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

[Thmoas-Guan]

Projective dimension of nontrivial projective module is zero.
And fix namespace for related lemmas.

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[github-actions]

PR summary 6ef8cc2731

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference

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Declarations diff

+ projectiveDimension_eq_zero_iff
+ projectiveDimension_eq_zero_of_projective
+ projective_iff_hasProjectiveDimensionLE_zero

You can run this locally as follows

## summary with just the declaration names:
./scripts/pr_summary/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/pr_summary/declarations_diff.sh long <optional_commit>

The doc-module for scripts/pr_summary/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/reporting/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).

[joelriou]

Could you add and use the following general lemma?

lemma projectiveDimension_eq_zero_iff (X : C) :
    projectiveDimension X = 0 ↔ Projective X ∧ ¬ Limits.IsZero X := by
  rw [← projectiveDimension_eq_bot_iff, projective_iff_hasProjectiveDimensionLT_one,
    ← projectiveDimension_lt_iff]
  refine ⟨fun h ↦ by simp [h], fun ⟨h₁, h₂⟩ ↦ ?_⟩
  generalize h : projectiveDimension X = d
  induction d with
  | bot => aesop
  | coe d => induction d <;> aesop

[Thmoas-Guan]

I think we have <0 iff = \bot for WithBot ENat, I'll update to see whether it works correctly.

[Thmoas-Guan]

I added Projective iff HasProjectiveDimensionLE _ 0 for convenience, what do you think about the current proof?

[joelriou]

Thanks!

bors merge

[mathlib-bors]

Pull request successfully merged into master.

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