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Found 186 declarations mentioning LinearMap.rTensor.

LinearMap.rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) : TensorProduct R N M →ₗ[R] TensorProduct R P M
LinearMap.rTensor_id 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] : LinearMap.rTensor M LinearMap.id = LinearMap.id
LinearMap.rTensor_def 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) : LinearMap.rTensor M f = TensorProduct.map f LinearMap.id
LinearMap.rTensor_id_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R N M) : (LinearMap.rTensor M LinearMap.id) x = x
LinearEquiv.coe_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) : ↑(LinearEquiv.rTensor M f) = LinearMap.rTensor M ↑f
LinearMap.rTensor_smul_action 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) (N : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (r : R) : LinearMap.rTensor M (DistribSMul.toLinearMap R N r) = DistribSMul.toLinearMap R (TensorProduct R N M) r
LinearMap.rTensor_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) (m : M) (n : N) : (LinearMap.rTensor M f) (n ⊗ₜ[R] m) = f n ⊗ₜ[R] m
LinearMap.lTensor_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : LinearMap.lTensor P g ∘ₗ LinearMap.rTensor N f = TensorProduct.map f g
LinearMap.rTensor_comp_lTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : LinearMap.rTensor Q f ∘ₗ LinearMap.lTensor M g = TensorProduct.map f g
LinearMap.rTensor_neg 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} {Q : Type u_5} [AddCommMonoid N] [AddCommGroup P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (f : N →ₗ[R] P) : LinearMap.rTensor Q (-f) = -LinearMap.rTensor Q f
LinearMap.rTensor_comp 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P) : LinearMap.rTensor M (g ∘ₗ f) = LinearMap.rTensor M g ∘ₗ LinearMap.rTensor M f
LinearMap.rTensor_zero 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] : LinearMap.rTensor M 0 = 0
LinearEquiv.coe_rTensor_symm 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N ≃ₗ[R] P) : ↑(LinearEquiv.rTensor M f).symm = LinearMap.rTensor M ↑f.symm
LinearMap.map_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R S] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (f' : S →ₗ[R] M) : TensorProduct.map f g ∘ₗ LinearMap.rTensor N f' = TensorProduct.map (f ∘ₗ f') g
LinearMap.rTensor_comp_map 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R S] [Module R P] [Module R Q] (f' : P →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) : LinearMap.rTensor Q f' ∘ₗ TensorProduct.map f g = TensorProduct.map (f' ∘ₗ f) g
LinearMap.lTensor_bij_iff_rTensor_bij 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) : Function.Bijective ⇑(LinearMap.lTensor M f) ↔ Function.Bijective ⇑(LinearMap.rTensor M f)
LinearMap.lTensor_inj_iff_rTensor_inj 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) : Function.Injective ⇑(LinearMap.lTensor M f) ↔ Function.Injective ⇑(LinearMap.rTensor M f)
LinearMap.lTensor_surj_iff_rTensor_surj 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) : Function.Surjective ⇑(LinearMap.lTensor M f) ↔ Function.Surjective ⇑(LinearMap.rTensor M f)
LinearMap.rTensor_mul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f g : Module.End R N) : LinearMap.rTensor M (f * g) = LinearMap.rTensor M f * LinearMap.rTensor M g
LinearMap.comm_comp_lTensor_comp_comm_eq 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N →ₗ[R] P) : ↑(TensorProduct.comm R Q P) ∘ₗ LinearMap.lTensor Q g ∘ₗ ↑(TensorProduct.comm R N Q) = LinearMap.rTensor Q g
LinearMap.comm_comp_rTensor_comp_comm_eq 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N →ₗ[R] P) : ↑(TensorProduct.comm R P Q) ∘ₗ LinearMap.rTensor Q g ∘ₗ ↑(TensorProduct.comm R Q N) = LinearMap.lTensor Q g
LinearMap.lTensor_comp_comm 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] P) : LinearMap.lTensor N f ∘ₗ ↑(TensorProduct.comm R M N) = ↑(TensorProduct.comm R P N) ∘ₗ LinearMap.rTensor N f
LinearMap.rTensor_comp_comm 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] P) : LinearMap.rTensor N f ∘ₗ ↑(TensorProduct.comm R N M) = ↑(TensorProduct.comm R N P) ∘ₗ LinearMap.lTensor N f
LinearMap.rTensor_add 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f g : N →ₗ[R] P) : LinearMap.rTensor M (f + g) = LinearMap.rTensor M f + LinearMap.rTensor M g
LinearMap.rTensor_pow 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : M →ₗ[R] M) (n : ℕ) : LinearMap.rTensor N f ^ n = LinearMap.rTensor N (f ^ n)
LinearMap.rTensor_smul 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (r : R) (f : N →ₗ[R] P) : LinearMap.rTensor M (r • f) = r • LinearMap.rTensor M f
LinearMap.rTensor_sub 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} {Q : Type u_5} [AddCommMonoid N] [AddCommGroup P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (f g : N →ₗ[R] P) : LinearMap.rTensor Q (f - g) = LinearMap.rTensor Q f - LinearMap.rTensor Q g
LinearMap.rTensor_comp_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (g : P →ₗ[R] Q) (f : N →ₗ[R] P) (x : TensorProduct R N M) : (LinearMap.rTensor M (g ∘ₗ f)) x = (LinearMap.rTensor M g) ((LinearMap.rTensor M f) x)
LinearMap.map_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R S] [Module R P] [Module R Q] (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (f' : S →ₗ[R] M) (x : TensorProduct R S N) : (TensorProduct.map f g) ((LinearMap.rTensor N f') x) = (TensorProduct.map (f ∘ₗ f') g) x
LinearMap.rTensor_map 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} {S : Type u_8} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [AddCommMonoid S] [Module R M] [Module R N] [Module R S] [Module R P] [Module R Q] (f' : P →ₗ[R] S) (f : M →ₗ[R] P) (g : N →ₗ[R] Q) (x : TensorProduct R M N) : (LinearMap.rTensor Q f') ((TensorProduct.map f g) x) = (TensorProduct.map (f' ∘ₗ f) g) x
LinearMap.lTensor_comm 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] P) (x : TensorProduct R M N) : (LinearMap.lTensor N f) ((TensorProduct.comm R M N) x) = (TensorProduct.comm R P N) ((LinearMap.rTensor N f) x)
LinearMap.rTensor_comm 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] {M : Type u_4} {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] P) (x : TensorProduct R N M) : (LinearMap.rTensor N f) ((TensorProduct.comm R N M) x) = (TensorProduct.comm R N P) ((LinearMap.lTensor N f) x)
LinearMap.coe_rTensorHom 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] : ⇑(LinearMap.rTensorHom M) = LinearMap.rTensor M
LinearMap.rTensor_comp_flip_mk 📋 Mathlib.LinearAlgebra.TensorProduct.Map
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : N →ₗ[R] P) (m : M) : LinearMap.rTensor M f ∘ₗ (TensorProduct.mk R N M).flip m = (TensorProduct.mk R P M).flip m ∘ₗ f
LinearMap.lid_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (f : N →ₗ[R] R) : ↑(TensorProduct.lid R M) ∘ₗ LinearMap.rTensor M f = TensorProduct.lift (LinearMap.lsmul R M ∘ₗ f)
TensorProduct.includeRight_lid 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
{R : Type u_1} [CommSemiring R] {M : Type u_4} [AddCommMonoid M] [Module R M] {S : Type u_10} [Semiring S] [Algebra R S] (m : TensorProduct R R M) : 1 ⊗ₜ[R] (TensorProduct.lid R M) m = (LinearMap.rTensor M (Algebra.algHom R R S).toLinearMap) m
