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Mathlib.RingTheory.Ideal.KrullsHeightTheorem | {
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Mathlib.RingTheory.KrullDimension.Regular | {
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Mathlib.RingTheory.KrullDimension.Regular | {
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Mathlib.RingTheory.KrullDimension.Regular | {
"line": 184,
"column": 83
} | {
"line": 184,
"column": 85
} | {
"line": 185,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nx : R\nreg : IsSMulRegular R x\nhx : x ∈ maximalIdeal R\n⊢ x ∈ Ring.jacobson R",
"ppTerm": "?m.35",
"assigned": true,
"usedConstants": [
"Eq.mpr",
"Semiring.toModule",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 205,
"column": 35
} | {
"line": 205,
"column": 37
} | {
"line": 205,
"column": 38
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nx : R\nrs' : List R\nih :\n ∀ {M : Type u_2} [inst : AddCommGroup M] [inst_1 : Module R M] [Module.Finite R M],\n Sequence.IsRegular M rs' → supportDim R (M ⧸ ofList rs' • ⊤) + ↑rs'.length = supportDim R M\nM : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 204,
"column": 37
} | {
"line": 204,
"column": 39
} | {
"line": 205,
"column": 6
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\ninst✝³ : IsLocalRing R\nx : R\nrs' : List R\nih :\n ∀ {M : Type u_2} [inst : AddCommGroup M] [inst_1 : Module R M] [Module.Finite R M],\n Sequence.IsRegular M rs' → supportDim R (M ⧸ ofList rs' • ⊤) + ↑rs'.length = supportDim R M\nM : ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 198,
"column": 87
} | {
"line": 198,
"column": 89
} | {
"line": 199,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝⁵ : CommRing R\ninst✝⁴ : IsNoetherianRing R\nM : Type u_2\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Module.Finite R M\ninst✝ : IsLocalRing R\nrs : List R\nreg : Sequence.IsRegular M rs\n⊢ supportDim R (M ⧸ ofList rs • ⊤) + ↑rs.length = supportDim R M",
"ppTerm": "?m... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 212,
"column": 53
} | {
"line": 212,
"column": 55
} | {
"line": 212,
"column": 56
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nrs : List R\nreg : Sequence.IsRegular R rs\n⊢ ofList rs = ofList rs • ⊤",
"ppTerm": "?m.37",
"assigned": true,
"usedConstants": [
"instHSMul",
"Semiring.toModule",
"instSMulOfMul",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 73,
"column": 47
} | {
"line": 73,
"column": 49
} | {
"line": 73,
"column": 50
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nJ : Submodule R M\nhJ : J ∈ s\n⊢ ∀ ⦃y : Submodule R M⦄,\n y ∈ {I ∈ s | (I.colon Set.u... | [] | by | [anonymous] | by |
Mathlib.RingTheory.KrullDimension.Regular | {
"line": 211,
"column": 90
} | {
"line": 211,
"column": 92
} | {
"line": 212,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝² : CommRing R\ninst✝¹ : IsNoetherianRing R\ninst✝ : IsLocalRing R\nrs : List R\nreg : Sequence.IsRegular R rs\n⊢ ringKrullDim (R ⧸ ofList rs) + ↑rs.length = ringKrullDim R",
"ppTerm": "?m.24",
"assigned": true,
"usedConstants": [
"Ideal.Quotient.commSemiring",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 84,
"column": 71
} | {
"line": 84,
"column": 73
} | {
"line": 85,
"column": 6
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).radical = (I'.col... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 87,
"column": 38
} | {
"line": 87,
"column": 40
} | {
"line": 87,
"column": 41
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).radical = (I'.col... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 75,
"column": 21
} | {
"line": 75,
"column": 23
} | {
"line": 75,
"column": 24
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\n⊢ 1 < q",
"ppTerm": "?m.77",
"assigned": true,
"usedConstants": [
"Eq.m... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 87,
"column": 54
} | {
"line": 87,
"column": 56
} | {
"line": 87,
"column": 57
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).radical = (I'.col... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 78,
