SystemOfParameters.lean`

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3 declarations, 0 contain sorry, 0 declared as axioms

Systems of parameters

For a local ring R of Krull dimension d, a system of parameters is a list of d elements of R generating an ideal whose radical is the maximal ideal. Over a Noetherian local ring this is the usual notion: a list of d elements of the maximal ideal with R ⧸ Ideal.ofList rs Artinian. Krull's height theorem gives a system of parameters over every Noetherian local ring, and that is the only case in which this definition is intended to be used.

Main declarations

IsLocalRing.IsSystemOfParameters R rs: the list rs is a system of parameters of R.

Imports (3)

Imported by 2 files

structure IsLocalRing.IsSystemOfParameters A system of parameters of a local ring R is a list of ringKrullDim R elements of R generating an ideal whose radical is the maximal ideal line 46 source
Source documentation
A system of parameters of a local ring R is a list of ringKrullDim R elements of R generating an ideal whose radical is the maximal ideal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, List
2 instances, coercions and auxiliary constants left out.
```
structure IsSystemOfParameters (rs : List R) : Prop where
…
```
theorem IsLocalRing.IsSystemOfParameters.mem_maximalIdeal The elements of a system of parameters lie in the maximal ideal line 55 source
Source documentation
The elements of a system of parameters lie in the maximal ideal.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, List, IsLocalRing.IsSystemOfParameters, Membership.mem, Ideal, IsLocalRing.maximalIdeal
Proof uses: Eq.rec, Ideal.radical, Ideal.ofList, Eq, Ideal.le_radical, Ideal.subset_span, Set.ofPred, IsLocalRing.IsSystemOfParameters.radical_eq
14 instances, coercions and auxiliary constants left out.
```
theorem IsSystemOfParameters.mem_maximalIdeal {rs : List R} (h : IsSystemOfParameters R rs)
    {r : R} (hr : r ∈ rs) : r ∈ maximalIdeal R :=
…
```
Mentions: IsLocalRing.IsSystemOfParameters
theorem IsLocalRing.isSystemOfParameters_nil Over a local ring of Krull dimension zero, the empty list is a system of parameters line 62 source
Source documentation
Over a local ring of Krull dimension zero, the empty list is a system of parameters.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, Ring.KrullDimLE, OfNat.ofNat, Nat, IsLocalRing.IsSystemOfParameters, List.nil
Proof uses: IsLocalRing.IsSystemOfParameters.mk, Eq.mpr, Eq, WithBot, ENat, List.length, ringKrullDim, id, congrFun', congrArg, CharP.cast_eq_zero, Eq.symm, Iff.mp, ringKrullDimZero_iff_ringKrullDim_eq_zero, Ideal, Ideal.radical, Ideal.ofList, IsLocalRing.maximalIdeal, Bot.bot, Ideal.ofList_nil and 1 more
27 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
```
theorem isSystemOfParameters_nil [Ring.KrullDimLE 0 R] :
    IsSystemOfParameters R ([] : List R) where
…
```
Mentions: IsLocalRing.IsSystemOfParameters