feat(Algebra): projective dimension of quotient regular sequence

[Thmoas-Guan]

In this PR, we proved that for finitely generated module over Noetherian local ring, quotient by regular sequence increase the projective dimension by exactly its length.

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• depends on: [Merged by Bors] - feat(Data/ENat): more lemma about WithBot ENat #36817

[github-actions]

PR summary 4ec5d95aa6

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.Regular.ProjectiveDimension (new file)	2210

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Declarations diff

+ add_one_le_iff'
+ add_one_le_natCast_iff
+ add_one_le_zero_iff
+ coe_eq_natCast
+ ext_vanish_of_for_all_finite
+ lt_zero_iff_eq_bot
+ projectiveDimension_eq_zero_of_projective
+ projectiveDimension_quotSMulTop_eq_succ_of_isSMulRegular
+ projectiveDimension_quotient_eq_length
+ projectiveDimension_quotient_regular_sequence

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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Increase in tech debt: (relative, absolute) = (2.00, 0.00)

Current number	Change	Type
6897	2	backward.isDefEq.respectTransparency

Current commit 99365ae914
Reference commit 4ec5d95aa6

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).

[erdOne]

Suggested change

	lemma ext_vanish_of_for_all_finite (M : ModuleCat.{v} R) (n : ℕ) [Module.Finite R M]
	(h : ∀ L : ModuleCat.{v} R, Module.Finite R L → Subsingleton (Ext.{w} M L n)) :
	∀ N : ModuleCat.{v} R, Subsingleton (Ext.{w} M N n) := by
	lemma subsingleton_ext_of_forall_finite (M : ModuleCat.{v} R) (n : ℕ) [Module.Finite R M]
	(h : ∀ L : ModuleCat.{v} R, Module.Finite R L → Subsingleton (Ext.{w} M L n))
	(N : ModuleCat.{v} R) : Subsingleton (Ext.{w} M N n) := by

[erdOne]

This should be its own instance.
(arguably should replace Ext.eq_zero_of_projective)

[mathlib4-merge-conflict-bot]

This pull request has conflicts, please merge master and resolve them.

[mathlib-merge-conflicts]

This pull request has conflicts, please merge master and resolve them.

[mathlib-dependent-issues]

This PR/issue depends on:

• [Merged by Bors] - feat(Data/ENat): more lemma about WithBot ENat #36817
  By Dependent Issues (🤖). Happy coding!