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moved dim_eq_bot_iff to krull.lean

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GTBarkley 2023-06-13 20:34:02 +00:00
parent afeeeb506f
commit f63286aff8
2 changed files with 29 additions and 4 deletions
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7
CommAlg/grant.lean
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@ -95,7 +95,7 @@ lemma krullDim_nonneg_of_nontrivial [Nontrivial R] : ∃ n : ℕ∞, Ideal.krull
-- lemma krullDim_ge_iff' (R : Type _) [CommRing R] {n : WithBot ℕ∞} : -- lemma krullDim_ge_iff' (R : Type _) [CommRing R] {n : WithBot ℕ∞} :
-- Ideal.krullDim R ≥ n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1 := sorry -- Ideal.krullDim R ≥ n ↔ ∃ c : List (PrimeSpectrum R), c.Chain' (· < ·) ∧ c.length = n + 1 := sorry
lemma prime_elim_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False := lemma primeSpectrum_empty_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False :=
x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem
lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
@ -107,15 +107,16 @@ lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R :=
by_contra hneg by_contra hneg
rw [not_isEmpty_iff] at hneg rw [not_isEmpty_iff] at hneg
rcases hneg with ⟨a, ha⟩ rcases hneg with ⟨a, ha⟩
exact prime_elim_of_subsingleton R ⟨a, ha⟩ exact primeSpectrum_empty_of_subsingleton R ⟨a, ha⟩
/-- A ring has Krull dimension -∞ if and only if it is the zero ring -/
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
unfold Ideal.krullDim unfold Ideal.krullDim
rw [←primeSpectrum_empty_iff, iSup_eq_bot] rw [←primeSpectrum_empty_iff, iSup_eq_bot]
constructor <;> intro h constructor <;> intro h
. rw [←not_nonempty_iff] . rw [←not_nonempty_iff]
rintro ⟨a, ha⟩ rintro ⟨a, ha⟩
specialize h ⟨a, ha⟩ -- specialize h ⟨a, ha⟩
tauto tauto
. rw [h.forall_iff] . rw [h.forall_iff]
trivial trivial

26
CommAlg/krull.lean
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@ -62,7 +62,31 @@ lemma krullDim_eq_height [LocalRing R] : krullDim R = height (closedPoint R) :=
#check height_le_krullDim #check height_le_krullDim
--some propositions that would be nice to be able to eventually --some propositions that would be nice to be able to eventually
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := sorry lemma primeSpectrum_empty_of_subsingleton (x : PrimeSpectrum R) [Subsingleton R] : False :=
x.1.ne_top_iff_one.1 x.2.1 <| Eq.substr (Subsingleton.elim 1 (0 : R)) x.1.zero_mem
lemma primeSpectrum_empty_iff : IsEmpty (PrimeSpectrum R) ↔ Subsingleton R := by
constructor
. contrapose
rw [not_isEmpty_iff, ←not_nontrivial_iff_subsingleton, not_not]
apply PrimeSpectrum.instNonemptyPrimeSpectrum
. intro h
by_contra hneg
rw [not_isEmpty_iff] at hneg
rcases hneg with ⟨a, ha⟩
exact primeSpectrum_empty_of_subsingleton ⟨a, ha⟩
/-- A ring has Krull dimension -∞ if and only if it is the zero ring -/
lemma dim_eq_bot_iff : krullDim R = ⊥ ↔ Subsingleton R := by
unfold Ideal.krullDim
rw [←primeSpectrum_empty_iff, iSup_eq_bot]
constructor <;> intro h
. rw [←not_nonempty_iff]
rintro ⟨a, ha⟩
specialize h ⟨a, ha⟩
tauto
. rw [h.forall_iff]
trivial
lemma dim_eq_zero_iff_field [IsDomain R] : krullDim R = 0 ↔ IsField R := by sorry lemma dim_eq_zero_iff_field [IsDomain R] : krullDim R = 0 ↔ IsField R := by sorry