comm_alg/CommAlg/sayantan(poly_over_field).lean

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import CommAlg.krull
import Mathlib.RingTheory.Ideal.Operations
import Mathlib.Order.Height
import Mathlib.RingTheory.PrincipalIdealDomain
import Mathlib.RingTheory.DedekindDomain.Basic
import Mathlib.AlgebraicGeometry.PrimeSpectrum.Basic
namespace Ideal
/-- The ring of polynomials over a field has dimension one. -/
lemma polynomial_over_field_dim_one {K : Type} [Nontrivial K] [Field K] : krullDim (Polynomial K) = 1 := by
rw [le_antisymm_iff]
let X := @Polynomial.X K _
constructor
· unfold krullDim
apply @iSup_le (WithBot ℕ∞) _ _ _ _
intro I
have PIR : IsPrincipalIdealRing (Polynomial K) := by infer_instance
by_cases I = ⊥
· rw [← height_bot_iff_bot] at h
simp only [WithBot.coe_le_one, ge_iff_le]
rw [h]
exact bot_le
· push_neg at h
have : I.asIdeal ≠ ⊥ := by
by_contra a
have : I = ⊥ := PrimeSpectrum.ext I ⊥ a
contradiction
have maxI := IsPrime.to_maximal_ideal this
have singleton : ∀P, P ∈ {J | J < I} ↔ P = ⊥ := by
intro P
constructor
· intro H
simp only [Set.mem_setOf_eq] at H
by_contra x
push_neg at x
have : P.asIdeal ≠ ⊥ := by
by_contra a
have : P = ⊥ := PrimeSpectrum.ext P ⊥ a
contradiction
have maxP := IsPrime.to_maximal_ideal this
have IneTop := IsMaximal.ne_top maxI
have : P ≤ I := le_of_lt H
rw [←PrimeSpectrum.asIdeal_le_asIdeal] at this
have : P.asIdeal = I.asIdeal := Ideal.IsMaximal.eq_of_le maxP IneTop this
have : P = I := PrimeSpectrum.ext P I this
replace H : P ≠ I := ne_of_lt H
contradiction
· intro pBot
simp only [Set.mem_setOf_eq, pBot]
exact lt_of_le_of_ne bot_le h.symm
replace singleton : {J | J < I} = {⊥} := Set.ext singleton
unfold height
sorry
· suffices : ∃I : PrimeSpectrum (Polynomial K), 1 ≤ (height I : WithBot ℕ∞)
· obtain ⟨I, h⟩ := this
have : (height I : WithBot ℕ∞) ≤ ⨆ (I : PrimeSpectrum (Polynomial K)), ↑(height I) := by
apply @le_iSup (WithBot ℕ∞) _ _ _ I
exact le_trans h this
have primeX : Prime Polynomial.X := @Polynomial.prime_X K _ _
have : IsPrime (span {X}) := by
refine (span_singleton_prime ?hp).mpr primeX
exact Polynomial.X_ne_zero
let P := PrimeSpectrum.mk (span {X}) this
unfold height
use P
have : ⊥ ∈ {J | J < P} := by
simp only [Set.mem_setOf_eq, bot_lt_iff_ne_bot]
suffices : P.asIdeal ≠ ⊥
· by_contra x
rw [PrimeSpectrum.ext_iff] at x
contradiction
by_contra x
simp only [span_singleton_eq_bot] at x
have := @Polynomial.X_ne_zero K _ _
contradiction
have : {J | J < P}.Nonempty := Set.nonempty_of_mem this
rwa [←Set.one_le_chainHeight_iff, ←WithBot.one_le_coe] at this