LinearMap.lTensor_rTensor_comp_assoc 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (x : M →ₗ[R] N) : LinearMap.lTensor P (LinearMap.rTensor Q x) ∘ₗ ↑(TensorProduct.assoc R P M Q) = ↑(TensorProduct.assoc R P N Q) ∘ₗ LinearMap.rTensor Q (LinearMap.lTensor P x)
LinearMap.rTensor_tensor 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (g : P →ₗ[R] Q) : LinearMap.rTensor (TensorProduct R M N) g = ↑(TensorProduct.assoc R Q M N) ∘ₗ LinearMap.rTensor N (LinearMap.rTensor M g) ∘ₗ ↑(TensorProduct.assoc R P M N).symm
LinearMap.rTensor_lTensor_comp_assoc_symm 📋 Mathlib.LinearAlgebra.TensorProduct.Associator
{R : Type u_1} [CommSemiring R] (M : Type u_4) {N : Type u_5} {P : Type u_6} {Q : Type u_7} [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R Q] [Module R P] (x : M →ₗ[R] N) : LinearMap.rTensor Q (LinearMap.lTensor P x) ∘ₗ ↑(TensorProduct.assoc R P M Q).symm = ↑(TensorProduct.assoc R P N Q).symm ∘ₗ LinearMap.lTensor P (LinearMap.rTensor Q x)
TensorProduct.AlgebraTensorModule.smul_eq_lsmul_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
{R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] (a : A) (x : TensorProduct R M N) : a • x = (LinearMap.rTensor N ((Algebra.lsmul R R M) a)) x
LinearMap.rTensor_baseChange 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
{R : Type u_1} {A : Type u_2} {B : Type u_3} {M : Type u_4} {N : Type u_5} [CommSemiring R] [Semiring A] [Algebra R A] [Semiring B] [Algebra R B] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (φ : A →ₐ[R] B) (t : TensorProduct R A M) (f : M →ₗ[R] N) : (LinearMap.rTensor N φ.toLinearMap) ((LinearMap.baseChange A f) t) = (LinearMap.baseChange B f) ((LinearMap.rTensor M φ.toLinearMap) t)
TensorProduct.AlgebraTensorModule.coe_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
{R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) : ⇑((TensorProduct.AlgebraTensorModule.rTensor R N) f) = ⇑(LinearMap.rTensor N (↑R f))
TensorProduct.AlgebraTensorModule.restrictScalars_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
{R : Type uR} {A : Type uA} {M : Type uM} {N : Type uN} {P : Type uP} [CommSemiring R] [Semiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module A P] [IsScalarTower R A P] (f : M →ₗ[A] P) : ↑R ((TensorProduct.AlgebraTensorModule.rTensor R N) f) = LinearMap.rTensor N (↑R f)
TensorProduct.AlgebraTensorModule.rTensor_tensor 📋 Mathlib.LinearAlgebra.TensorProduct.Tower
(R : Type uR) (A : Type uA) {M : Type uM} {N : Type uN} {P : Type uP} (P' : Type uP') [CommSemiring R] [CommSemiring A] [Algebra R A] [AddCommMonoid M] [Module R M] [Module A M] [IsScalarTower R A M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module A P] [AddCommMonoid P'] [Module A P'] [Module R P] [IsScalarTower R A P] [Module R P'] [IsScalarTower R A P'] (g : P →ₗ[A] P') : LinearMap.rTensor (TensorProduct R M N) g = ↑(TensorProduct.AlgebraTensorModule.assoc R A A P' M N) ∘ₗ TensorProduct.AlgebraTensorModule.map (LinearMap.rTensor M g) LinearMap.id ∘ₗ ↑(TensorProduct.AlgebraTensorModule.assoc R A A P M N).symm
TensorProduct.directSumRight_comp_rTensor 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
(R : Type u) [CommSemiring R] {ι₁ : Type v₁} [DecidableEq ι₁] {M₁ : ι₁ → Type w₁} {M₁' : Type w₁'} {M₂' : Type w₂'} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [AddCommMonoid M₁'] [AddCommMonoid M₂'] [(i₁ : ι₁) → Module R (M₁ i₁)] [Module R M₁'] [Module R M₂'] (f : M₁' →ₗ[R] M₂') : ↑(TensorProduct.directSumRight R R M₂' M₁) ∘ₗ LinearMap.rTensor (DirectSum ι₁ fun i => M₁ i) f = (DirectSum.lmap fun x => LinearMap.rTensor (M₁ x) f) ∘ₗ ↑(TensorProduct.directSumRight R R M₁' M₁)
TensorProduct.directSumLeft_symm_of 📋 Mathlib.LinearAlgebra.DirectSum.TensorProduct
(R : Type u) [CommSemiring R] (S : Type u_1) [Semiring S] [Algebra R S] {ι₁ : Type v₁} [DecidableEq ι₁] {M₁ : ι₁ → Type w₁} {M₂' : Type w₂'} [(i₁ : ι₁) → AddCommMonoid (M₁ i₁)] [AddCommMonoid M₂'] [(i₁ : ι₁) → Module R (M₁ i₁)] [Module R M₂'] [(i₁ : ι₁) → Module S (M₁ i₁)] [∀ (i₁ : ι₁), IsScalarTower R S (M₁ i₁)] {i : ι₁} (x : TensorProduct R (M₁ i) M₂') : (TensorProduct.directSumLeft R S M₁ M₂').symm ((DirectSum.of (fun i => TensorProduct R (M₁ i) M₂') i) x) = (LinearMap.rTensor M₂' (DirectSum.lof R ι₁ M₁ i)) x
TensorProduct.finsuppLeft_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
{R : Type u_1} {S : Type u_2} [CommSemiring R] [Semiring S] [Algebra R S] {M : Type u_3} [AddCommMonoid M] [Module R M] [Module S M] [IsScalarTower R S M] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (t : TensorProduct R (ι →₀ M) N) (i : ι) : ((TensorProduct.finsuppLeft R S M N ι) t) i = (LinearMap.rTensor N (Finsupp.lapply i)) t
TensorProduct.finsuppScalarLeft_apply 📋 Mathlib.LinearAlgebra.DirectSum.Finsupp
{R : Type u_1} [CommSemiring R] {N : Type u_4} [AddCommMonoid N] [Module R N] {ι : Type u_5} [DecidableEq ι] (pn : TensorProduct R (ι →₀ R) N) (i : ι) : ((TensorProduct.finsuppScalarLeft R N ι) pn) i = (TensorProduct.lid R N) ((LinearMap.rTensor N (Finsupp.lapply i)) pn)
TensorProduct.equivFinsuppOfBasisLeft_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Basis
{R : Type u_1} {M : Type u_3} {N : Type u_4} {ι : Type u_5} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq ι] (ℬ : Module.Basis ι R M) (x : TensorProduct R M N) (i : ι) : ((TensorProduct.equivFinsuppOfBasisLeft ℬ) x) i = (TensorProduct.lid R N) ((LinearMap.rTensor N (ℬ.coord i)) x)
Submodule.map_range_rTensor_subtype_lid 📋 Mathlib.RingTheory.TensorProduct.Basic
{R : Type u_1} {Q : Type u_2} [CommSemiring R] [AddCommMonoid Q] [Module R Q] {I : Submodule R R} : Submodule.map (↑(TensorProduct.lid R Q)) (LinearMap.rTensor Q I.subtype).range = I • ⊤
Submodule.exists_fg_le_subset_range_rTensor_subtype 📋 Mathlib.RingTheory.TensorProduct.Finite
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (s : Set (TensorProduct R N M)) (hs : s.Finite) : ∃ J, ∃ (_ : J.FG), s ⊆ ↑(LinearMap.rTensor M J.subtype).range
Submodule.exists_fg_le_eq_rTensor_subtype 📋 Mathlib.RingTheory.TensorProduct.Finite
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (x : TensorProduct R N M) : ∃ J, ∃ (_ : J.FG), ∃ y, x = (LinearMap.rTensor M J.subtype) y
Submodule.exists_fg_le_subset_range_rTensor_inclusion 📋 Mathlib.RingTheory.TensorProduct.Finite
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {I : Submodule R N} (s : Set (TensorProduct R (↥I) M)) (hs : s.Finite) : ∃ J, ∃ (_ : J.FG) (hle : J ≤ I), s ⊆ ↑(LinearMap.rTensor M (Submodule.inclusion hle)).range
Submodule.exists_fg_le_eq_rTensor_inclusion 📋 Mathlib.RingTheory.TensorProduct.Finite
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {I : Submodule R N} (x : TensorProduct R (↥I) M) : ∃ J, ∃ (_ : J.FG) (hle : J ≤ I), ∃ y, x = (LinearMap.rTensor M (Submodule.inclusion hle)) y
Polynomial.X_pow_smul_rTensor_monomial 📋 Mathlib.Algebra.Polynomial.AlgebraMap
{R : Type u} {S : Type v} [CommSemiring R] [CommSemiring S] [Algebra R S] {N : Type u_3} [AddCommMonoid N] [Module R N] (k : ℕ) (sn : TensorProduct R S N) : Polynomial.X ^ k • (LinearMap.rTensor N (↑R (Polynomial.monomial 0))) sn = (LinearMap.rTensor N (↑R (Polynomial.monomial k))) sn
