feat: regular local rings are domains

[JarodAlper]

We have added three new files in Mathlib/RingTheory/LocalRing

• EmbeddingDimension.lean
• LocalRingDimension.lean
• RegularLocalRings.lean
  We have added two lemmas and an instance in Mathlib/RingTheory/LocalRing/RingHom/Basic.lean

This work was done by Jarod Alper and Brian Nugent.

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[github-actions]

PR summary 5a0d04196f

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.LocalRing.EmbeddingDimension (new file)	1781
Mathlib.RingTheory.LocalRing.LocalRingDimension (new file)	1807
Mathlib.RingTheory.LocalRing.RegularLocalRings (new file)	1825

---

Declarations diff

+ Ideal.toCotangent_out
+ IsLocalRing.ContangentSpace_extend_singleton_basis
+ IsLocalRing.IsDiscreteValuationRing_iff_IsRegularLocalRing_ringKrullDim_one
+ IsLocalRing.IsNoetherianRing_span_singleton_IsPrime_not_minimalPrimes_IsDomain
+ IsLocalRing.embDim
+ IsLocalRing.embDim_eq_spanFinrank_maximalIdeal
+ IsLocalRing.embDim_quot_singleton
+ IsLocalRing.embDim_quot_singleton'
+ IsLocalRing.embDim_quotient_span_singleton.{u}
+ IsLocalRing.exists_finset_card_eq_ringKrullDim
+ IsLocalRing.quotient_singleton_ringKrullDim
+ IsLocalRing.quotient_span_singleton_spanFinrank
+ IsLocalRing.ringKrullDim_le_embDim
+ IsLocalRing.ringKrullDim_le_spanFinrank
+ IsLocalRing.ringKrullDim_quotient_succ_eq_of_nonZeroDivisor
+ IsNoetherianRing.Ideal_spanRank_eq_spanFinrank
+ IsRegularLocalRing
+ IsRegularLocalRing.IsDomain
+ IsRegularLocalRing.IsDomain_Induction
+ IsRegularLocalRing.quotient_span_singleton_IsRegularLocalRing
+ IsRegularLocalRing.ringKrullDim_zero_IsField
+ Module.Finrank_eq_spanFinrankOfTop
+ Submodule.SpanFinRankOfSubmodule_eq_spanFinrankOfTop
+ Submodule.spanRank_eq_spanRank_Top
+ Submodule.spanRank_injective_map.{u}
+ Submodule.spanRank_injective_map_le'.{u}
+ Submodule.spanRank_injective_map_le.{u}
+ finRingKrullDim
+ finRingKrullDim_ringKrullDim
+ inst_IsRegularLocalRing_IsDomain
+ instance {R : Type*} [CommRing R] [IsLocalRing R] [IsNoetherianRing R] :
+ instance {R : Type*} [CommRing R] [IsLocalRing R] {I : Ideal R} [Nontrivial (R ⧸ I)] :
+ le_minimalPrimes
+ quotient_comap_maximalIdeal
+ quotient_map_maximalIdeal

You can run this locally as follows

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

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

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

---

No changes to technical debt.

You can run this locally as

./scripts/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).

[mathlib4-merge-conflict-bot]

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