"column": 27
} | {
"line": 78,
"column": 29
} | {
"line": 78,
"column": 30
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\n⊢ 1 ≤ q ^ n",
"ppTerm":... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 89,
"column": 38
} | {
"line": 89,
"column": 40
} | {
"line": 89,
"column": 41
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).radical = (I'.col... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 89,
"column": 54
} | {
"line": 89,
"column": 56
} | {
"line": 89,
"column": 57
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).radical = (I'.col... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 70
} | {
"line": 91,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).ra... | [] | rw [← hI, colon_finsetInf,
radical_finset_inf (i := I') (by simp [hI']) (by simp), id_eq] at hIJ
rw [hIJ, ← hJ, colon_finsetInf,
radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.RingTheory.Lasker | {
"line": 86,
"column": 6
} | {
"line": 89,
"column": 70
} | {
"line": 91,
"column": 0
} | [
{
"pp": "case refine_3\nR : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\nI J I' : Submodule R M\nhI' : I' ∈ s\nhI : {I ∈ s | (I.colon Set.univ).ra... | [] | rw [← hI, colon_finsetInf,
radical_finset_inf (i := I') (by simp [hI']) (by simp), id_eq] at hIJ
rw [hIJ, ← hJ, colon_finsetInf,
radical_finset_inf (i := J') (by simp [hJ']) (by simp), id_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.RingTheory.Lasker | {
"line": 65,
"column": 92
} | {
"line": 65,
"column": 94
} | {
"line": 66,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\n⊢ ∃ t,\n t.inf id = N ∧\n (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 96,
"column": 51
} | {
"line": 96,
"column": 53
} | {
"line": 97,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns : Finset (Submodule R M)\nhs : s.inf id = N\nhs' : ∀ ⦃J : Submodule R M⦄, J ∈ s → J.IsPrimary\n⊢ ∃ t,\n t.inf id = N ∧\n (∀ ⦃J : Submodule R M⦄, J ∈ t → J.IsPrimary) ∧\n ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 112,
"column": 71
} | {
"line": 112,
"column": 73
} | {
"line": 113,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nh : IsLasker R M\nN : Submodule R M\n⊢ ∃ t, N.IsMinimalPrimaryDecomposition t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Submodule.IsMinimalPrimaryDecomposition.mk",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 132,
"column": 98
} | {
"line": 132,
"column": 100
} | {
"line": 133,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\nx : M\n⊢ (N.colon {x}).radical = {x_1 ∈ t | x ∉ x_1}.inf fun q ↦ (q.colon Set.univ).radical",
"ppTerm": "?m.69",
... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 130,
"column": 86
} | {
"line": 130,
"column": 88
} | {
"line": 131,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\n⊢ (fun J ↦ (J.colon Set.univ).radical) '' ↑t = N.associatedPrimes",
"ppTerm": "?m.33",
"assigned": true,
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 156,
"column": 55
} | {
"line": 156,
"column": 57
} | {
"line": 157,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\nq : Submodule R M\nhq : q ∈ t\n⊢ (q.colon Set.univ).radical ∈ N.associatedPrimes",
"ppTerm": "?m.36",
"assign... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 183,
"column": 63
} | {
"line": 183,
"column": 65
} | {
"line": 184,
"column": 6
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nhq : (q.colon Set.univ).radical = ↑p\nS : Submonoid R := ⨅ q ∈ s₀, (↑q)... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 189,
"column": 27
} | {
"line": 189,
"column": 29
} | {
"line": 190,
"column": 6
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nS : Submonoid R := ⨅ q ∈ s₀, (↑q).primeCompl\nf : M →ₗ[R] LocalizedModu... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 91,
"column": 94
} | {
"line": 91,
"column": 96
} | {
"line": 92,
"column": 4
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis✝ : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ q ^ n\nΦₙ : ℤ[X... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 171,
