| 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 | |