TensorProduct.instDirectedSystemCoeLinearMapIdRTensor 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] [DirectedSystem G fun x1 x2 x3 => ⇑(f x1 x2 x3)] : DirectedSystem (fun x => TensorProduct R (G x) M) fun i j h => ⇑(LinearMap.rTensor M (f i j h))
TensorProduct.fromDirectLimit 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] : (Module.DirectLimit (fun x => TensorProduct R (G x) M) fun i j h => LinearMap.rTensor M (f i j h)) →ₗ[R] TensorProduct R (Module.DirectLimit G f) M
TensorProduct.toDirectLimit 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] : TensorProduct R (Module.DirectLimit G f) M →ₗ[R] Module.DirectLimit (fun x => TensorProduct R (G x) M) fun i j h => LinearMap.rTensor M (f i j h)
TensorProduct.directLimitLeft 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] : TensorProduct R (Module.DirectLimit G f) M ≃ₗ[R] Module.DirectLimit (fun x => TensorProduct R (G x) M) fun i j h => LinearMap.rTensor M (f i j h)
TensorProduct.fromDirectLimit_of_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) {M : Type u_4} [AddCommMonoid M] [Module R M] {i : ι} (g : G i) (m : M) : (TensorProduct.fromDirectLimit f M) ((Module.DirectLimit.of R ι (fun x => TensorProduct R (G x) M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)) = (Module.DirectLimit.of R ι G f i) g ⊗ₜ[R] m
TensorProduct.toDirectLimit_tmul_of 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) {M : Type u_4} [AddCommMonoid M] [Module R M] {i : ι} (g : G i) (m : M) : (TensorProduct.toDirectLimit f M) ((Module.DirectLimit.of R ι G f i) g ⊗ₜ[R] m) = (Module.DirectLimit.of R ι (fun x => TensorProduct R (G x) M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)
TensorProduct.directLimitLeft_tmul_of 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] {i : ι} (g : G i) (m : M) : (TensorProduct.directLimitLeft f M) ((Module.DirectLimit.of R ι G f i) g ⊗ₜ[R] m) = (Module.DirectLimit.of R ι (fun i => TensorProduct R (G i) M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)
TensorProduct.directLimitLeft_rTensor_of 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] {i : ι} (x : TensorProduct R (G i) M) : (TensorProduct.directLimitLeft f M) ((LinearMap.rTensor M (Module.DirectLimit.of R ι G f i)) x) = (Module.DirectLimit.of R ι (fun i => TensorProduct R (G i) M) (fun i j h => LinearMap.rTensor M (f i j h)) i) x
TensorProduct.directLimitLeft_symm_of_tmul 📋 Mathlib.LinearAlgebra.TensorProduct.DirectLimit
{R : Type u_1} [CommSemiring R] {ι : Type u_2} [DecidableEq ι] [Preorder ι] {G : ι → Type u_3} [(i : ι) → AddCommMonoid (G i)] [(i : ι) → Module R (G i)] (f : (i j : ι) → i ≤ j → G i →ₗ[R] G j) (M : Type u_4) [AddCommMonoid M] [Module R M] {i : ι} (g : G i) (m : M) : (TensorProduct.directLimitLeft f M).symm ((Module.DirectLimit.of R ι (fun x => TensorProduct R (G x) M) (fun i j h => LinearMap.rTensor M (f i j h)) i) (g ⊗ₜ[R] m)) = (Module.DirectLimit.of R ι G f i) g ⊗ₜ[R] m
Submodule.FG.exists_rTensor_fg_inclusion_eq 📋 Mathlib.Algebra.Colimit.TensorProduct
{R : Type u_1} {M : Type u_2} {P : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid P] [Module R P] {N : Submodule R P} (hN : N.FG) {x y : TensorProduct R (↥N) M} (eq : (LinearMap.rTensor M N.subtype) x = (LinearMap.rTensor M N.subtype) y) : ∃ N', N'.FG ∧ ∃ (h : N ≤ N'), (LinearMap.rTensor M (Submodule.inclusion h)) x = (LinearMap.rTensor M (Submodule.inclusion h)) y
LinearMap.rTensor_surjective 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} (Q : Type u_5) [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] {g : N →ₗ[R] P} (hg : Function.Surjective ⇑g) : Function.Surjective ⇑(LinearMap.rTensor Q g)
rTensor_mkQ 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (Q : Type u_5) [AddCommGroup Q] [Module R Q] (N : Submodule R M) : (LinearMap.rTensor Q N.mkQ).ker = (LinearMap.rTensor Q N.subtype).range
LinearMap.rTensor_range 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} (Q : Type u_5) [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] {g : N →ₗ[R] P} : (LinearMap.rTensor Q g).range = (LinearMap.rTensor Q g.range.subtype).range
LinearMap.rTensor_exact_iff_lTensor_exact 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} [CommSemiring R] {M : Type u_2} {N : Type u_3} {P : Type u_4} (Q : Type u_5) [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] {f : M →ₗ[R] N} {g : N →ₗ[R] P} : Function.Exact ⇑(LinearMap.rTensor Q f) ⇑(LinearMap.rTensor Q g) ↔ Function.Exact ⇑(LinearMap.lTensor Q f) ⇑(LinearMap.lTensor Q g)
rTensor.toFun 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) : TensorProduct R N Q ⧸ (LinearMap.rTensor Q f).range →ₗ[R] TensorProduct R P Q
rTensor_exact 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) : Function.Exact ⇑(LinearMap.rTensor Q f) ⇑(LinearMap.rTensor Q g)
rTensor.inverse 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) : TensorProduct R P Q →ₗ[R] TensorProduct R N Q ⧸ (LinearMap.rTensor Q f).range
rTensor.inverse_of_rightInverse 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] {h : P → N} (hfg : Function.Exact ⇑f ⇑g) (hgh : Function.RightInverse h ⇑g) : TensorProduct R P Q →ₗ[R] TensorProduct R N Q ⧸ (LinearMap.rTensor Q f).range
rTensor.equiv 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) : (TensorProduct R N Q ⧸ (LinearMap.rTensor Q f).range) ≃ₗ[R] TensorProduct R P Q
rTensor.linearEquiv_of_rightInverse 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) {h : P → N} (hgh : Function.RightInverse h ⇑g) : (TensorProduct R N Q ⧸ (LinearMap.rTensor Q f).range) ≃ₗ[R] TensorProduct R P Q
Ideal.map_includeLeft_eq 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} [CommSemiring R] {A : Type u_2} {B : Type u_3} [Semiring A] [Semiring B] [Algebra R A] [Algebra R B] (I : Ideal A) : Submodule.restrictScalars R (Ideal.map Algebra.TensorProduct.includeLeft I) = (LinearMap.rTensor B (Submodule.restrictScalars R I).subtype).range
TensorProduct.map_ker 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) {M' : Type u_6} {N' : Type u_7} {P' : Type u_8} [AddCommGroup M'] [AddCommGroup N'] [AddCommGroup P'] [Module R M'] [Module R N'] [Module R P'] {f' : M' →ₗ[R] N'} {g' : N' →ₗ[R] P'} (hfg' : Function.Exact ⇑f' ⇑g') (hg' : Function.Surjective ⇑g') : (TensorProduct.map g g').ker = (LinearMap.lTensor N f').range ⊔ (LinearMap.rTensor N' f).range
le_comap_range_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} [CommSemiring R] {N : Type u_3} {P : Type u_4} {Q : Type u_5} [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R N] [Module R P] [Module R Q] (g : N →ₗ[R] P) (q : Q) : g.range ≤ Submodule.comap ((TensorProduct.mk R P Q).flip q) (LinearMap.rTensor Q g).range
rTensor.inverse_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) : rTensor.inverse Q hfg hg ∘ₗ LinearMap.rTensor Q g = (LinearMap.rTensor Q f).range.mkQ
rTensor.inverse_of_rightInverse_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) {h : P → N} (hgh : Function.RightInverse h ⇑g) : rTensor.inverse_of_rightInverse Q hfg hgh ∘ₗ LinearMap.rTensor Q g = (LinearMap.rTensor Q f).range.mkQ
rTensor.inverse_apply 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) (hg : Function.Surjective ⇑g) (y : TensorProduct R N Q) : (rTensor.inverse Q hfg hg) ((LinearMap.rTensor Q g) y) = Submodule.Quotient.mk y