"column": 60
} | {
"line": 171,
"column": 62
} | {
"line": 172,
"column": 2
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\nq : Submodule R M\nhqp : q.IsPrimary\np : ↑N.associatedPrimes\nhq : (q.colon Set.univ).radical = ↑p\n⊢ comap (mkLinearMap (⨅ q ∈ s₀, (... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 211,
"column": 19
} | {
"line": 211,
"column": 46
} | {
"line": 211,
"column": 47
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\ns : Finset (Submodule R M)\nhs : s ⊆ t\nhs' : Finset.image (fun q ... | [
"R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\ns : Finset (Submodule R M)\nhs : s ⊆ t\nhs' : Finset.image (fun q ↦ (q.colon S... | ← localized₀FrameHom_apply, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.RingTheory.LittleWedderburn | {
"line": 118,
"column": 23
} | {
"line": 118,
"column": 25
} | {
"line": 119,
"column": 4
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ q ^ n\nΦₙ : ℤ[X]... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 213,
"column": 72
} | {
"line": 213,
"column": 74
} | {
"line": 214,
"column": 4
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\ns : Finset (Submodule R M)\nhs : s ⊆ t\nhs' : Finset.image (fun q ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 208,
"column": 47
} | {
"line": 208,
"column": 49
} | {
"line": 209,
"column": 2
} | [
{
"pp": "R : Type u_3\nM : Type u_4\ninst✝² : CommRing R\ninst✝¹ : AddCommMonoid M\ninst✝ : Module R M\nN : Submodule R M\nt : Finset (Submodule R M)\nht : N.IsMinimalPrimaryDecomposition t\ns₀ : Finset ↑N.associatedPrimes\nhs₀ : IsLowerSet ↑s₀\ns : Finset (Submodule R M)\nhs : s ⊆ t\nhs' : Finset.image (fun q ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 126,
"column": 27
} | {
"line": 126,
"column": 29
} | {
"line": 126,
"column": 30
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis✝¹ : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ q ^ n\nΦₙ : ℤ[... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 229,
"column": 34
} | {
"line": 229,
"column": 36
} | {
"line": 230,
"column": 4
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nt : Finset (Ideal R)\nht : Submodule.IsMinimalPrimaryDecomposition I t\np : Ideal R\nhp : p ∈ I.minimalPrimes\n⊢ t.inf radical ≤ p",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Ideal.radical_mono",
"Eq.mpr",
"... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 227,
"column": 38
} | {
"line": 227,
"column": 40
} | {
"line": 228,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝ : CommSemiring R\nI : Ideal R\nt : Finset (Ideal R)\nht : Submodule.IsMinimalPrimaryDecomposition I t\n⊢ I.minimalPrimes ⊆ radical '' ↑t",
"ppTerm": "?m.18",
"assigned": true,
"usedConstants": [
"Ideal.radical_mono",
"Eq.mpr",
"Submodule",
"le_ref... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 274,
"column": 22
} | {
"line": 274,
"column": 24
} | {
"line": 274,
"column": 25
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : InfIrred N\na : R\nb : M\nhab : a • b ∈ N\nn : ℕ\na✝ b✝ : M\nhx : a✝ ∈ {x | a ^ n • x ∈ N}\nhy : b✝ ∈ {x | a ^ n • x ∈ N}\n⊢ a✝ + b✝ ∈ {x | a ^ n • x ∈ N}",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 275,
"column": 17
} | {
"line": 275,
"column": 19
} | {
"line": 275,
"column": 20
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : InfIrred N\na : R\nb : M\nhab : a • b ∈ N\nn : ℕ\n⊢ 0 ∈ {x | a ^ n • x ∈ N}",
"ppTerm": "?m.62",
"assigned": true,
"usedConstants": [
"Submod... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 276,
"column": 23
} | {
"line": 276,
"column": 25
} | {
"line": 276,
"column": 26
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh✝ : InfIrred N\na : R\nb : M\nhab : a • b ∈ N\nn : ℕ\nx : R\ny : M\nh : y ∈ {x | a ^ n • x ∈ N}\n⊢ x • y ∈ {x | a ^ n • x ∈ N}",
"ppTerm": "?m.66",
"assig... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 277,
"column": 26
} | {
"line": 277,
"column": 28
