mathlib4 feat(Algebra): maximal Cohen Macaulay module

In this PR, we defined the concept of maximal Cohen Macaulay module, and proved that finitely generated maximal Cohen Macaulay module over regular local ring is free.

[ ] depends on: #28683
[ ] depends on: #26218

Sep 11 '25 01:09 Thmoas-Guan

PR summary 86622378a5

Import changes exceeding 2%

%	File
+42.10%	`Mathlib.RingTheory.Regular.Category`
+26.90%	`Mathlib.RingTheory.Regular.Depth`

Import changes for modified files

Dependency changes

File	Base Count	Head Count	Change
Mathlib.RingTheory.Regular.Category	1209	1718	+509 (+42.10%)
Mathlib.RingTheory.Regular.Depth	1569	1991	+422 (+26.90%)

Import changes for all files

Files	Import difference
`Mathlib.RingTheory.Regular.Depth`	422
`Mathlib.RingTheory.Regular.Category`	509
`Mathlib.RingTheory.RegularLocalRing.Defs` (new file)	1585
`Mathlib.Algebra.Category.ModuleCat.Ext.DimensionShifting` (new file)	1717
`Mathlib.Algebra.Category.ModuleCat.Ext.Finite` (new file)	1718
`Mathlib.RingTheory.RegularLocalRing.Basic` (new file)	1904
`Mathlib.RingTheory.Regular.Ischebeck` (new file)	2116
`Mathlib.RingTheory.CohenMacaulay.Basic` (new file)	2147
`Mathlib.RingTheory.CohenMacaulay.Maximal` (new file)	2151

Declarations diff

+ CategoryTheory.InjectivePresentation.shortComplex + CategoryTheory.InjectivePresentation.shortComplex_shortExact + Ideal.depth + Ideal.depth_eq_of_iso + Ideal.depth_quotSMulTop_succ_eq_moduleDepth + Ideal.quotient_smul_top_lt_of_le_smul_top + IsCohenMacaulayLocalRing + IsCohenMacaulayLocalRing.of_isLocalRing_of_isCohenMacaulayRing + IsCohenMacaulayRing + IsCohenMacaulayRing.of_isCohenMacaulayLocalRing + IsLocalRing.ResidueField.map_bijective_of_surjective + IsLocalRing.ResidueField.map_injective + 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.ideal_depth_eq_sSup_length_regular + IsLocalRing.ideal_depth_le_depth + IsLocalRing.spanFinrank_maximalIdeal_eq_finrank_cotangentSpace + IsRegularLocalRing + IsRegularLocalRing.iff_finrank_cotangentSpace + LinearMap.shortComplexKer + LinearMap.shortExact_shortComplexKer + LinearMapOfSemiLinearMapAlgebraMap + Module.exists_finite_presentation + Module.finite_shrink + Module.free_shrink + ModuleCat.IsCohenMacaulay + ModuleCat.IsCohenMacaulay_of_iso + ModuleCat.IsMaximalCohenMacaulay + ModuleCat.depth_eq_supportDim_of_cohenMacaulay + ModuleCat.depth_eq_supportDim_unbot_of_cohenMacaulay + ModuleCat.finite_ext + ModuleCat.isCohenMacaulay_iff + ModuleCat.projective_shortComplex + ModuleCat.projective_shortComplex_shortExact + SemiLinearMapAlgebraMapOfLinearMap + Submodule.comap_lt_top_of_lt_range + Submodule.smul_top_eq_comap_smul_top_of_surjective + WithBot.add_le_add_natCast_left_iff + WithBot.add_le_add_natCast_right_iff + WithBot.add_le_add_one_left_iff + WithBot.add_le_add_one_right_iff + WithBot.add_natCast_cancel + WithBot.add_one_cancel + WithBot.natCast_add_cancel + WithBot.one_add_cancel + WithTop.add_cast_cancel + WithTop.add_le_add_cast_iff_left + WithTop.add_le_add_cast_iff_right + WithTop.cast_add_cancel + 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 + exist_nat_eq + exists_isRegular_of_exists_subsingleton_ext + exists_isRegular_tfae + extClass_postcomp_surjective_of_projective_X₂ + extClass_precomp_surjective_of_projective_X₂ + ext_subsingleton_of_exists_isRegular + ext_subsingleton_of_lt_moduleDepth + free_of_isMaximalCohenMacaulay_of_isRegularLocalRing + free_of_quotSMulTop_free + ideal_depth_quotient_regular_sequence_add_length_eq_ideal_depth + instance [IsCohenMacaulayLocalRing R] : (ModuleCat.of R R).IsCohenMacaulay + instance [IsNoetherianRing R] [IsLocalRing R] [Small.{v} R] + instance [Small.{v} R] (M : ModuleCat.{v} R) : Module.Free R M.projective_shortComplex.X₂ + instance {M : ModuleCat.{v} R} (ip : InjectivePresentation M) : Injective ip.shortComplex.X₂ := ip.2 + isCohenMacaulayLocalRing_def + isCohenMacaulayLocalRing_iff + isCohenMacaulayLocalRing_localization_atPrime + isCohenMacaulayLocalRing_of_isRegularLocalRing + isCohenMacaulayLocalRing_of_ringEquiv + isCohenMacaulayLocalRing_of_ringKrullDim_le_depth + isCohenMacaulayRing_def + isCohenMacaulayRing_def' + isCohenMacaulayRing_iff + isCohenMacaulayRing_of_ringEquiv + isCohenMacaulay_of_isMaximalCohenMacaulay + isDomain_of_isRegularLocalRing + isField_of_isRegularLocalRing_of_dimension_zero + 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 + isMaximalCohenMacaulay_def + isRegularLocalRing_def + isRegular_of_span_eq_maximalIdeal + localize_at_prime_depth_eq_of_isCohenMacaulay + localize_at_prime_isCohenMacaulay_of_isCohenMacaulay + 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 + mono_postcomp_mk₀_of_mono + mono_precomp_mk₀_of_epi + of_ringEquiv + of_span_eq + postcomp_mk₀_injective_of_mono + pow_mono_of_mono + precomp_mk₀_injective_of_epi + quotSMulTop_isCohenMacaulay_iff_isCohenMacaulay + quotSMulTop_isLocalizedModule_map + quotSMulTop_nontrivial + quotient_isRegularLocalRing_tfae + 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 + quotient_span_singleton + ringKrullDim_le_spanFinrank_maximalIdeal + ring_depth_invariant + ring_depth_uLift + smul_id_postcomp_eq_zero_of_mem_ann + subsingleton_of_injective + subsingleton_of_projective

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).