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Explanation legend Lean4 Domain Explanation Language (L4DEL)

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L4DEL: Lean4 Domain Explanation Language

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4 declarations, 0 contain sorry, 0 declared as axioms

The dimension theorem for Noetherian local rings

For a Noetherian local ring (R, 𝔪), the following three quantities coincide (Atiyah–Macdonald, Thm. 11.14; Stacks Project, tag 00KQ):

dim R — the Krull dimension (Mathlib's ringKrullDim R);
d(R) — the growth degree of the Hilbert–Samuel function n ↦ ℓ(R ⧸ 𝔪ⁿ) (growthDeg R);
δ(R) — the least number of generators of an ideal of definition, i.e. of an 𝔪-primary ideal (minGenPrimary R).

The proof is the classical cycle dim ≤ d ≤ δ ≤ dim:

dim ≤ d — Artin–Rees induction (FLT/Slop/DimensionTheorem/DimLeGrowth.lean);
d ≤ δ — monomial counting (FLT/Slop/DimensionTheorem/GrowthLeDelta.lean);
δ ≤ dim — existence of systems of parameters, from the converse of Krull's height theorem (FLT/Slop/DimensionTheorem/DimEqDelta.lean); the direct inequality dim ≤ δ (Krull's height theorem) is also proved there, so dim = δ does not depend on the Hilbert–Samuel machinery.

Main results

DimensionTheorem.dimension_theorem — dim R = d(R) and dim R = δ(R).
DimensionTheorem.ringKrullDim_eq_growthDeg, ringKrullDim_le_growthDeg, growthDeg_eq_minGenPrimary — the individual identities.

Formal scope

This development identifies dimension with the polynomial-growth degree of the Hilbert–Samuel function; it does not construct the eventual Hilbert–Samuel polynomial itself (the Hilbert–Serre theorem), for which the growth-order and polynomial-degree definitions of d(R) agree. The results depend only on the standard axioms (propext, Classical.choice, Quot.sound); the development is sorry-free.

Imports (3)

Imported by 2 files

theorem DimensionTheorem.ringKrullDim_le_growthDeg dim R ≤ d(R) line 58 source
Source documentation
dim R ≤ d(R).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, LE.le, WithBot, ENat, ringKrullDim, DimensionTheorem.growthDeg
Proof uses: DimensionTheorem.ringKrullDim_le_of_growthLE, DimensionTheorem.growthLE_hilbertSamuel_growthDeg
10 instances, coercions and auxiliary constants left out.
```
theorem ringKrullDim_le_growthDeg :
    ringKrullDim R ≤ (growthDeg R : WithBot ℕ∞) :=
…
```
Mentions: DimensionTheorem.growthDeg
theorem DimensionTheorem.ringKrullDim_eq_growthDeg The dimension theorem for Noetherian local rings, first equality: the Krull dimension equals the growth degree of the Hilbert–Samuel function line 64 source
Source documentation
The dimension theorem for Noetherian local rings, first equality: the Krull dimension equals the growth degree of the Hilbert–Samuel function.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Eq, WithBot, ENat, ringKrullDim, DimensionTheorem.growthDeg
Proof uses: le_antisymm, DimensionTheorem.ringKrullDim_le_growthDeg, Trans.trans, LE.le, DimensionTheorem.minGenPrimary, cast, Nat, Eq.symm, Iff.mpr, Nat.cast_le, DimensionTheorem.growthDeg_le_minGenPrimary, DimensionTheorem.minGenPrimary_le_ringKrullDim
49 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem ringKrullDim_eq_growthDeg :
    ringKrullDim R = (growthDeg R : WithBot ℕ∞) := by
…
```
Mentions: DimensionTheorem.growthDeg
theorem DimensionTheorem.dimension_theorem The dimension theorem for Noetherian local rings: dim R = d(R) = δ(R) line 74 source
Source documentation
The dimension theorem for Noetherian local rings: dim R = d(R) = δ(R).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, And, Eq, WithBot, ENat, ringKrullDim, DimensionTheorem.growthDeg, DimensionTheorem.minGenPrimary
Proof uses: And.intro, DimensionTheorem.ringKrullDim_eq_growthDeg, DimensionTheorem.ringKrullDim_eq_minGenPrimary
10 instances, coercions and auxiliary constants left out.
```
theorem dimension_theorem :
    ringKrullDim R = (growthDeg R : WithBot ℕ∞) ∧
      ringKrullDim R = (minGenPrimary R : WithBot ℕ∞) :=
…
```
Mentions: DimensionTheorem.growthDeg, DimensionTheorem.minGenPrimary
theorem DimensionTheorem.growthDeg_eq_minGenPrimary The three quantities as natural numbers: d(R) = δ(R) line 80 source
Source documentation
The three quantities as natural numbers: d(R) = δ(R).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, IsNoetherianRing, Eq, Nat, DimensionTheorem.growthDeg, DimensionTheorem.minGenPrimary
Proof uses: WithBot, ENat, ringKrullDim, DimensionTheorem.ringKrullDim_eq_growthDeg, DimensionTheorem.ringKrullDim_eq_minGenPrimary, Eq.trans, Eq.symm, cast
13 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
```
theorem growthDeg_eq_minGenPrimary : growthDeg R = minGenPrimary R := by
…
```
Mentions: DimensionTheorem.growthDeg, DimensionTheorem.minGenPrimary