DimEqDelta.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

`dim R = δ(R)`

The Krull dimension of a Noetherian local ring equals the minimal number of generators of an ideal of definition (an 𝔪-primary ideal) — the equivalence (1) ↔ (3) of the dimension theorem (Stacks Project, tag 00KQ).

Both inequalities are assembled from Krull's height theorem and its converse, which are already in Mathlib (Mathlib/RingTheory/Ideal/KrullsHeightTheorem.lean, He–Jiang–Merten–Wang–Yang–Zhang, 2025):

dim R ≤ δ(R) (ringKrullDim_le_minGenPrimary): if I is an ideal of definition then 𝔪 is minimal over I (maximalIdeal_mem_minimalPrimes), so Krull's height theorem (Ideal.height_le_spanRank_toENat_of_mem_minimalPrimes) bounds dim R = height 𝔪 by the number of generators of I.
δ(R) ≤ dim R (minGenPrimary_le_ringKrullDim): the converse of the height theorem (Ideal.exists_spanRank_eq_and_height_eq) produces J ≤ 𝔪 generated by height 𝔪 elements with height J = height 𝔪. Every minimal prime of J then has height dim R, hence equals 𝔪, so J.radical = 𝔪: J is a system of parameters for R.

The classical proof of the second inequality constructs a system of parameters by prime avoidance (Atiyah–Macdonald, Ch. 11); here that content is packaged inside Mathlib's exists_spanRank_eq_and_height_eq.

Main result

DimensionTheorem.ringKrullDim_eq_minGenPrimary : ringKrullDim R = δ(R) for R Noetherian local.

Imports (1)

FLT.Slop.DimensionTheorem.Defs

Imported by 2 files

lemma DimensionTheorem.maximalIdeal_mem_minimalPrimes If I has radical 𝔪, then 𝔪 is minimal over I line 52 source
Source documentation
If I has radical 𝔪, then 𝔪 is minimal over I.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, Ideal, Eq, Ideal.radical, IsLocalRing.maximalIdeal, Membership.mem, Set, Ideal.minimalPrimes
Proof uses: Eq.mpr, id, congrArg, Eq.symm, Ideal.radical_minimalPrimes, Singleton.singleton, Ideal.minimalPrimes_eq_subsingleton_self, Set.mem_singleton
9 instances, coercions and auxiliary constants left out.

Tactic trace: 2 steps
Open to load the checked before-and-after goal states.
```
lemma maximalIdeal_mem_minimalPrimes {I : Ideal R}
    (h : I.radical = maximalIdeal R) : maximalIdeal R ∈ I.minimalPrimes := by
…
```
theorem DimensionTheorem.ringKrullDim_le_minGenPrimary Dimension theorem, easy inequality: dim R ≤ δ(R), from Krull's height theorem line 59 source
Source documentation
Dimension theorem, easy inequality: dim R ≤ δ(R), from Krull's height theorem.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, LE.le, WithBot, ENat, ringKrullDim, DimensionTheorem.minGenPrimary
Proof uses: Set.Nonempty, Nat, Set.ofPred, Exists, Ideal, And, Eq, Ideal.radical, IsLocalRing.maximalIdeal, Submodule.spanFinrank, Exists.intro, Membership.mem, Set, And.intro, Ideal.IsPrime.radical, Ideal.IsMaximal.isPrime, rfl, Nat.sInf_mem, Ideal.height, OrderRingHom and 20 more
43 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
```
theorem ringKrullDim_le_minGenPrimary :
    ringKrullDim R ≤ (minGenPrimary R : WithBot ℕ∞) := by
…
```
Mentions: DimensionTheorem.minGenPrimary
theorem DimensionTheorem.minGenPrimary_le_ringKrullDim Dimension theorem, converse inequality: δ(R) ≤ dim R; there exists a system of parameters, i.e. an 𝔪-primary ideal generated by dim R elements line 81 source
Source documentation
Dimension theorem, converse inequality: δ(R) ≤ dim R; there exists a system of parameters, i.e. an 𝔪-primary ideal generated by dim R elements.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, LE.le, WithBot, ENat, DimensionTheorem.minGenPrimary, ringKrullDim
Proof uses: Ne, Ideal, IsLocalRing.maximalIdeal, Top.top, Ideal.IsMaximal.ne_top, And, Eq, Cardinal, Submodule.spanRank, Ideal.height, Exists, Ideal.exists_spanRank_eq_and_height_eq, Iff.mp, top_le_iff, Eq.rec, Submodule.spanFinrank, Cardinal.ofENat_eq_nat, Eq.mpr, id, congrArg and 35 more
77 instances, coercions and auxiliary constants left out.

Tactic trace: 21 steps
Open to load the checked before-and-after goal states.
```
theorem minGenPrimary_le_ringKrullDim :
    (minGenPrimary R : WithBot ℕ∞) ≤ ringKrullDim R := by
…
```
Mentions: DimensionTheorem.minGenPrimary
theorem DimensionTheorem.ringKrullDim_eq_minGenPrimary dim R = δ(R) for a Noetherian local ring line 116 source
Source documentation
dim R = δ(R) for a Noetherian local ring.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Eq, WithBot, ENat, ringKrullDim, DimensionTheorem.minGenPrimary
Proof uses: le_antisymm, DimensionTheorem.ringKrullDim_le_minGenPrimary, DimensionTheorem.minGenPrimary_le_ringKrullDim
16 instances, coercions and auxiliary constants left out.
```
theorem ringKrullDim_eq_minGenPrimary :
    ringKrullDim R = (minGenPrimary R : WithBot ℕ∞) :=
…
```
Mentions: DimensionTheorem.minGenPrimary