rTensor.inverse_of_rightInverse_apply 📋 Mathlib.LinearAlgebra.TensorProduct.RightExactness
{R : Type u_1} {M : Type u_2} {N : Type u_3} {P : Type u_4} [CommRing R] [AddCommGroup M] [AddCommGroup N] [AddCommGroup P] [Module R M] [Module R N] [Module R P] {f : M →ₗ[R] N} {g : N →ₗ[R] P} (Q : Type u_5) [AddCommGroup Q] [Module R Q] (hfg : Function.Exact ⇑f ⇑g) {h : P → N} (hgh : Function.RightInverse h ⇑g) (y : TensorProduct R N Q) : (rTensor.inverse_of_rightInverse Q hfg hgh) ((LinearMap.rTensor Q g) y) = Submodule.Quotient.mk y
TensorProduct.exists_finite_submodule_left_of_setFinite 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] (s : Set (TensorProduct R M N)) (hs : s.Finite) : ∃ M', Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor N M'.subtype).range
TensorProduct.exists_finite_submodule_left_of_setFinite' 📋 Mathlib.LinearAlgebra.TensorProduct.Finiteness
{R : Type u_1} {M : Type u_2} {N : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [Module R M] [Module R N] {M₁ : Submodule R M} {N₁ : Submodule R N} (s : Set (TensorProduct R ↥M₁ ↥N₁)) (hs : s.Finite) : ∃ M', ∃ (hM : M' ≤ M₁), Module.Finite R ↥M' ∧ s ⊆ ↑(LinearMap.rTensor (↥N₁) (Submodule.inclusion hM)).range
Module.Flat.rTensor_preserves_injective_linearMap 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [Module.Flat R M] (f : N →ₗ[R] P) (hf : Function.Injective ⇑f) : Function.Injective ⇑(LinearMap.rTensor M f)
Module.Flat.iff_rTensor_preserves_injective_linearMapₛ 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] [Small.{v', u} R] : Module.Flat R M ↔ ∀ ⦃N N' : Type v'⦄ [inst : AddCommMonoid N] [inst_1 : AddCommMonoid N'] [inst_2 : Module R N] [inst_3 : Module R N'] (f : N →ₗ[R] N'), Function.Injective ⇑f → Function.Injective ⇑(LinearMap.rTensor M f)
Module.Flat.iff_rTensor_preserves_injective_linearMap 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃N N' : Type (max u v)⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : Module R N] [inst_3 : Module R N'] (f : N →ₗ[R] N'), Function.Injective ⇑f → Function.Injective ⇑(LinearMap.rTensor M f)
Module.Flat.iff_rTensor_preserves_injective_linearMap' 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Small.{v', u} R] : Module.Flat R M ↔ ∀ ⦃N N' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : Module R N] [inst_3 : Module R N'] (f : N →ₗ[R] N'), Function.Injective ⇑f → Function.Injective ⇑(LinearMap.rTensor M f)
Module.Flat.iff_rTensor_injectiveₛ 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] : Module.Flat R M ↔ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] (N : Submodule R P), Function.Injective ⇑(LinearMap.rTensor M N.subtype)
Module.Flat.mk 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommSemiring R] [AddCommMonoid M] [Module R M] (out : ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] [Module.Finite R P] (N : Submodule R P), N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)) : Module.Flat R M
Module.Flat.out 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid M} {inst✝² : Module R M} [self : Module.Flat R M] ⦃P : Type u⦄ [AddCommMonoid P] [Module R P] [Module.Finite R P] (N : Submodule R P) : N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)
Module.flat_iff 📋 Mathlib.RingTheory.Flat.Basic
(R : Type u) (M : Type v) [CommSemiring R] [AddCommMonoid M] [Module R M] : Module.Flat R M ↔ ∀ ⦃P : Type u⦄ [inst : AddCommMonoid P] [inst_1 : Module R P] [Module.Finite R P] (N : Submodule R P), N.FG → Function.Injective ⇑(LinearMap.rTensor M N.subtype)
Module.Flat.rTensor_exact 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] [Module.Flat R M] ⦃N : Type u_1⦄ ⦃N' : Type u_2⦄ ⦃N'' : Type u_3⦄ [AddCommGroup N] [AddCommGroup N'] [AddCommGroup N''] [Module R N] [Module R N'] [Module R N''] ⦃f : N →ₗ[R] N'⦄ ⦃g : N' →ₗ[R] N''⦄ (exact : Function.Exact ⇑f ⇑g) : Function.Exact ⇑(LinearMap.rTensor M f) ⇑(LinearMap.rTensor M g)
Module.Flat.iff_rTensor_exact 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃N N' N'' : Type (max u v)⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N''] [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄ ⦃g : N' →ₗ[R] N''⦄, Function.Exact ⇑f ⇑g → Function.Exact ⇑(LinearMap.rTensor M f) ⇑(LinearMap.rTensor M g)
Module.Flat.iff_rTensor_exact' 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] [Small.{v', u} R] : Module.Flat R M ↔ ∀ ⦃N N' N'' : Type v'⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : AddCommGroup N''] [inst_3 : Module R N] [inst_4 : Module R N'] [inst_5 : Module R N''] ⦃f : N →ₗ[R] N'⦄ ⦃g : N' →ₗ[R] N''⦄, Function.Exact ⇑f ⇑g → Function.Exact ⇑(LinearMap.rTensor M f) ⇑(LinearMap.rTensor M g)
LinearMap.rTensor_injective_of_fg 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] {f : N →ₗ[R] P} (h : ∀ (N' : Submodule R N) (P' : Submodule R P), N'.FG → P'.FG → ∀ (h : N' ≤ Submodule.comap f P'), Function.Injective ⇑(LinearMap.rTensor M (f.restrict h))) : Function.Injective ⇑(LinearMap.rTensor M f)
LinearMap.rTensor_injective_iff_subtype 📋 Mathlib.RingTheory.Flat.Basic
{R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} {Q : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [AddCommMonoid P] [Module R P] [AddCommMonoid Q] [Module R Q] {f : N →ₗ[R] P} (hf : Function.Injective ⇑f) (e : P ≃ₗ[R] Q) : Function.Injective ⇑(LinearMap.rTensor M f) ↔ Function.Injective ⇑(LinearMap.rTensor M (↑e ∘ₗ f).range.subtype)
SemimoduleCat.hom_whiskerRight 📋 Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
{R : Type u} [CommSemiring R] {L M : SemimoduleCat R} (f : L ⟶ M) (N : SemimoduleCat R) : SemimoduleCat.Hom.hom (CategoryTheory.MonoidalCategoryStruct.whiskerRight f N) = LinearMap.rTensor (↑N) (SemimoduleCat.Hom.hom f)
ModuleCat.hom_whiskerRight 📋 Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
{R : Type u} [CommRing R] {L M : ModuleCat R} (f : L ⟶ M) (N : ModuleCat R) : ModuleCat.Hom.hom (CategoryTheory.MonoidalCategoryStruct.whiskerRight f N) = LinearMap.rTensor (↑N) (ModuleCat.Hom.hom f)
ModuleCat.MonoidalCategory.whiskerRight_def 📋 Mathlib.Algebra.Category.ModuleCat.Monoidal.Basic
{R : Type u} [CommRing R] {X₁✝ X₂✝ : ModuleCat R} (f : X₁✝ ⟶ X₂✝) (M : ModuleCat R) : CategoryTheory.MonoidalCategoryStruct.whiskerRight f M = ModuleCat.ofHom (LinearMap.rTensor (↑M) (ModuleCat.Hom.hom f))
Coalgebra.rTensor_counit_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (LinearMap.rTensor A CoalgebraStruct.counit) (CoalgebraStruct.comul a) = 1 ⊗ₜ[R] a
Coalgebra.coassoc 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : ↑(TensorProduct.assoc R A A A) ∘ₗ LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul
Coalgebra.rTensor_counit_comp_comul 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : AddCommMonoid A} {inst✝² : Module R A} [self : Coalgebra R A] : LinearMap.rTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R R A) 1
Coalgebra.coassoc_symm 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] : ↑(TensorProduct.assoc R A A A).symm ∘ₗ LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul
Coalgebra.coassoc_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (TensorProduct.assoc R A A A) ((LinearMap.rTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)) = (LinearMap.lTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)
