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[Merged by Bors] - feat(Algebra/ModuleCat): lemma for projective dimension eq zero #37351

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[Merged by Bors] - feat(Algebra/ModuleCat): lemma for projective dimension eq zero #37351

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1b2a710
add projective dimension eq zero lemma
Thmoas-Guan Mar 30, 2026
f7d2504
Merge branch 'master' into projectiveDimension-eq-zero
Thmoas-Guan Apr 3, 2026
ee32f16
add lemma
Thmoas-Guan Apr 3, 2026
c83495b
use lemma
Thmoas-Guan Apr 3, 2026
0aa607b
fix namespace
Thmoas-Guan Apr 4, 2026
eee15c4
add deprecation
Thmoas-Guan Apr 4, 2026
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11 changes: 8 additions & 3 deletions Mathlib/Algebra/Category/ModuleCat/ProjectiveDimension.lean
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Expand Up @@ -20,15 +20,15 @@ public import Mathlib.CategoryTheory.Abelian.Projective.Dimension

universe v v' u u'

variable {R : Type u} [CommRing R] [Small.{v} R]
variable {R : Type u} [CommRing R]

open CategoryTheory Abelian Module

namespace CategoryTheory

section

variable {R' : Type u'} [CommRing R'] [Small.{v'} R'] (e : R ≃+* R')
variable [Small.{v} R] {R' : Type u'} [CommRing R'] [Small.{v'} R'] (e : R ≃+* R')

variable {M : ModuleCat.{v} R} {N : ModuleCat.{v'} R'}

Expand Down Expand Up @@ -81,7 +81,7 @@ end

section

variable [Small.{v'} R] {M : ModuleCat.{v} R} {N : ModuleCat.{v'} R}
variable [Small.{v} R] [Small.{v'} R] {M : ModuleCat.{v} R} {N : ModuleCat.{v'} R}

lemma hasProjectiveDimensionLE_of_linearEquiv (e : M ≃ₗ[R] N)
(n : ℕ) [HasProjectiveDimensionLE M n] : HasProjectiveDimensionLE N n :=
Expand All @@ -94,3 +94,8 @@ lemma projectiveDimension_eq_of_linearEquiv (e : M ≃ₗ[R] N) :
end

end CategoryTheory

lemma projectiveDimension_eq_zero_of_projective (M : ModuleCat.{v} R) [Nontrivial M]
[Projective M] : projectiveDimension M = 0 := by
simpa [projectiveDimension_eq_zero_iff, ModuleCat.isZero_iff_subsingleton,
not_subsingleton_iff_nontrivial] using ⟨‹_›, ‹_›⟩
9 changes: 9 additions & 0 deletions Mathlib/CategoryTheory/Abelian/Projective/Dimension.lean
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Expand Up @@ -154,6 +154,9 @@ lemma projective_iff_hasProjectiveDimensionLT_one :
exact ⟨fun _ ↦ inferInstance, fun _ ↦ projective_iff_subsingleton_ext_one.2
(HasProjectiveDimensionLT.subsingleton X 1 1 (by rfl))⟩

lemma projective_iff_hasProjectiveDimensionLE_zero : Projective X ↔ HasProjectiveDimensionLE X 0 :=
projective_iff_hasProjectiveDimensionLT_one

instance (priority := low) [HasProjectiveDimensionLT X 1] : Projective X :=
projective_iff_hasProjectiveDimensionLT_one.mpr ‹_›

Expand Down Expand Up @@ -321,6 +324,12 @@ lemma projectiveDimension_ne_top_iff (X : C) :
simp only [ne_eq, WithBot.coe_eq_top, ENat.coe_ne_top, not_false_eq_true, true_iff]
exact ⟨d, by simpa only [← projectiveDimension_le_iff] using hd.le⟩

lemma projectiveDimension_eq_zero_iff (X : C) :
projectiveDimension X = 0 ↔ Projective X ∧ ¬ Limits.IsZero X := by
rw [← projectiveDimension_eq_bot_iff, projective_iff_hasProjectiveDimensionLE_zero,
← projectiveDimension_le_iff, ← WithBot.lt_zero_iff_eq_bot, not_lt, Nat.cast_zero,
le_antisymm_iff]

end CategoryTheory

end ProjectiveDimension
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