feat(Algebra): cohen macaulay local ring is catenary

[Thmoas-Guan]

We prove Cohen Macaulay local ring is catenary.
Following Theorem 31, Chater 6, Commutative Algebra, Matsumura

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• depends on: feat(Algebra): definition of cohen macaulay #26218

[github-actions]

PR summary 2ff88851d5

Import changes exceeding 2%

%	File
+25.29%	Mathlib.RingTheory.Regular.Depth

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.Regular.Depth	1815	2274	+459 (+25.29%)
Mathlib.RingTheory.Regular.Category	1480	1343	-137 (-9.26%)

Import changes for all files

Files	Import difference
Mathlib.RingTheory.Regular.Category	-137
Mathlib.RingTheory.Regular.Depth	459
Mathlib.RingTheory.Depth.Basic (new file)	2273
Mathlib.RingTheory.Depth.Ischebeck (new file)	2312
Mathlib.RingTheory.CohenMacaulay.Basic (new file)	2417
Mathlib.RingTheory.CohenMacaulay.Catenary (new file)	2418

---

Declarations diff

+ CategoryTheory.Abelian.Ext.pow_mono_of_mono
+ Ext.smul_id_postcomp_eq_zero_of_mem_ann
+ Ideal.depth
+ Ideal.depth_eq_height
+ Ideal.depth_eq_of_iso
+ Ideal.depth_le_height
+ Ideal.depth_quotSMulTop_succ_eq_moduleDepth
+ Ideal.exist_regular_sequence_length_eq_height
+ Ideal.height_add_ringKrullDim_quotient_eq_ringKrullDim
+ Ideal.ofList_height_eq_length_of_isWeaklyRegular
+ Ideal.ofList_height_eq_length_of_isWeaklyRegular'
+ Ideal.ofList_height_le_length
+ Ideal.ofList_spanFinrank_le_length
+ Ideal.primeHeight_add_ringKrullDim_quotient_eq_ringKrullDim
+ Ideal.primeHeight_eq_ringKrullDim_localization
+ IsCohenMacaulayLocalRing
+ IsCohenMacaulayLocalRing.of_isLocalRing_of_isCohenMacaulayRing
+ IsCohenMacaulayRing
+ IsCohenMacaulayRing.of_isCohenMacaulayLocalRing
+ IsLocalRing.Ideal.ofList_height_le_length'
+ IsLocalRing.depth
+ IsLocalRing.depth_eq_of_algebraMap_surjective
+ IsLocalRing.depth_eq_of_iso
+ IsLocalRing.depth_eq_of_ringEquiv
+ IsLocalRing.depth_eq_sSup_length_regular
+ IsLocalRing.depth_quotSMulTop_succ_eq_moduleDepth
+ IsLocalRing.depth_quotient_regular_sequence_add_length_eq_depth
+ IsLocalRing.depth_quotient_regular_succ_eq_depth
+ IsLocalRing.depth_quotient_span_regular_succ_eq_depth
+ IsLocalRing.height_eq_height_maximalIdeal_of_maximalIdeal_mem_minimalPrimes
+ IsLocalRing.ideal_depth_eq_sSup_length_regular
+ IsLocalRing.ideal_depth_le_depth
+ LinearMap.exact_lsmul_smul_top_mkQ
+ LinearMapOfSemiLinearMapAlgebraMap
+ ModuleCat.IsCohenMacaulay
+ ModuleCat.IsCohenMacaulay_of_iso
+ ModuleCat.depth_eq_supportDim_of_cohenMacaulay
+ ModuleCat.depth_eq_supportDim_unbot_of_cohenMacaulay
+ ModuleCat.exists_isRegular_of_exists_subsingleton_ext
+ ModuleCat.exists_isRegular_tfae
+ ModuleCat.isCohenMacaulay_iff
+ ModuleCat.subsingleton_ext_of_exists_isRegular
+ SemiLinearMapAlgebraMapOfLinearMap
+ Submodule.comap_lt_top_of_lt_range
+ associatedPrimes_self_eq_minimalPrimes
+ associated_prime_eq_minimalPrimes_isCohenMacaulay
+ associated_prime_minimal_of_isCohenMacaulay
+ depth_eq_dim_quotient_associated_prime_of_isCohenMacaulay
+ depth_le_ringKrullDim