Coalgebra.coassoc_symm_apply 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [Coalgebra R A] (a : A) : (TensorProduct.assoc R A A A).symm ((LinearMap.lTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)) = (LinearMap.rTensor A CoalgebraStruct.comul) (CoalgebraStruct.comul a)
Coalgebra.mk 📋 Mathlib.RingTheory.Coalgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [AddCommMonoid A] [Module R A] [toCoalgebraStruct : CoalgebraStruct R A] (coassoc : ↑(TensorProduct.assoc R A A A) ∘ₗ LinearMap.rTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul = LinearMap.lTensor A CoalgebraStruct.comul ∘ₗ CoalgebraStruct.comul) (rTensor_counit_comp_comul : LinearMap.rTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R R A) 1) (lTensor_counit_comp_comul : LinearMap.lTensor A CoalgebraStruct.counit ∘ₗ CoalgebraStruct.comul = (TensorProduct.mk R A R).flip 1) : Coalgebra R A
HopfAlgebra.mul_antipode_rTensor_comul 📋 Mathlib.RingTheory.HopfAlgebra.Basic
{R : Type u} {A : Type v} {inst✝ : CommSemiring R} {inst✝¹ : Semiring A} [self : HopfAlgebra R A] : LinearMap.mul' R A ∘ₗ LinearMap.rTensor A (HopfAlgebraStruct.antipode R) ∘ₗ CoalgebraStruct.comul = Algebra.linearMap R A ∘ₗ CoalgebraStruct.counit
HopfAlgebra.mk 📋 Mathlib.RingTheory.HopfAlgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [toHopfAlgebraStruct : HopfAlgebraStruct R A] (mul_antipode_rTensor_comul : LinearMap.mul' R A ∘ₗ LinearMap.rTensor A (HopfAlgebraStruct.antipode R) ∘ₗ CoalgebraStruct.comul = Algebra.linearMap R A ∘ₗ CoalgebraStruct.counit) (mul_antipode_lTensor_comul : LinearMap.mul' R A ∘ₗ LinearMap.lTensor A (HopfAlgebraStruct.antipode R) ∘ₗ CoalgebraStruct.comul = Algebra.linearMap R A ∘ₗ CoalgebraStruct.counit) : HopfAlgebra R A
HopfAlgebra.mul_antipode_rTensor_comul_apply 📋 Mathlib.RingTheory.HopfAlgebra.Basic
{R : Type u} {A : Type v} [CommSemiring R] [Semiring A] [HopfAlgebra R A] (a : A) : (LinearMap.mul' R A) ((LinearMap.rTensor A (HopfAlgebraStruct.antipode R)) (CoalgebraStruct.comul a)) = (algebraMap R A) (CoalgebraStruct.counit a)
dualTensorHom_comp_rTensor_dualMap 📋 Mathlib.LinearAlgebra.Contraction
{R : Type u_2} {M : Type u_3} {N : Type u_4} {P : Type u_5} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] (f : M →ₗ[R] N) : dualTensorHom R M P ∘ₗ LinearMap.rTensor P f.dualMap = LinearMap.lcomp R P f ∘ₗ dualTensorHom R N P
contractLeft_assoc_coevaluation' 📋 Mathlib.LinearAlgebra.Coevaluation
(K : Type u) [Field K] (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V] : LinearMap.lTensor V (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K V (Module.Dual K V) V) ∘ₗ LinearMap.rTensor V (coevaluation K V) = ↑(TensorProduct.rid K V).symm ∘ₗ ↑(TensorProduct.lid K V)
contractLeft_assoc_coevaluation 📋 Mathlib.LinearAlgebra.Coevaluation
(K : Type u) [Field K] (V : Type v) [AddCommGroup V] [Module K V] [FiniteDimensional K V] : LinearMap.rTensor (Module.Dual K V) (contractLeft K V) ∘ₗ ↑(TensorProduct.assoc K (Module.Dual K V) V (Module.Dual K V)).symm ∘ₗ LinearMap.lTensor (Module.Dual K V) (coevaluation K V) = ↑(TensorProduct.lid K (Module.Dual K V)).symm ∘ₗ ↑(TensorProduct.rid K (Module.Dual K V))
rTensor_injective_iff_lcomp_surjective 📋 Mathlib.Algebra.Module.CharacterModule
{R : Type uR} [CommRing R] {A : Type uA} [AddCommGroup A] {A' : Type u_1} [AddCommGroup A'] {B : Type uB} [AddCommGroup B] [Module R A] [Module R A'] [Module R B] {f : A →ₗ[R] A'} : Function.Injective ⇑(LinearMap.rTensor B f) ↔ Function.Surjective ⇑(LinearMap.lcomp R (CharacterModule B) f)
CharacterModule.dual_rTensor_conj_homEquiv 📋 Mathlib.Algebra.Module.CharacterModule
{R : Type uR} [CommRing R] {A : Type uA} [AddCommGroup A] (A' : Type u_1) [AddCommGroup A'] {B : Type uB} [AddCommGroup B] [Module R A] [Module R A'] [Module R B] (f : A →ₗ[R] A') : ↑CharacterModule.homEquiv.symm ∘ₗ CharacterModule.dual (LinearMap.rTensor B f) ∘ₗ ↑CharacterModule.homEquiv = LinearMap.lcomp R (CharacterModule B) f
TensorProduct.quotTensorEquivQuotSMul_comp_mkQ_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Quotient
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] (I : Ideal R) : ↑(TensorProduct.quotTensorEquivQuotSMul M I) ∘ₗ LinearMap.rTensor M (Submodule.mkQ I) = (I • ⊤).mkQ ∘ₗ ↑(TensorProduct.lid R M)
Ideal.subtype_rTensor_range 📋 Mathlib.RingTheory.TensorProduct.Quotient
{R : Type u_1} [CommRing R] (M : Type u_2) [AddCommGroup M] [Module R M] (I : Ideal R) : (↑(TensorProduct.lid R M) ∘ₗ LinearMap.rTensor M (Submodule.subtype I)).range = I • ⊤
Module.FaithfullyFlat.zero_iff_rTensor_zero 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] [h : Module.FaithfullyFlat R M] {N : Type u_1} [AddCommGroup N] [Module R N] {N' : Type u_2} [AddCommGroup N'] [Module R N'] (f : N →ₗ[R] N') : f = 0 ↔ LinearMap.rTensor M f = 0
Module.FaithfullyFlat.iff_zero_iff_rTensor_zero 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] : Module.FaithfullyFlat R M ↔ Module.Flat R M ∧ ∀ {N : Type (max u v)} [inst : AddCommGroup N] [inst_1 : Module R N] {N' : Type (max u v)} [inst_2 : AddCommGroup N'] [inst_3 : Module R N'] (f : N →ₗ[R] N'), LinearMap.rTensor M f = 0 ↔ f = 0
Module.FaithfullyFlat.range_le_ker_of_exact_rTensor 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] {N1 : Type u_1} [AddCommGroup N1] [Module R N1] {N2 : Type u_2} [AddCommGroup N2] [Module R N2] {N3 : Type u_3} [AddCommGroup N3] [Module R N3] (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3) [fl : Module.FaithfullyFlat R M] (ex : Function.Exact ⇑(LinearMap.rTensor M l12) ⇑(LinearMap.rTensor M l23)) : l12.range ≤ l23.ker
Module.FaithfullyFlat.rTensor_reflects_exact 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] {N1 : Type u_1} [AddCommGroup N1] [Module R N1] {N2 : Type u_2} [AddCommGroup N2] [Module R N2] {N3 : Type u_3} [AddCommGroup N3] [Module R N3] (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3) [fl : Module.FaithfullyFlat R M] (ex : Function.Exact ⇑(LinearMap.rTensor M l12) ⇑(LinearMap.rTensor M l23)) : Function.Exact ⇑l12 ⇑l23
Module.FaithfullyFlat.rTensor_exact_iff_exact 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] {N1 : Type u_1} [AddCommGroup N1] [Module R N1] {N2 : Type u_2} [AddCommGroup N2] [Module R N2] {N3 : Type u_3} [AddCommGroup N3] [Module R N3] (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3) [Module.FaithfullyFlat R M] : Function.Exact ⇑(LinearMap.rTensor M l12) ⇑(LinearMap.rTensor M l23) ↔ Function.Exact ⇑l12 ⇑l23
Module.FaithfullyFlat.iff_exact_iff_rTensor_exact 📋 Mathlib.RingTheory.Flat.FaithfullyFlat.Basic
(R : Type u) (M : Type v) [CommRing R] [AddCommGroup M] [Module R M] : Module.FaithfullyFlat R M ↔ ∀ {N1 : Type (max u v)} [inst : AddCommGroup N1] [inst_1 : Module R N1] {N2 : Type (max u v)} [inst_2 : AddCommGroup N2] [inst_3 : Module R N2] {N3 : Type (max u v)} [inst_4 : AddCommGroup N3] [inst_5 : Module R N3] (l12 : N1 →ₗ[R] N2) (l23 : N2 →ₗ[R] N3), Function.Exact ⇑l12 ⇑l23 ↔ Function.Exact ⇑(LinearMap.rTensor M l12) ⇑(LinearMap.rTensor M l23)
Module.Flat.injective_characterModule_iff_rTensor_preserves_injective_linearMap 📋 Mathlib.RingTheory.Flat.Tensor
