TensorProduct.lean`

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5 declarations, 0 contain sorry, 0 declared as axioms

Tensor product compatibilities for the finite adele ring

Compatibilities between tensor products and the local components of the finite adele ring of a Dedekind domain.

Imports (5)

Imported by 2 files

def IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent Documented declaration: localcomponent line 54 source
Source documentation
If φ : 𝔸_K^f ⊗[K] V → 𝔸_K^f ⊗[K] V is 𝔸_K^f-linear and p : HeightOneSpectrum (𝓞 K) then localcomponent R K V p φ : Kₚ ⊗[K] V →[K] Kₚ ⊗[K] V is the associated map φₚ defined as Kₚ ⊗[K] V --(single)--> 𝔸_K^f ⊗ V --(φ)--> 𝔸_K^f ⊗ V --(eval)--> Kₚ ⊗ V. This map morally satisfies φ = Πₚ φₚ but because source of φ isn't literally a restricted product we cannot make this assertion.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, DecidableEq, IsDedekindDomain.HeightOneSpectrum, Field, Algebra, IsFractionRing, AddCommGroup, Module, FiniteDimensional, TopologicalSpace, TensorProduct, IsDedekindDomain.FiniteAdeleRing, IsTopologicalAddGroup, IsModuleTopology, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ContinuousLinearMap, RingHom.id
Proof uses: ContinuousLinearMap.comp, ContinuousLinearMap.rTensor', ContinuousAlgHom.toContinuousLinearMap, IsDedekindDomain.FiniteAdeleRing.evalContinuousAlgebraMap, LinearMap, ContinuousLinearMap.mk, IsDedekindDomain.FiniteAdeleRing.singleContinuousLinearMap
73 instances, coercions and auxiliary constants left out.
```
noncomputable def TensorProduct.localcomponent (p : HeightOneSpectrum R)
    (φ : FiniteAdeleRing R K ⊗[K] V →L[FiniteAdeleRing R K]
      FiniteAdeleRing R K ⊗[K] V) :
    p.adicCompletion K ⊗[K] V →L[K] p.adicCompletion K ⊗[K] V :=
…
```
lemma IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent_id_apply line 71 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, DecidableEq, IsDedekindDomain.HeightOneSpectrum, Field, Algebra, IsFractionRing, AddCommGroup, Module, FiniteDimensional, TopologicalSpace, TensorProduct, IsDedekindDomain.FiniteAdeleRing, IsTopologicalAddGroup, IsModuleTopology, IsDedekindDomain.HeightOneSpectrum.adicCompletion, Eq, ContinuousLinearMap, RingHom.id, IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent and 1 more
Proof uses: LinearMap, LinearMap.comp, ContinuousAlgHom.toContinuousLinearMap, IsDedekindDomain.FiniteAdeleRing.evalContinuousAlgebraMap, IsDedekindDomain.FiniteAdeleRing.singleContinuousLinearMap, LinearMap.id, LinearMap.ext, IsDedekindDomain.FiniteAdeleRing.evalContinuousAlgebraMap_singleContinuousLinearMap, of_eq_true, Eq.trans, True, congrFun', congrArg, ContinuousLinearMap.comp, ContinuousLinearMap.mk, LinearMap.rTensor, ContinuousLinearMap.comp.congr_simp, ContinuousLinearMap.rTensor', id, Eq.refl and 3 more
81 instances, coercions and auxiliary constants left out.

Tactic trace: 4 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent_comp_apply line 83 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, DecidableEq, IsDedekindDomain.HeightOneSpectrum, Field, Algebra, IsFractionRing, AddCommGroup, Module, FiniteDimensional, TopologicalSpace, TensorProduct, IsDedekindDomain.FiniteAdeleRing, IsTopologicalAddGroup, IsModuleTopology, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ContinuousLinearMap, RingHom.id, Eq, IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent and 1 more
Proof uses: LinearMap, LinearMap.rTensor, LinearMap.comp, IsDedekindDomain.FiniteAdeleRing.singleContinuousLinearMap, ContinuousAlgHom.toContinuousLinearMap, IsDedekindDomain.FiniteAdeleRing.evalContinuousAlgebraMap, HSMul.hSMul, IsDedekindDomain.FiniteAdeleRing.localIdempotent, LinearMap.lsmul, congr_arg, IsDedekindDomain.FiniteAdeleRing.singleContinuousAlgebraMap_comp_evalContinuousLinearMap, TensorProduct.inductionOn, of_eq_true, TensorProduct.tmul, HMul.hMul, Eq.trans, True, congrFun', congrArg, id and 16 more
121 instances, coercions and auxiliary constants left out.

