting Rˣ ≃* μ_{q-1}(R) × (1 + 𝔪), whose inverse is multiplication.
A ring R that is moreover integrally closed in an R-algebra A has the same (q - 1)-st
roots of unity as A, since roots of unity are integral. This gives the corresponding
identification μ_{q-1}(A) ≃* kˣ; the case of a fraction ring of R is the one used for local
fields.
Main results #
TauCeti.teichmuller: the Teichmüller liftkˣ →* Rˣ.TauCeti.teichmuller_eq_iff: the characterization of its values.TauCeti.eq_teichmuller: it is the only multiplicative section of reduction.TauCeti.range_teichmuller: its image is exactlyμ_{q-1}(R).TauCeti.rootsOfUnityMulEquivUnitsResidueField: reduction is an isomorphismμ_{q-1}(R) ≃* kˣ.TauCeti.isComplement'_rootsOfUnity_ker_unitsMap_residue,TauCeti.unitsMulEquivRootsOfUnityProdKerResidue: the Teichmüller splittingRˣ ≃* μ_{q-1}(R) × (1 + 𝔪).TauCeti.rootsOfUnityAlgebraMulEquivUnitsResidueField: ifRis integrally closed in anR-algebraA, thenμ_{q-1}(A) ≃* kˣ.
References #
- J.-P. Serre, Corps Locaux, II §4.
- J. Neukirch, Algebraic Number Theory, II §5.
A finite field has at least two elements, so the exponent q - 1 is nonzero.
Reduction identifies the (q - 1)-st roots of unity of R with the units of its residue
field.
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Instances For
The Teichmüller lift of a Henselian local ring R with finite residue field k of
cardinality q: the multiplicative section of reduction that sends x to the unique
(q - 1)-st root of unity above x.
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The Teichmüller lift is the inverse of the roots-of-unity equivalence, read in Rˣ. This
unfolds the definition of teichmuller; stating it once lets the proofs below work with the
equivalence.
The Teichmüller lift takes its values in the (q - 1)-st roots of unity.
The simplifier-normalized form of the torsion property of the Teichmüller lift, stated with
Fintype.card rather than Nat.card.
The Teichmüller lift is a section of reduction.
The Teichmüller lift is a section of reduction, read in the unit group of the residue field.
The Teichmüller lift of x is the unique root of unity of order dividing q - 1 above
x.
The Teichmüller lift is injective, being a section of reduction.
The Teichmüller lift is the only multiplicative section of reduction. No torsion assumption on
the section is needed: a monoid hom out of kˣ, a group of order q - 1, automatically takes
(q - 1)-st roots of unity as values.
The image of the Teichmüller lift is μ_{q-1}(R).
A Henselian local ring with residue field of cardinality q has exactly q - 1 roots of
unity of order dividing q - 1.
The Teichmüller splitting of the unit group #
The Teichmüller lift is a section of reduction Rˣ → kˣ, so Rˣ is the internal direct product
of μ_{q-1}(R), the image of the lift, and the kernel 1 + 𝔪 of reduction on units, the
principal units.
The Teichmüller splitting, as complementary subgroups: every unit of R is uniquely the
product of a (q - 1)-st root of unity and a principal unit, that is a unit reducing to 1 in
the residue field.
The Teichmüller splitting of the unit group, Rˣ ≃* μ_{q-1}(R) × (1 + 𝔪): a unit u
goes to the Teichmüller representative ω(ū) of its residue class together with the principal
unit ω(ū)⁻¹ * u, and the inverse is multiplication.
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- One or more equations did not get rendered due to their size.
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The inverse of the Teichmüller splitting is multiplication, (ζ, v) ↦ ζ * v.
The root-of-unity component of a unit u is the Teichmüller representative of its residue
class.
The principal-unit component of a unit u is u divided by the Teichmüller representative
of its residue class.
If R is integrally closed in an R-algebra A, reduction identifies μ_{q-1}(A) with
the unit group of the residue field: the roots of unity of A are integral over R, hence
already lie in R.
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- One or more equations did not get rendered due to their size.
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The forward direction of the equivalence, read in A: the Teichmüller lift of the value at a
root of unity u of A is u itself.
The value of the equivalence at a root of unity u of A is the reduction of the element of
R that u comes from.