} | {
"line": 278,
"column": 4
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : InfIrred N\na : R\nb : M\nhab : a • b ∈ N\nf : ℕ → Submodule R M := fun n ↦ { carrier := {x | a ^ n • x ∈ N}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\n... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 131,
"column": 74
} | {
"line": 131,
"column": 76
} | {
"line": 132,
"column": 4
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\nval✝ : Fintype D\nZ : Subring D := Subring.center D\nhZ : Z ≠ ⊤\nthis✝² : Field ↥Z := hD.field ⋯\nq : ℕ := card ↥Z\ncard_Z : q = card ↥Z\nhq : 1 < q\nn : ℕ := finrank (↥Z) D\ncard_D : card D = q ^ n\nh1qn : 1 ≤ q ^ n\nΦₙ : ℤ[... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Lasker | {
"line": 283,
"column": 83
} | {
"line": 283,
"column": 85
} | {
"line": 283,
"column": 86
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\na : R\nb : M\nhab : a • b ∈ N\nf : ℕ → Submodule R M := fun n ↦ { carrier := {x | a ^ n • x ∈ N}, add_mem' := ⋯, zero_mem' := ⋯, smul_mem' := ⋯ }\nhf : Monotone f\... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 67,
"column": 89
} | {
"line": 67,
"column": 91
} | {
"line": 68,
"column": 2
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nhD : InductionHyp D\n⊢ Subring.center D = ⊤",
"ppTerm": "?m.6",
"assigned": true,
"usedConstants": [
"Int.instAddCommGroup",
"Subring.mem_center_iff",
"Fintype.card_congr",
"Iff.mpr",
"instPowNat",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 150,
"column": 44
} | {
"line": 150,
"column": 46
} | {
"line": 151,
"column": 4
} | [
{
"pp": "n : ℕ\nIH : ∀ m < n, ∀ (D : Type u_1) [inst : DivisionRing D] [Finite D] (val : Fintype D), card D = m → Subring.center D = ⊤\nD : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nval✝ : Fintype D\nhn : card D = n\nR : Subring D\nhR : R < ⊤\nx y : D\nhx : x ∈ R\nhy : y ∈ R\nthis : ⟨y, hy⟩ ∈ Subring... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 144,
"column": 67
} | {
"line": 144,
"column": 69
} | {
"line": 145,
"column": 2
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\n⊢ Subring.center D = ⊤",
"ppTerm": "?m.5",
"assigned": true,
"usedConstants": [
"Subring.mem_center_iff",
"Fintype.divisionRingOfIsDomain",
"Eq.mpr",
"Semigroup.toMul",
"SubsemiringClass.toSemiring._proof... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 168,
"column": 26
} | {
"line": 168,
"column": 28
} | {
"line": 168,
"column": 29
} | [
{
"pp": "D : Type u_1\ninst✝¹ : DivisionRing D\ninst✝ : Finite D\nx y : D\n⊢ x * y = y * x",
"ppTerm": "?m.9",
"assigned": true,
"usedConstants": [
"Subring.mem_center_iff",
"Semigroup.toMul",
"HMul.hMul",
"Subring.instSetLike",
"Ring.toNonAssocRing",
"congrArg",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LittleWedderburn | {
"line": 173,
"column": 95
} | {
"line": 173,
"column": 97
} | {
"line": 174,
"column": 2
} | [
{
"pp": "D : Type u_1\ninst✝² : Finite D\ninst✝¹ : Ring D\ninst✝ : IsDomain D\n⊢ IsField D",
"ppTerm": "?m.3",
"assigned": true,
"usedConstants": [
"Fintype.divisionRingOfIsDomain",
"littleWedderburn",
"Classical.propDecidable",
"Field.toIsField",
"Nonempty.intro",
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Lasker | {
"line": 269,
"column": 86
} | {
"line": 269,
"column": 88
} | {
"line": 270,
"column": 2
} | [
{
"pp": "R : Type u_1\nM : Type u_2\ninst✝³ : CommRing R\ninst✝² : AddCommGroup M\ninst✝¹ : Module R M\ninst✝ : IsNoetherian R M\nN : Submodule R M\nh : InfIrred N\n⊢ N.IsPrimary",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"Iff.mpr",
"Eq.mpr",
"Submodule.pointwiseDist... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 52,
"column": 88
} | {
"line": 52,
"column": 90
} | {
"line": 53,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsLocalIso R S\n⊢ Ideal.span {g | IsStandardOpenImmersion R (Localization.Away g)} = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Ideal.subset_span",
"Fals... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 61,
"column": 82
} | {