+ depth_le_ringKrullDim_associatedPrime
+ depth_le_supportDim
+ depth_ne_top
+ depth_quotient_regular_sequence_add_length_eq_depth
+ ext_subsingleton_of_lt_moduleDepth
+ ideal_depth_quotient_regular_sequence_add_length_eq_ideal_depth
+ instance [IsCohenMacaulayLocalRing R] : (ModuleCat.of R R).IsCohenMacaulay
+ isCohenMacaulayLocalRing_def
+ isCohenMacaulayLocalRing_iff
+ isCohenMacaulayLocalRing_localization_atPrime
+ isCohenMacaulayLocalRing_of_ringEquiv
+ isCohenMacaulayLocalRing_of_ringKrullDim_le_depth
+ isCohenMacaulayRing_def
+ isCohenMacaulayRing_def'
+ isCohenMacaulayRing_iff
+ isCohenMacaulayRing_of_ringEquiv
+ isLocalization_at_prime_prime_depth_le_depth
+ isLocalize_at_prime_depth_eq_of_isCohenMacaulay
+ isLocalize_at_prime_dim_eq_prime_depth_of_isCohenMacaulay
+ isLocalize_at_prime_isCohenMacaulay_of_isCohenMacaulay
+ isLocalizedModule_quotSMulTop_isLocalizedModule_map
+ isRegular_of_maximalIdeal_mem_ofList_minimalPrimes
+ isRegular_of_ofList_height_eq_length_of_isCohenMacaulayLocalRing
+ localize_at_prime_depth_eq_of_isCohenMacaulay
+ localize_at_prime_isCohenMacaulay_of_isCohenMacaulay
+ maximalIdeal_mem_minimalPrimes_of_surjective
+ maximalIdeal_mem_ofList_append_minimalPrimes_of_ofList_height_eq_length
+ moduleDepth
+ moduleDepth_eq_depth_of_supp_eq
+ moduleDepth_eq_find
+ moduleDepth_eq_iff
+ moduleDepth_eq_moduleDepth_shrink
+ moduleDepth_eq_of_iso_fst
+ moduleDepth_eq_of_iso_snd
+ moduleDepth_eq_sSup_length_regular
+ moduleDepth_eq_sup_nat
+ moduleDepth_eq_top_iff
+ moduleDepth_eq_zero_of_hom_nontrivial
+ moduleDepth_ge_depth_sub_dim
+ moduleDepth_ge_min_of_shortExact_fst_fst
+ moduleDepth_ge_min_of_shortExact_fst_snd
+ moduleDepth_ge_min_of_shortExact_snd_fst
+ moduleDepth_ge_min_of_shortExact_snd_snd
+ moduleDepth_ge_min_of_shortExact_trd_fst
+ moduleDepth_ge_min_of_shortExact_trd_snd
+ moduleDepth_lt_top_iff
+ moduleDepth_quotSMulTop_succ_eq_moduleDepth
+ moduleDepth_quotient_regular_sequence_add_length_eq_moduleDepth
+ quotSMulTop_isCohenMacaulay_iff_isCohenMacaulay
+ quotSMulTop_isLocalizedModule_map
+ quotient_prime_ringKrullDim_ne_bot
+ quotient_regular_isCohenMacaulay_iff_isCohenMacaulay
+ quotient_regular_sequence_isCohenMacaulay_iff_isCohenMacaulay
+ quotient_regular_smul_top_isCohenMacaulay_iff_isCohenMacaulay
+ quotient_span_regular_isCohenMacaulay_iff_isCohenMacaulay
+ ringKrullDim_quotient_eq_iSup_quotient_minimalPrimes
+ ring_depth_invariant
+ ring_depth_uLift
+ smulShortComplex_f_eq_smul_id
+ smulShortComplex_f_hom_eq_smul_id

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.

---

Increase in tech debt: (relative, absolute) = (16.00, 0.00)

Current number	Change	Type
6431	16	backward.isDefEq.respectTransparency

Current commit bd211ac0e7
Reference commit 2ff88851d5

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).

[mathlib4-dependent-issues-bot]

This PR/issue depends on:

• feat(Algebra): definition of cohen macaulay #26218
  By Dependent Issues (🤖). Happy coding!

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