feat(RingTheory): regular local ring is domain

[Thmoas-Guan]

In this PR, we proved for a regular local ring R,
1 : for a finite set S in the maximal Ideal of R, it can be extended to a regular system of parameters iff they are linear independent in the cotangent space iff R/span S is regular local ring of dimesion dim R - |S|

2 : is domain
3 : regular system of parameter form regular sequence.

---

• depends on: feat(RingTheory): add lemma for span rank of maximal ideal under surjection #40182
• depends on: feat(Algebra/Module): lemma for spanFinrank eq one #40813
• depends on: feat(RingTheory/Ideal): prime avoidance for maximal ideal #40814

[github-actions]

PR summary c7517fded0

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.RegularLocalRing.Basic (new file)	2193

---

Declarations diff (regex)

+ FiniteRingKrullDim.ringKrullDim_eq_nat
+ Ideal.span_singleton_mul_eq_self_of_isPrime
+ IsDiscreteValuationRing.of_isRegularLocalRing_of_ringKrullDim_eq_one
+ IsLocalRing.spanFinrank_maximalIdeal_add_finrank_eq_of_surjective
+ IsLocalRing.spanFinrank_maximalIdeal_quotient
+ instance : RingHomSurjective (residue R) := ⟨residue_surjective⟩
+ instance [IsRegularLocalRing R] : IsDomain R := isDomain_of_isRegularLocalRing R
+ isDomain_of_isRegularLocalRing
+ isRegular_of_span_eq_maximalIdeal
+ quotient_isRegularLocalRing_tfae
+ quotient_span_singleton
+ spanFinrank_eq_one_iff
+ subset_iUnion_iff_mem_of_isMaximal_of_finite
+ toCotangentSpace

You can run this locally as follows

## from your `mathlib4` directory:
git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci

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

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

The doc-module for scripts/pr_summary/declarations_diff.sh in the mathlib-ci repository contains some details about this script.

Declarations diff (Lean)

✅ Lean-aware diff — post-build, computed from the Lean environment (commit c7517fd).

• +15 new declarations
• −0 removed declarations

+FiniteRingKrullDim.ringKrullDim_eq_nat
+Ideal.span_singleton_mul_eq_self_of_isPrime
+Ideal.subset_iUnion_iff_mem_of_isMaximal_of_finite
+IsDiscreteValuationRing.of_isRegularLocalRing_of_ringKrullDim_eq_one
+IsLocalRing.CotangentSpace.congr_simp
+IsLocalRing.instRingHomSurjectiveResidueFieldResidue
+IsLocalRing.spanFinrank_maximalIdeal_add_finrank_eq_of_surjective
+IsLocalRing.spanFinrank_maximalIdeal_quotient
+IsLocalRing.toCotangentSpace
+Submodule.spanFinrank_eq_one_iff
+instIsDomainOfIsRegularLocalRing
+isDomain_of_isRegularLocalRing
+isRegular_of_span_eq_maximalIdeal
+quotient_isRegularLocalRing_tfae
+quotient_span_singleton

---

Increase in strong tech debt: (relative, absolute) = (1.00, 0.00)

Current number	Change	Type (strong)
5666	1	backward.isDefEq.respectTransparency

No changes to weak technical debt.

Current commit c7517fded0
Reference commit 07f4b8dcd0

This script lives in the mathlib-ci repository. To run it locally, from your mathlib4 directory:

git clone https://github.com/leanprover-community/mathlib-ci.git ../mathlib-ci
../mathlib-ci/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).

[Thmoas-Guan]

This PR is depending some lemma developed from Krull heights theorem.

[mathlib4-merge-conflict-bot]

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

[mathlib4-merge-conflict-bot]

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

[mathlib4-merge-conflict-bot]

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

[mathlib4-merge-conflict-bot]

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

[mathlib-dependent-issues]

This PR/issue depends on:

• feat(RingTheory): add lemma for span rank of maximal ideal under surjection #40182
• feat(Algebra/Module): lemma for spanFinrank eq one #40813
• feat(RingTheory/Ideal): prime avoidance for maximal ideal #40814
  By Dependent Issues (🤖). Happy coding!

[mathlib-merge-conflicts]

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

[Thmoas-Guan]

I added IsLocalRing.spanFinrank_maximalIdeal_add_finrank_eq_of_surjective that would also be useful elsewhere, do we have a good place for this?