"line": 61,
"column": 84
} | {
"line": 62,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\n⊢ IsLocalIso R S ↔ Ideal.span {g | IsStandardOpenImmersion R (Localization.Away g)} = ⊤",
"ppTerm": "?m.17",
"assigned": true,
"usedConstants": [
"Semiring.toModule",
"OreLocalizat... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 66,
"column": 48
} | {
"line": 66,
"column": 50
} | {
"line": 67,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝³ : CommSemiring R\ninst✝² : CommSemiring S\ninst✝¹ : Algebra R S\ninst✝ : IsStandardOpenImmersion R S\nq : Ideal S\nhq : q.IsPrime\n⊢ ∃ g ∉ q, IsStandardOpenImmersion R (Localization.Away g)",
"ppTerm": "?m.16",
"assigned": true,
"usedConstants": [
"N... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 77,
"column": 59
} | {
"line": 77,
"column": 61
} | {
"line": 77,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : ⨆ i, PrimeSpectrum.basicOpen (f i) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LocalIso | {
"line": 75,
"column": 35
} | {
"line": 75,
"column": 37
} | {
"line": 76,
"column": 4
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : Ideal.span (Set.range f) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝² : ∀ (i :... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.FinitePresentation | {
"line": 33,
"column": 37
} | {
"line": 33,
"column": 39
} | {
"line": 34,
"column": 2
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ns : Set R\nhs : Ideal.span s = ⊤\nMₚ : ↑s → Type u_3\ninst✝⁷ : (g : ↑s) → AddCommGroup (Mₚ g)\ninst✝⁶ : (g : ↑s) → Module R (Mₚ g)\nRₚ : ↑s → Type u_4\ninst✝⁵ : (g : ↑s) → CommRing (Rₚ g)\ninst✝⁴ : (g : ↑s) ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.FinitePresentation | {
"line": 33,
"column": 37
} | {
"line": 43,
"column": 96
} | {
"line": 45,
"column": 0
} | [
{
"pp": "R : Type u_1\ninst✝¹⁰ : CommRing R\nM : Type u_2\ninst✝⁹ : AddCommGroup M\ninst✝⁸ : Module R M\ns : Set R\nhs : Ideal.span s = ⊤\nMₚ : ↑s → Type u_3\ninst✝⁷ : (g : ↑s) → AddCommGroup (Mₚ g)\ninst✝⁶ : (g : ↑s) → Module R (Mₚ g)\nRₚ : ↑s → Type u_4\ninst✝⁵ : (g : ↑s) → CommRing (Rₚ g)\ninst✝⁴ : (g : ↑s) ... | [] | by
have : Module.Finite R M :=
Module.Finite.of_localizationSpan' (Rₚ := Rₚ) s hs ϕ (fun _ ↦ inferInstance)
obtain ⟨n, f, fsurj⟩ := Module.Finite.exists_fin' R M
rw [← Module.FinitePresentation.fg_ker_iff f fsurj]
refine f.ker.of_localizationSpan' s hs (Rₚ := Rₚ)
(fun g ↦ TensorProduct.mk R (Rₚ g) (Fin ... | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 47,
"column": 33
} | {
"line": 47,
"column": 35
} | {
"line": 47,
"column": 36
} | [
{
"pp": "R : Type u\ninst✝¹⁴ : CommRing R\nM : Type v\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\nS : Submonoid R\ninst✝¹¹ : Small.{v, u} R\ninst✝¹⁰ : IsNoetherianRing R\nRₛ : Type u'\ninst✝⁹ : Small.{v', u'} Rₛ\ninst✝⁸ : CommRing Rₛ\ninst✝⁷ : Algebra R Rₛ\nMₛ : Type v'\ninst✝⁶ : AddCommGroup Mₛ\ninst✝⁵ : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 50,
"column": 60
} | {
"line": 50,
"column": 62
} | {
"line": 51,
"column": 2
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹³ : CommRing A\ninst✝¹² : CommRing B\ninst✝¹¹ : Algebra A B\nG : Type u\ninst✝¹⁰ : Group G\ninst✝⁹ : MulSemiringAction G B\ninst✝⁸ : SMulCommClass G A B\ninst✝⁷ : TopologicalSpace G\ninst✝⁶ : CompactSpace G\ninst✝⁵ : TotallyDisconnectedSpace G\ninst✝⁴ : IsTopologicalGr... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 68,
"column": 79
} | {
"line": 68,
"column": 81
} | {
"line": 69,
"column": 6
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 89,
"column": 80
} | {
"line": 89,
"column": 82
} | {
"line": 90,
"column": 6
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh✝ : Ideal.span (Set.range f) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝² : ∀ (i ... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 42,
"column": 79
} | {
"line": 42,
"column": 81
} | {
"line": 43,
"column": 2
} | [
{