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Injective R (CharacterModule M) ↔ ∀ ⦃N N' : Type v⦄ [inst : AddCommGroup N] [inst_1 : AddCommGroup N'] [inst_2 : Module R N] [inst_3 : Module R N'] (f : N →ₗ[R] N'), Function.Injective ⇑f → Function.Injective ⇑(LinearMap.rTensor M f)
Module.Flat.iff_rTensor_injective' 📋 Mathlib.RingTheory.Flat.Tensor
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ (I : Ideal R), Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype I))
Module.Flat.iff_rTensor_injective 📋 Mathlib.RingTheory.Flat.Tensor
{R : Type u} {M : Type v} [CommRing R] [AddCommGroup M] [Module R M] : Module.Flat R M ↔ ∀ ⦃I : Ideal R⦄, I.FG → Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype I))
Lie.Derivation.ofDerivation_apply 📋 Mathlib.Algebra.Lie.Derivation.BaseChange
{R : Type u_1} [CommRing R] {A : Type u_2} [CommRing A] [Algebra R A] {L : Type u_3} [LieRing L] [LieAlgebra R L] (d : Derivation R A A) (x : TensorProduct R A L) : ((Lie.Derivation.ofDerivation L) d) x = (LinearMap.rTensor L ↑d) x
DirectSum.decomposeTensor_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Decomposition
{ι : Type u_1} {R : Type u_2} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) (N : Type u_5) [AddCommMonoid N] [Module R N] {i : ι} : DirectSum.decomposeTensor ℳ N i = (LinearMap.rTensor N (ℳ i).subtype).range
DirectSum.subtype_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Decomposition
{ι : Type u_1} {R : Type u_2} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) (N : Type u_5) [AddCommMonoid N] [Module R N] [DecidableEq ι] [DirectSum.Decomposition ℳ] (i : ι) : Function.Injective ⇑(LinearMap.rTensor N (ℳ i).subtype)
DirectSum.val_decomposeTensorEquiv_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Decomposition
{ι : Type u_1} {R : Type u_2} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) (N : Type u_5) [AddCommMonoid N] [Module R N] [DecidableEq ι] [DirectSum.Decomposition ℳ] {i : ι} (x : TensorProduct R (↥(ℳ i)) N) : ↑((DirectSum.decomposeTensorEquiv ℳ N i) x) = (LinearMap.rTensor N (ℳ i).subtype) x
DirectSum.decomposeTensorEquiv_apply 📋 Mathlib.LinearAlgebra.TensorProduct.Decomposition
{ι : Type u_1} {R : Type u_2} {M : Type u_3} [CommSemiring R] [AddCommMonoid M] [Module R M] (ℳ : ι → Submodule R M) (N : Type u_5) [AddCommMonoid N] [Module R N] [DecidableEq ι] [DirectSum.Decomposition ℳ] {i : ι} (x : TensorProduct R (↥(ℳ i)) N) : (DirectSum.decomposeTensorEquiv ℳ N i) x = ⟨(LinearMap.rTensor N (ℳ i).subtype) x, ⋯⟩
TensorProduct.rTensor_injective_of_forall_vanishesTrivially 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] (hMN : ∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → TensorProduct.VanishesTrivially R m n) (M' : Submodule R M) : Function.Injective ⇑(LinearMap.rTensor N M'.subtype)
TensorProduct.forall_vanishesTrivially_iff_forall_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] : (∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → TensorProduct.VanishesTrivially R m n) ↔ ∀ (M' : Submodule R M), Function.Injective ⇑(LinearMap.rTensor N M'.subtype)
TensorProduct.forall_vanishesTrivially_iff_forall_fg_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] : (∀ {l : ℕ} {m : Fin l → M} {n : Fin l → N}, ∑ i, m i ⊗ₜ[R] n i = 0 → TensorProduct.VanishesTrivially R m n) ↔ ∀ (M' : Submodule R M), M'.FG → Function.Injective ⇑(LinearMap.rTensor N M'.subtype)
TensorProduct.vanishesTrivially_of_sum_tmul_eq_zero_of_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] {ι : Type u_4} [Fintype ι] {m : ι → M} {n : ι → N} (hm : Function.Injective ⇑(LinearMap.rTensor N (Submodule.span R (Set.range m)).subtype)) (hmn : ∑ i, m i ⊗ₜ[R] n i = 0) : TensorProduct.VanishesTrivially R m n
TensorProduct.vanishesTrivially_iff_sum_tmul_eq_zero_of_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] {ι : Type u_4} [Fintype ι] {m : ι → M} {n : ι → N} (hm : Function.Injective ⇑(LinearMap.rTensor N (Submodule.span R (Set.range m)).subtype)) : TensorProduct.VanishesTrivially R m n ↔ ∑ i, m i ⊗ₜ[R] n i = 0
TensorProduct.rTensor_injective_of_forall_fg_rTensor_injective 📋 Mathlib.LinearAlgebra.TensorProduct.Vanishing
(R : Type u_1) [CommRing R] {M : Type u_2} [AddCommGroup M] [Module R M] {N : Type u_3} [AddCommGroup N] [Module R N] (hMN : ∀ (M' : Submodule R M), M'.FG → Function.Injective ⇑(LinearMap.rTensor N M'.subtype)) (M' : Submodule R M) : Function.Injective ⇑(LinearMap.rTensor N M'.subtype)
Module.Flat.tfae_equational_criterion 📋 Mathlib.RingTheory.Flat.EquationalCriterion
(R : Type u_1) (M : Type u_2) [CommRing R] [AddCommGroup M] [Module R M] : [Module.Flat R M, ∀ (I : Ideal R), Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype I)), ∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i ⊗ₜ[R] x i = 0 → TensorProduct.VanishesTrivially R f x, ∀ {l : ℕ} {f : Fin l → R} {x : Fin l → M}, ∑ i, f i • x i = 0 → Module.IsTrivialRelation f x, ∀ {l : ℕ} {f : Fin l →₀ R} {x : (Fin l →₀ R) →ₗ[R] M}, x f = 0 → ∃ k a y, x = y ∘ₗ a ∧ a f = 0].TFAE
Ideal.pi_mkQ_rTensor 📋 Mathlib.RingTheory.Ideal.Quotient.ChineseRemainder
{R : Type u_1} [CommRing R] {ι : Type u_2} (M : Type u_3) [AddCommGroup M] [Module R M] (I : ι → Ideal R) [Fintype ι] [DecidableEq ι] : LinearMap.rTensor M (LinearMap.pi fun i => Submodule.mkQ (I i)) = ↑(TensorProduct.piLeft R M fun i => R ⧸ I i).symm ∘ₗ (LinearMap.pi fun i => (TensorProduct.mk R (R ⧸ I i) M) 1) ∘ₗ ↑(TensorProduct.lid R M)
QuotSMulTop.equivTensorQuot_naturality 📋 Mathlib.RingTheory.QuotSMulTop
{R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') : ↑(QuotSMulTop.equivTensorQuot r M') ∘ₗ (QuotSMulTop.map r) f = LinearMap.rTensor (R ⧸ Ideal.span {r}) f ∘ₗ ↑(QuotSMulTop.equivTensorQuot r M)
QuotSMulTop.equivTensorQuot_naturality_mk 📋 Mathlib.RingTheory.QuotSMulTop
{R : Type u_2} [CommRing R] (r : R) {M : Type u_1} {M' : Type u_3} [AddCommGroup M] [Module R M] [AddCommGroup M'] [Module R M'] (f : M →ₗ[R] M') (x : M) : (QuotSMulTop.equivTensorQuot r M') (((QuotSMulTop.map r) f) (Submodule.Quotient.mk x)) = (LinearMap.rTensor (R ⧸ Ideal.span {r}) f) ((QuotSMulTop.equivTensorQuot r M) (Submodule.Quotient.mk x))
lTensor_injective_of_exact_of_exact_of_rTensor_injective 📋 Mathlib.RingTheory.LocalRing.Module
{R : Type u_1} [CommRing R] {M₁ : Type u_5} {M₂ : Type u_6} {M₃ : Type u_7} {N₁ : Type u_8} {N₂ : Type u_9} {N₃ : Type u_10} [AddCommGroup M₁] [Module R M₁] [AddCommGroup M₂] [Module R M₂] [AddCommGroup M₃] [Module R M₃] [AddCommGroup N₁] [Module R N₁] [AddCommGroup N₂] [Module R N₂] [AddCommGroup N₃] [Module R N₃] {f₁ : M₁ →ₗ[R] M₂} {f₂ : M₂ →ₗ[R] M₃} {g₁ : N₁ →ₗ[R] N₂} {g₂ : N₂ →ₗ[R] N₃} (hfexact : Function.Exact ⇑f₁ ⇑f₂) (hfsurj : Function.Surjective ⇑f₂) (hgexact : Function.Exact ⇑g₁ ⇑g₂) (hgsurj : Function.Surjective ⇑g₂) (hfinj : Function.Injective ⇑(LinearMap.rTensor N₃ f₁)) (hginj : Function.Injective ⇑(LinearMap.lTensor M₂ g₁)) : Function.Injective ⇑(LinearMap.lTensor M₃ g₁)
Module.free_of_maximalIdeal_rTensor_injective 📋 Mathlib.RingTheory.LocalRing.Module
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [IsLocalRing R] [Module.FinitePresentation R M] (H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype (IsLocalRing.maximalIdeal R)))) : Module.Free R M
Module.exists_basis_of_span_of_maximalIdeal_rTensor_injective 📋 Mathlib.RingTheory.LocalRing.Module
