AdicTopology.lean`

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21 declarations, 0 contain sorry, 0 declared as axioms

Local rings with the adic topology

The class IsLocalRing.IsAdicTopology R recording that the topology on a local ring R is the adic topology associated to its maximal ideal, together with basic openness lemmas.

Imports (13)

Imported by 5 files

class IsLocalRing.IsAdicTopology Documented declaration: IsAdicTopology line 41 source
Source documentation
IsAdicTopology R says that the topology on the local topological ring R is the maximal ideal-adic one, that is, that a basis of neighbourhoods of 0 in R is given by powers of the maximal ideal of R.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing
5 instances, coercions and auxiliary constants left out.
```
class IsAdicTopology (R) [CommRing R] [IsLocalRing R]
    [TopologicalSpace R] [IsTopologicalRing R] : Prop where
…
```
instance IsLocalRing.(instance at line 47) line 47 source
lemma IsLocalRing.isOpen_maximalIdeal_pow'' line 51 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, Nat, IsOpen, Ideal, HPow.hPow, IsLocalRing.maximalIdeal
Proof uses: And.left, Set, Membership.mem, Filter, nhds, OfNat.ofNat, Exists, LE.le, Iff.mp, IsAdic, And, isAdic_iff, IsLocalRing.IsAdicTopology.isAdic
31 instances, coercions and auxiliary constants left out.
lemma IsLocalRing.isOpen_maximalIdeal' line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, IsOpen, Ideal, IsLocalRing.maximalIdeal
Proof uses: Eq.rec, HPow.hPow, Nat, OfNat.ofNat, Eq, IsLocalRing.isOpen_maximalIdeal_pow'', pow_one
26 instances, coercions and auxiliary constants left out.
lemma IsLocalRing.hasBasis_maximalIdeal_pow line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, Filter.HasBasis, Nat, nhds, OfNat.ofNat, True, Ideal, HPow.hPow, IsLocalRing.maximalIdeal
Proof uses: Eq.rec, Ideal.adicTopology, Eq, Ideal.hasBasis_nhds_zero_adic, Eq.symm, IsLocalRing.IsAdicTopology.isAdic
38 instances, coercions and auxiliary constants left out.

Tactic trace: 11 steps
Open to load the checked before-and-after goal states.
instance IsLocalRing.(instance at line 62) line 62 source
instance IsLocalRing.(instance at line 64) line 64 source
instance IsLocalRing.(instance at line 72) line 72 source
Source comment
This is actually an iff
```
instance (priority := 100) [IsArtinianRing R] : DiscreteTopology R := by
…
```
lemma IsLocalRing.Submodule.isCompact_of_fg line 79 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, CompactSpace, Submodule, Submodule.FG, IsCompact
Proof uses: ContinuousAdd, IsModuleTopology.toContinuousAdd, Finset, Eq, Submodule.span, LinearMap.range, Subtype, Membership.mem, RingHom.id, Fintype.linearCombination, of_eq_true, Eq.trans, True, congrFun', congrArg, Set.range, Fintype.range_linearCombination, Set, Subtype.range_coe_subtype, eq_self and 14 more
58 instances, coercions and auxiliary constants left out.

Tactic trace: 8 steps
Open to load the checked before-and-after goal states.
lemma IsLocalRing.Ideal.isCompact_of_fg line 93 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, CompactSpace, Ideal, Ideal.FG, IsCompact
Proof uses: IsLocalRing.Submodule.isCompact_of_fg
19 instances, coercions and auxiliary constants left out.
lemma IsLocalRing.IsModuleTopology.compactSpace line 97 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, AddCommGroup, Module, IsModuleTopology, CompactSpace, Module.Finite
Proof uses: CompactSpace.mk, IsLocalRing.Submodule.isCompact_of_fg, Top.top, Submodule, Module.Finite.fg_top
35 instances, coercions and auxiliary constants left out.
lemma IsLocalRing.isCompact_of_isNoetherianRing line 105 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, TopologicalSpace, IsTopologicalRing, IsNoetherianRing, CompactSpace, Ideal, IsCompact
Proof uses: IsLocalRing.Ideal.isCompact_of_fg, IsNoetherian.noetherian
16 instances, coercions and auxiliary constants left out.
lemma IsLocalRing.isOpen_iff_finite_quotient' line 109 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, CompactSpace, Ideal, Iff, IsOpen, Finite, HasQuotient.Quotient
Proof uses: Iff.intro, AddSubgroup.quotient_finite_of_isOpen, Submodule.toAddSubgroup, Nat, LE.le, HPow.hPow, IsLocalRing.maximalIdeal, Exists, IsLocalRing.exists_maximalIdeal_pow_le_of_isArtinianRing_quotient, Submodule, AddSubgroup.isOpen_mono, IsLocalRing.isOpen_maximalIdeal_pow''
54 instances, coercions and auxiliary constants left out.

Tactic trace: 19 steps
Open to load the checked before-and-after goal states.
instance IsLocalRing.(instance at line 119) line 119 source
instance IsLocalRing.(instance at line 122) line 122 source
instance IsLocalRing.(instance at line 128) line 128 source
lemma IsLocalRing.compactSpace_of_finite_residueField line 145 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, IsNoetherianRing, Finite, IsLocalRing.ResidueField, IsAdicComplete, IsLocalRing.maximalIdeal, CompactSpace
Proof uses: RingHom, Nat, HasQuotient.Quotient, Ideal, HPow.hPow, Algebra.algebraMap, Ideal.finite_quotient_pow, IsNoetherian.noetherian, Continuous, id, continuous_algebraMap, Topology.IsClosedEmbedding, Topology.IsClosedEmbedding.mk, Topology.IsEmbedding.mk, Eq.mpr, Topology.IsInducing, Eq, Filter, nhds, OfNat.ofNat and 91 more
143 instances, coercions and auxiliary constants left out.

Tactic trace: 48 steps
Open to load the checked before-and-after goal states.
instance IsLocalRing.(instance at line 199) line 199 source
Source documentation
Any profinite noetherian ring has the 𝔪-adic topology.
```
instance (priority := 100) [IsNoetherianRing R]
…
```
lemma IsLocalRing.Continuous.of_isLocalHom line 209 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsLocalRing, TopologicalSpace, IsTopologicalRing, IsLocalRing.IsAdicTopology, RingHom, IsLocalHom, Continuous
Proof uses: continuous_of_continuousAt_zero, id, ContinuousAt, OfNat.ofNat, Eq.mpr, Filter.Tendsto, nhds, Eq, congrArg, map_zero, Iff.mpr, Nat, True, Exists, And, Membership.mem, Set, Ideal, HPow.hPow, IsLocalRing.maximalIdeal and 28 more
76 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
abbrev IsLocalRing.withIdeal The default WithIdeal structure on a local ring R, picking out the maximal ideal line 222 source
Source documentation
The default WithIdeal structure on a local ring R, picking out the maximal ideal.

standard axioms: propext what this means
Statement uses: CommRing, IsLocalRing, WithIdeal
Proof uses: WithIdeal.mk, IsLocalRing.maximalIdeal
4 instances, coercions and auxiliary constants left out.
```
abbrev withIdeal {R} [CommRing R] [IsLocalRing R] : WithIdeal R := ⟨maximalIdeal R⟩
…
```
instance IsLocalRing.(instance at line 226) line 226 source