Tactic trace: 7 steps
Open to load the checked before-and-after goal states.
lemma IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent_apply Documented declaration: localcomponent_apply line 117 source
Source documentation
If φ : 𝔸_K^f ⊗ V → 𝔸_K^f ⊗ V is 𝔸_K^f-linear and φₚ is its local component at a place p then for all x : 𝔸_K^f ⊗ V we have (evalₚ ⊗ id_V) (φ x) = φₚ ((evalₚ ⊗ id_V) x), or, more colloquiually, (φ x)ₚ = φₚ (xₚ).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, DecidableEq, IsDedekindDomain.HeightOneSpectrum, Field, Algebra, IsFractionRing, AddCommGroup, Module, FiniteDimensional, TopologicalSpace, TensorProduct, IsDedekindDomain.FiniteAdeleRing, IsTopologicalAddGroup, IsModuleTopology, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ContinuousLinearMap, RingHom.id, Eq, ContinuousLinearMap.rTensor' and 6 more
Proof uses: id, Eq.mpr, IsDedekindDomain.FiniteAdeleRing.singleContinuousLinearMap, ContinuousLinearMap.comp, congrArg, Eq.symm, ContinuousLinearMap.rTensor'_comp_apply, LinearMap, LinearMap.rTensor, LinearMap.lsmul, IsDedekindDomain.FiniteAdeleRing.localIdempotent, IsDedekindDomain.FiniteAdeleRing.singleContinuousAlgebraMap_comp_evalContinuousLinearMap, HSMul.hSMul, TensorProduct.inductionOn, LinearMap.rTensor_tmul, of_eq_true, HAdd.hAdd, Eq.trans, True, congr and 13 more
145 instances, coercions and auxiliary constants left out.

Tactic trace: 12 steps
Open to load the checked before-and-after goal states.
```
lemma TensorProduct.localcomponent_apply
    (φ : FiniteAdeleRing R K ⊗[K] V →L[FiniteAdeleRing R K] FiniteAdeleRing R K ⊗[K] V)
    (x : FiniteAdeleRing R K ⊗[K] V) (p : HeightOneSpectrum R) :
    (ContinuousLinearMap.rTensor' V
      (evalContinuousAlgebraMap R K p).toContinuousLinearMap) (φ x) =
    TensorProduct.localcomponent R K V p φ ((ContinuousLinearMap.rTensor' V
      (evalContinuousAlgebraMap R K p).toContinuousLinearMap) x) := by
…
```
Mentions: ContinuousLinearMap.rTensor', IsDedekindDomain.FiniteAdeleRing.evalContinuousAlgebraMap, IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent
def IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponentEquiv Documented declaration: localcomponentEquiv line 146 source
Source documentation
If φ : 𝔸_K^f ⊗[K] V → 𝔸_K^f ⊗[K] V is 𝔸_K^f-linear and p : HeightOneSpectrum (𝓞 K) then localcomponent R K V p φ : Kₚ ⊗[K] V →[K] Kₚ ⊗[K] V is the associated map φₚ defined as Kₚ ⊗[K] V --(single)--> 𝔸_K^f ⊗ V --(φ)--> 𝔸_K^f ⊗ V --(eval)--> Kₚ ⊗ V. This map morally satisfies φ = Πₚ φₚ but because source of φ isn't literally a restricted product we cannot make this assertion.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommRing, IsDedekindDomain, DecidableEq, IsDedekindDomain.HeightOneSpectrum, Field, Algebra, IsFractionRing, AddCommGroup, Module, FiniteDimensional, TopologicalSpace, TensorProduct, IsDedekindDomain.FiniteAdeleRing, IsTopologicalAddGroup, IsModuleTopology, IsDedekindDomain.HeightOneSpectrum.adicCompletion, ContinuousLinearEquiv, RingHom.id
Proof uses: ContinuousLinearMap, IsDedekindDomain.FiniteAdeleRing.TensorProduct.localcomponent, ContinuousLinearEquiv.mk, LinearEquiv.mk, ContinuousLinearEquiv.symm
74 instances, coercions and auxiliary constants left out.

Tactic trace: 6 steps
Open to load the checked before-and-after goal states.
```
noncomputable def TensorProduct.localcomponentEquiv (p : HeightOneSpectrum R)
    (φ : FiniteAdeleRing R K ⊗[K] V ≃L[FiniteAdeleRing R K]
      FiniteAdeleRing R K ⊗[K] V) :
    p.adicCompletion K ⊗[K] V ≃L[K] p.adicCompletion K ⊗[K] V where
…
```