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [IsLocalRing R] [Module.FinitePresentation R M] (H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype (IsLocalRing.maximalIdeal R)))) {ι : Type u} (v : ι → M) (hv : Submodule.span R (Set.range v) = ⊤) : ∃ κ a b, ∀ (i : κ), b i = v (a i)
Module.exists_basis_of_basis_baseChange 📋 Mathlib.RingTheory.LocalRing.Module
{R : Type u_1} {M : Type u_2} [CommRing R] [AddCommGroup M] [Module R M] [IsLocalRing R] [Module.FinitePresentation R M] {ι : Type u_5} (v : ι → M) (hli : LinearIndependent (IsLocalRing.ResidueField R) (⇑((TensorProduct.mk R (IsLocalRing.ResidueField R) M) 1) ∘ v)) (hsp : Submodule.span (IsLocalRing.ResidueField R) (Set.range (⇑((TensorProduct.mk R (IsLocalRing.ResidueField R) M) 1) ∘ v)) = ⊤) (H : Function.Injective ⇑(LinearMap.rTensor M (Submodule.subtype (IsLocalRing.maximalIdeal R)))) : ∃ b, ∀ (i : ι), b i = v i
Submodule.mulMap_comp_rTensor 📋 Mathlib.LinearAlgebra.TensorProduct.Submodule
{R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {M : Submodule R S} (N : Submodule R S) {M' : Submodule R S} (hM : M' ≤ M) : M.mulMap N ∘ₗ LinearMap.rTensor (↥N) (Submodule.inclusion hM) = M'.mulMap N
Submodule.mulLeftMap_eq_mulMap_comp 📋 Mathlib.LinearAlgebra.TensorProduct.Submodule
{R : Type u} {S : Type v} [CommSemiring R] [Semiring S] [Algebra R S] {M : Submodule R S} (N : Submodule R S) {ι : Type u_1} [DecidableEq ι] (m : ι → ↥M) : Submodule.mulLeftMap N m = M.mulMap N ∘ₗ LinearMap.rTensor (↥N) (Finsupp.linearCombination R m) ∘ₗ ↑(TensorProduct.finsuppScalarLeft R (↥N) ι).symm
Module.Invertible.rTensor_bijective_iff 📋 Mathlib.RingTheory.PicardGroup
{R : Type u} (M : Type v) {N : Type u_1} {P : Type u_2} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module.Invertible R M] {f : N →ₗ[R] P} : Function.Bijective ⇑(LinearMap.rTensor M f) ↔ Function.Bijective ⇑f
Module.Invertible.rTensor_injective_iff 📋 Mathlib.RingTheory.PicardGroup
{R : Type u} (M : Type v) {N : Type u_1} {P : Type u_2} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module.Invertible R M] {f : N →ₗ[R] P} : Function.Injective ⇑(LinearMap.rTensor M f) ↔ Function.Injective ⇑f
Module.Invertible.rTensor_surjective_iff 📋 Mathlib.RingTheory.PicardGroup
{R : Type u} (M : Type v) {N : Type u_1} {P : Type u_2} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [Module R M] [Module R N] [Module R P] [Module.Invertible R M] {f : N →ₗ[R] P} : Function.Surjective ⇑(LinearMap.rTensor M f) ↔ Function.Surjective ⇑f
Module.Invertible.rightCancelEquiv_comp_rTensor_comp_symm 📋 Mathlib.RingTheory.PicardGroup
{R : Type u} {M : Type v} {N : Type u_1} {P : Type u_2} {Q : Type u_3} [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (e : TensorProduct R M N ≃ₗ[R] R) (f : P →ₗ[R] Q) : ↑(Module.Invertible.rightCancelEquiv Q e) ∘ₗ LinearMap.rTensor N (LinearMap.rTensor M f) ∘ₗ ↑(Module.Invertible.rightCancelEquiv P e).symm = f
Module.Invertible.rTensorEquiv_symm_apply_apply 📋 Mathlib.RingTheory.PicardGroup
{R : Type u} {M : Type v} {N : Type u_1} (P : Type u_2) (Q : Type u_3) [CommSemiring R] [AddCommMonoid M] [AddCommMonoid N] [AddCommMonoid P] [AddCommMonoid Q] [Module R M] [Module R N] [Module R P] [Module R Q] (e : TensorProduct R M N ≃ₗ[R] R) (a : TensorProduct R P M →ₗ[R] TensorProduct R Q M) (x : P) : ((Module.Invertible.rTensorEquiv P Q e).symm a) x = ((Module.Invertible.rightCancelEquiv Q e).congrRight (LinearMap.rTensor N a)) ((TensorProduct.assoc R P M N).symm (x ⊗ₜ[R] e.symm 1))
LinearMap.intrinsicStar_rTensor 📋 Mathlib.Algebra.Star.LinearMap
{R : Type u_5} {E : Type u_6} {F : Type u_7} {G : Type u_8} [CommSemiring R] [StarRing R] [AddCommMonoid E] [StarAddMonoid E] [Module R E] [StarModule R E] [AddCommMonoid F] [StarAddMonoid F] [Module R F] [StarModule R F] [AddCommMonoid G] [StarAddMonoid G] [Module R G] [StarModule R G] (f : WithConv (E →ₗ[R] F)) : star (WithConv.toConv (LinearMap.rTensor G f.ofConv)) = WithConv.toConv (LinearMap.rTensor G (star f).ofConv)
LinearIsometry.toLinearMap_rTensor 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
{𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (f : E →ₗᵢ[𝕜] F) : (LinearIsometry.rTensor G f).toLinearMap = LinearMap.rTensor G f.toLinearMap
ContinuousLinearMap.toLinearMap_rTensor 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
{𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} (G : Type u_4) [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (f : E →L[𝕜] F) : ↑(ContinuousLinearMap.rTensor G f) = LinearMap.rTensor G ↑f
LinearIsometry.rTensor_apply 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
{𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (f : E →ₗᵢ[𝕜] F) (x : TensorProduct 𝕜 E G) : (LinearIsometry.rTensor G f) x = (LinearMap.rTensor G f.toLinearMap) x
ContinuousLinearMap.rTensor_apply 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
{𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] (f : E →L[𝕜] F) (x : TensorProduct 𝕜 E G) : (ContinuousLinearMap.rTensor G f) x = (LinearMap.rTensor G ↑f) x
LinearMap.adjoint_rTensor 📋 Mathlib.Analysis.InnerProductSpace.TensorProduct
{𝕜 : Type u_1} {E : Type u_2} {F : Type u_3} {G : Type u_4} [RCLike 𝕜] [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [NormedAddCommGroup G] [InnerProductSpace 𝕜 G] [FiniteDimensional 𝕜 E] [FiniteDimensional 𝕜 F] [FiniteDimensional 𝕜 G] (f : E →ₗ[𝕜] F) : LinearMap.adjoint (LinearMap.rTensor G f) = LinearMap.rTensor G (LinearMap.adjoint f)
TensorProduct.gradedMul_def 📋 Mathlib.LinearAlgebra.TensorProduct.Graded.External
(R : Type u_5) {ι : Type u_6} [CommSemiring ι] [Module ι (Additive ℤˣ)] [DecidableEq ι] (𝒜 : ι → Type u_7) (ℬ : ι → Type u_8) [CommRing R] [(i : ι) → AddCommGroup (𝒜 i)] [(i : ι) → AddCommGroup (ℬ i)] [(i : ι) → Module R (𝒜 i)] [(i : ι) → Module R (ℬ i)] [DirectSum.GRing 𝒜] [DirectSum.GRing ℬ] [DirectSum.GAlgebra R 𝒜] [DirectSum.GAlgebra R ℬ] : TensorProduct.gradedMul R 𝒜 ℬ = TensorProduct.curry (TensorProduct.map (LinearMap.mul' R (DirectSum ι fun i => 𝒜 i)) (LinearMap.mul' R (DirectSum ι fun i => ℬ i)) ∘ₗ ↑(TensorProduct.assoc R (DirectSum ι fun i => 𝒜 i) (DirectSum ι fun i => 𝒜 i) (TensorProduct R (DirectSum ι fun i => ℬ i) (DirectSum ι fun i => ℬ i))).symm ∘ₗ LinearMap.lTensor (DirectSum ι fun i => 𝒜 i) (↑(TensorProduct.assoc R (DirectSum ι fun i => 𝒜 i) (DirectSum ι fun i => ℬ i) (DirectSum ι fun i => ℬ i)) ∘ₗ LinearMap.rTensor (DirectSum ι fun i => ℬ i) ↑(TensorProduct.gradedComm R ℬ 𝒜) ∘ₗ ↑(TensorProduct.assoc R (DirectSum ι ℬ) (DirectSum ι 𝒜) (DirectSum ι ℬ)).symm) ∘ₗ ↑(TensorProduct.assoc R (DirectSum ι 𝒜) (DirectSum ι ℬ) (TensorProduct R (DirectSum ι 𝒜) (DirectSum ι ℬ))))
Representation.IntertwiningMap.toLinearMap_rTensor 📋 Mathlib.RepresentationTheory.Intertwining
{A : Type u_1} {G : Type u_2} {V : Type u_3} {W : Type u_4} {U : Type u_5} [CommSemiring A] [Monoid G] [AddCommMonoid V] [AddCommMonoid W] [AddCommMonoid U] [Module A V] [Module A W] [Module A U] {ρ : Representation A G V} {σ : Representation A G W} {τ : Representation A G U} (f : σ.IntertwiningMap τ) : (Representation.IntertwiningMap.rTensor ρ f).toLinearMap = LinearMap.rTensor V f.toLinearMap
Representation.ind_apply 📋 Mathlib.RepresentationTheory.Induced
