File size: 993 Bytes
b797a5b | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 | import OAI.NumberTheory.PiExponent.LocalAlgebra.RegularSequenceLowExt
import OAI.NumberTheory.PiExponent.Polynomials.PolynomialLocalRegularParameters
namespace OAI
noncomputable section
namespace PiExponentSiegelAux.W30
open CategoryTheory CategoryTheory.Abelian
universe u
theorem polynomialLocal_residueExt_below_height
(K : Type u) [Field K] (n : ℕ)
(Q : Ideal (MvPolynomial (Fin n) K)) [Q.IsPrime]
(i : ℕ) (hi : (i : ℕ∞) < Q.height) :
Subsingleton (Ext
(ModuleCat.of (Localization.AtPrime Q)
(IsLocalRing.ResidueField (Localization.AtPrime Q)))
(ModuleCat.of (Localization.AtPrime Q) (Localization.AtPrime Q)) i) := by
obtain ⟨h, hh, xs, hlen, hreg, hmax⟩ :=
PiExponentJets.PolynomialLocalResidueResolution.exists_regularParameters_actual_height K n Q
apply residueFieldExt_below_regular_length xs hreg hmax i
rw [hlen]
exact ENat.natCast_lt_natCast.mp (by simpa only [hh] using hi)
end PiExponentSiegelAux.W30
end
end OAI
|