"pp": "R : Type u\ninst✝¹⁴ : CommRing R\nM : Type v\ninst✝¹³ : AddCommGroup M\ninst✝¹² : Module R M\nS : Submonoid R\ninst✝¹¹ : Small.{v, u} R\ninst✝¹⁰ : IsNoetherianRing R\nRₛ : Type u'\ninst✝⁹ : Small.{v', u'} Rₛ\ninst✝⁸ : CommRing Rₛ\ninst✝⁷ : Algebra R Rₛ\nMₛ : Type v'\ninst✝⁶ : AddCommGroup Mₛ\ninst✝⁵ : ... | [] | by | [anonymous] | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 74,
"column": 16
} | {
"line": 74,
"column": 18
} | {
"line": 74,
"column": 19
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 73,
"column": 50
} | {
"line": 73,
"column": 52
} | {
"line": 74,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝⁸ : CommSemiring R\ninst✝⁷ : CommSemiring S\ninst✝⁶ : Algebra R S\nι : Type u_3\nf : ι → S\nh : Ideal.span (Set.range f) = ⊤\nT : ι → Type u_4\ninst✝⁵ : (i : ι) → CommSemiring (T i)\ninst✝⁴ : (i : ι) → Algebra R (T i)\ninst✝³ : (i : ι) → Algebra S (T i)\ninst✝² : ∀ (i :... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 101,
"column": 59
} | {
"line": 101,
"column": 61
} | {
"line": 101,
"column": 62
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\ns : Set S\nh✝ : Ideal.span s = ⊤\nh : ∀ x ∈ s, IsLocalIso R (Localization.Away x)\n⊢ Ideal.span (Set.range fun i ↦ ↑i) = ⊤",
"ppTerm": "?m.31",
"assigned": true,
"usedConstants": [
"Eq.m... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.Invariant.Profinite | {
"line": 75,
"column": 20
} | {
"line": 75,
"column": 22
} | {
"line": 75,
"column": 23
} | [
{
"pp": "A : Type u_1\nB : Type u_2\ninst✝¹⁵ : CommRing A\ninst✝¹⁴ : CommRing B\ninst✝¹³ : Algebra A B\nG : Type u\ninst✝¹² : Group G\ninst✝¹¹ : MulSemiringAction G B\ninst✝¹⁰ : SMulCommClass G A B\ninst✝⁹ : TopologicalSpace G\ninst✝⁸ : CompactSpace G\ninst✝⁷ : TotallyDisconnectedSpace G\ninst✝⁶ : IsTopological... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 100,
"column": 74
} | {
"line": 100,
"column": 76
} | {
"line": 101,
"column": 2
} | [
{
"pp": "R : Type u_1\nS : Type u_2\ninst✝² : CommSemiring R\ninst✝¹ : CommSemiring S\ninst✝ : Algebra R S\ns : Set S\nh✝ : Ideal.span s = ⊤\nh : ∀ x ∈ s, IsLocalIso R (Localization.Away x)\n⊢ IsLocalIso R S",
"ppTerm": "?m.20",
"assigned": true,
"usedConstants": [
"Algebra.IsLocalIso.of_span_... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 79,
"column": 74
} | {
"line": 79,
"column": 76
} | {
"line": 79,
"column": 77
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nH : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)\nI : Ideal R\nx✝¹ : FinitePresentation R ↥I := finit... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalIso | {
"line": 110,
"column": 70
} | {
"line": 110,
"column": 72
} | {
"line": 111,
"column": 4
} | [
{
"pp": "ι : Type u_3\nR : Type u_4\nS : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → CommRing (S i)\ninst✝² : (i : ι) → Algebra R (S i)\ninst✝¹ : Finite ι\ninst✝ : ∀ (i : ι), IsLocalIso R (S i)\nthis : (i : ι) → Algebra ((i : ι) → S i) (S i) := fun i ↦ (Pi.evalAlgHom R S i).toAlgebra\ni : ι\n⊢ IsL... | [] | by | Lean.Elab.Tactic.evalWithAnnotateState | by |
Mathlib.RingTheory.LocalIso | {
"line": 107,
"column": 32
} | {
"line": 107,
"column": 34
} | {
"line": 108,
"column": 2
} | [
{
"pp": "ι : Type u_3\nR : Type u_4\nS : ι → Type u_5\ninst✝⁴ : CommSemiring R\ninst✝³ : (i : ι) → CommRing (S i)\ninst✝² : (i : ι) → Algebra R (S i)\ninst✝¹ : Finite ι\ninst✝ : ∀ (i : ι), IsLocalIso R (S i)\n⊢ IsLocalIso R ((i : ι) → S i)",
"ppTerm": "?m.15",
"assigned": true,
"usedConstants": [
... | [] | by | [anonymous] | by |
Mathlib.RingTheory.LocalProperties.Injective | {
"line": 84,
"column": 41
} | {
"line": 84,
"column": 43
} | {
"line": 84,
"column": 44
} | [
{
"pp": "R : Type u\ninst✝⁴ : CommRing R\nM : Type v\ninst✝³ : AddCommGroup M\ninst✝² : Module R M\ninst✝¹ : Small.{v, u} R\ninst✝ : IsNoetherianRing R\nH : ∀ (I : Ideal R) (x : I.IsMaximal), Injective (Localization.AtPrime I) (LocalizedModule I.primeCompl M)\nI : Ideal R\nx✝¹ : FinitePresentation R ↥I := finit... | [] | by | [anonymous] | by |
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