IsResidueAlgebra.lean`

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12 declarations, 0 contain sorry, 0 declared as axioms

Residue algebras

The typeclass IsResidueAlgebra 𝓞 A expressing that an 𝓞-algebra A has the same residue field as 𝓞, that is, the induced map on residue fields is bijective.

Imports (3)

Imported by 2 files

class IsResidueAlgebra IsResidueAlgebra 𝓞 indicates that A [Algebra 𝓞 A] has the same residue field as 𝓞 line 35 source
Source documentation
IsResidueAlgebra 𝓞 indicates that A [Algebra 𝓞 A] has the same residue field as 𝓞. It is sufficient for the natural map 𝓞 to 𝓴 A to be surjective. The actual ≃+* between residue fields is given in IsResidueAlgebra.ringEquiv.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing
2 instances, coercions and auxiliary constants left out.
```
class IsResidueAlgebra : Prop where
…
```
lemma IsResidueAlgebra.algebraMap_surjective line 42 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, Function.Surjective, IsLocalRing.ResidueField, RingHom, Algebra.algebraMap
Proof uses: IsResidueAlgebra.isSurjective'
11 instances, coercions and auxiliary constants left out.
lemma IsResidueAlgebra.algebraMap_bijective line 45 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, IsLocalHom, RingHom, Algebra.algebraMap, Function.Bijective, IsLocalRing.ResidueField
Proof uses: Function.Surjective, IsLocalRing.residue, IsLocalRing.residue_surjective, IsResidueAlgebra.algebraMap_surjective, And.intro, Function.Injective, RingHom.injective, Iff.mp, Function.comp, Function.Surjective.of_comp_iff
28 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
def IsResidueAlgebra.algEquiv The isomorphism of residue fields for a residue algebra line 53 source
Source documentation
The isomorphism of residue fields for a residue algebra.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, IsLocalHom, RingHom, Algebra.algebraMap, AlgEquiv, IsLocalRing.ResidueField
Proof uses: AlgEquiv.ofBijective, IsScalarTower.toAlgHom, IsResidueAlgebra.algebraMap_bijective
22 instances, coercions and auxiliary constants left out.
```
noncomputable def algEquiv : 𝓴 𝓞 ≃ₐ[𝓞] 𝓴 A :=
…
```
instance IsResidueAlgebra.(instance at line 56) line 56 source
instance IsResidueAlgebra.(instance at line 60) line 60 source
lemma IsResidueAlgebra.of_restrictScalars line 74 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, IsScalarTower
Proof uses: IsResidueAlgebra.mk, Function.Surjective.of_comp, IsLocalRing.ResidueField, RingHom, Algebra.algebraMap, Eq.mpr, Function.Surjective, Function.comp, RingHom.comp, id, Eq, congrArg, Eq.symm, RingHom.coe_comp, IsScalarTower.algebraMap_eq, IsResidueAlgebra.algebraMap_surjective
14 instances, coercions and auxiliary constants left out.

Tactic trace: 3 steps
Open to load the checked before-and-after goal states.
lemma IsResidueAlgebra.exists_sub_mem_maximalIdeal line 86 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, Exists, Membership.mem, Ideal, IsLocalRing.maximalIdeal, HSub.hSub, RingHom, Algebra.algebraMap
Proof uses: Eq, IsLocalRing.ResidueField, IsLocalRing.residue, IsResidueAlgebra.algebraMap_surjective, Exists.intro, Eq.mpr, HasQuotient.Quotient, Ideal.Quotient.mk, id, congrArg, Eq.symm, propext, Ideal.Quotient.eq
39 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
def IsResidueAlgebra.preimage line 94 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra
Proof uses: Exists.choose, Membership.mem, Ideal, IsLocalRing.maximalIdeal, HSub.hSub, RingHom, Algebra.algebraMap, IsResidueAlgebra.exists_sub_mem_maximalIdeal
17 instances, coercions and auxiliary constants left out.
lemma IsResidueAlgebra.preimage_spec line 96 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, Membership.mem, Ideal, IsLocalRing.maximalIdeal, HSub.hSub, RingHom, Algebra.algebraMap, IsResidueAlgebra.preimage
Proof uses: Exists.choose_spec, IsResidueAlgebra.exists_sub_mem_maximalIdeal
30 instances, coercions and auxiliary constants left out.
lemma IsResidueAlgebra.residue_preimage line 99 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, Eq, IsLocalRing.ResidueField, RingHom, IsLocalRing.residue, Algebra.algebraMap, IsResidueAlgebra.preimage
Proof uses: Eq.symm, HasQuotient.Quotient, Ideal, IsLocalRing.maximalIdeal, Ideal.Quotient.mk, Iff.mpr, Membership.mem, HSub.hSub, Ideal.Quotient.eq, IsResidueAlgebra.preimage_spec
27 instances, coercions and auxiliary constants left out.
lemma IsResidueAlgebra.residue_preimage_eq_iff line 103 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, Algebra, IsLocalRing, IsResidueAlgebra, IsLocalHom, RingHom, Algebra.algebraMap, IsLocalRing.ResidueField, Iff, Eq, IsLocalRing.residue, IsResidueAlgebra.preimage, IsLocalRing.ResidueField.map
Proof uses: Eq.mpr, id, congrArg, Eq.symm, propext, Function.Injective.eq_iff, And.left, Function.Injective, Function.Surjective, IsResidueAlgebra.algebraMap_bijective, IsLocalRing.ResidueField.map_residue, IsResidueAlgebra.residue_preimage, Iff.rfl
21 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.