{k : Type u_1} {G : Type u_2} {H : Type u_3} [CommRing k] [Group G] [Group H] (φ : G →* H) {A : Type u_4} [AddCommGroup A] [Module k A] (ρ : Representation k G A) (h : H) : (Representation.ind φ ρ) h = Representation.Coinvariants.map (Representation.tprod (MonoidHom.comp (Representation.leftRegular k H) φ) ρ) (Representation.tprod (MonoidHom.comp (Representation.leftRegular k H) φ) ρ) { toLinearMap := LinearMap.rTensor A (MonoidAlgebra.mapDomainLinearMap k k fun x => x * h⁻¹), isIntertwining' := ⋯ }
Submodule.isCoideal_iff_comul_mem 📋 Mathlib.RingTheory.Coalgebra.Quotient
{R : Type u_1} {C : Type u_2} [CommRing R] [AddCommGroup C] [Module R C] [CoalgebraStruct R C] (I : Submodule R C) : I.IsCoideal ↔ (∀ x ∈ I, CoalgebraStruct.counit x = 0) ∧ ∀ x ∈ I, CoalgebraStruct.comul x ∈ (LinearMap.lTensor C I.subtype).range ⊔ (LinearMap.rTensor C I.subtype).range
DirectedSystem.rTensor 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
(R : Type u) (N : Type u_2) [CommSemiring R] [AddCommMonoid N] [Module R N] {ι : Type u_3} [Preorder ι] {F : ι → Type u_4} [(i : ι) → AddCommMonoid (F i)] [(i : ι) → Module R (F i)] {f : ⦃i j : ι⦄ → i ≤ j → F i →ₗ[R] F j} (D : DirectedSystem F fun x x_1 h => ⇑(f h)) : DirectedSystem (fun i => TensorProduct R (F i) N) fun x x_1 h => ⇑(LinearMap.rTensor N (f h))
TensorProduct.exists_of_fg 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u} {M : Type u_1} {N : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (u : TensorProduct R M N) : ∃ P, P.FG ∧ u ∈ (LinearMap.rTensor N P.subtype).range
TensorProduct.Algebra.exists_of_fg 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u_1} {S : Type u_2} {N : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid N] [Module R N] (u : TensorProduct R S N) : ∃ A, A.FG ∧ u ∈ (LinearMap.rTensor N A.val.toLinearMap).range
TensorProduct.eq_zero_of_fg_of_subtype_eq_zero 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u} {M : Type u_1} {N : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {P : Submodule R M} (hP : P.FG) {t : TensorProduct R (↥P) N} (h : (LinearMap.rTensor N P.subtype) t = 0) : ∃ Q, ∃ (hPQ : P ≤ Q), Q.FG ∧ (LinearMap.rTensor N (Submodule.inclusion hPQ)) t = 0
Submodule.FG.rTensor.directedSystem 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
(R : Type u) (M : Type u_1) (N : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] : DirectedSystem (fun P => TensorProduct R (↥↑P) N) fun ⦃x x_1⦄ h => ⇑(LinearMap.rTensor N (Submodule.inclusion h))
Submodule.FG.rTensor.directLimit 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
(R : Type u) (M : Type u_1) (N : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq { P // P.FG }] : (Module.DirectLimit (fun P => TensorProduct R (↥↑P) N) fun ⦃P Q⦄ h => LinearMap.rTensor N (Submodule.inclusion h)) ≃ₗ[R] TensorProduct R M N
TensorProduct.eq_of_fg_of_subtype_eq 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u} {M : Type u_1} {N : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {P : Submodule R M} (hP : P.FG) {t t' : TensorProduct R (↥P) N} (h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P.subtype) t') : ∃ Q, ∃ (hPQ : P ≤ Q), Q.FG ∧ (LinearMap.rTensor N (Submodule.inclusion hPQ)) t = (LinearMap.rTensor N (Submodule.inclusion hPQ)) t'
TensorProduct.eq_of_fg_of_subtype_eq' 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u} {M : Type u_1} {N : Type u_2} [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] {P : Submodule R M} (hP : P.FG) {t : TensorProduct R (↥P) N} {P' : Submodule R M} (hP' : P'.FG) {t' : TensorProduct R (↥P') N} (h : (LinearMap.rTensor N P.subtype) t = (LinearMap.rTensor N P'.subtype) t') : ∃ Q, ∃ (hPQ : P ≤ Q) (hP'Q : P' ≤ Q), Q.FG ∧ (LinearMap.rTensor N (Submodule.inclusion hPQ)) t = (LinearMap.rTensor N (Submodule.inclusion hP'Q)) t'
Submodule.exists_fg_of_baseChange_eq_zero 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u_1} {S : Type u_2} {M : Type u_3} {N : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] (f : M →ₗ[R] N) {t : TensorProduct R S M} (ht : (LinearMap.baseChange S f) t = 0) : ∃ A, ∃ (_ : A.FG), ∃ u, (LinearMap.baseChange (↥A) f) u = 0 ∧ (LinearMap.rTensor M A.val.toLinearMap) u = t
TensorProduct.Algebra.eq_of_fg_of_subtype_eq 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u_1} {S : Type u_2} {N : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid N] [Module R N] {A : Subalgebra R S} (hA : A.FG) {t t' : TensorProduct R (↥A) N} (h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A.val.toLinearMap) t') : ∃ B, ∃ (hAB : A ≤ B), B.FG ∧ (LinearMap.rTensor N (Subalgebra.inclusion hAB).toLinearMap) t = (LinearMap.rTensor N (Subalgebra.inclusion hAB).toLinearMap) t'
TensorProduct.Algebra.eq_of_fg_of_subtype_eq' 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
{R : Type u_1} {S : Type u_2} {N : Type u_4} [CommSemiring R] [Semiring S] [Algebra R S] [AddCommMonoid N] [Module R N] {A : Subalgebra R S} (hA : A.FG) {t : TensorProduct R (↥A) N} {A' : Subalgebra R S} (hA' : A'.FG) {t' : TensorProduct R (↥A') N} (h : (LinearMap.rTensor N A.val.toLinearMap) t = (LinearMap.rTensor N A'.val.toLinearMap) t') : ∃ B, ∃ (hAB : A ≤ B) (hA'B : A' ≤ B), B.FG ∧ (LinearMap.rTensor N (Subalgebra.inclusion hAB).toLinearMap) t = (LinearMap.rTensor N (Subalgebra.inclusion hA'B).toLinearMap) t'
Submodule.FG.rTensor.directLimit_apply' 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
(R : Type u) (M : Type u_1) (N : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq { P // P.FG }] {P : Submodule R M} (hP : P.FG) (u : TensorProduct R (↥P) N) : (Submodule.FG.rTensor.directLimit R M N) ((Module.DirectLimit.of R { P // P.FG } (fun P => TensorProduct R (↥↑P) N) (fun ⦃x x_1⦄ h => LinearMap.rTensor N (Submodule.inclusion h)) ⟨P, hP⟩) u) = (LinearMap.rTensor N P.subtype) u
Submodule.FG.rTensor.directLimit_apply 📋 Mathlib.RingTheory.TensorProduct.DirectLimitFG
(R : Type u) (M : Type u_1) (N : Type u_2) [CommSemiring R] [AddCommMonoid M] [Module R M] [AddCommMonoid N] [Module R N] [DecidableEq { P // P.FG }] {P : { P // P.FG }} (u : TensorProduct R (↥↑P) N) : (Submodule.FG.rTensor.directLimit R M N) ((Module.DirectLimit.of R { P // P.FG } (fun P => TensorProduct R (↥↑P) N) (fun ⦃x x_1⦄ h => LinearMap.rTensor N (Submodule.inclusion h)) P) u) = (LinearMap.rTensor N (↑P).subtype) u
PolynomialLaw.isCompat_apply 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {S : Type v} [CommSemiring S] [Algebra R S] (f : M →ₚₗ[R] N) {T : Type w} [CommSemiring T] [Algebra R T] (h : S →ₐ[R] T) (t : TensorProduct R S M) : (LinearMap.rTensor N h.toLinearMap) (PolynomialLaw.toFun S f t) = PolynomialLaw.toFun T f ((LinearMap.rTensor M h.toLinearMap) t)
PolynomialLaw.isCompat_apply' 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {f : M →ₚₗ[R] N} {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') (x : TensorProduct R S M) : (LinearMap.rTensor N φ.toLinearMap) (f.toFun' S x) = f.toFun' S' ((LinearMap.rTensor M φ.toLinearMap) x)
PolynomialLaw.isCompat 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {S : Type v} [CommSemiring S] [Algebra R S] (f : M →ₚₗ[R] N) {T : Type w} [CommSemiring T] [Algebra R T] (h : S →ₐ[R] T) : ⇑(LinearMap.rTensor N h.toLinearMap) ∘ PolynomialLaw.toFun S f = PolynomialLaw.toFun T f ∘ ⇑(LinearMap.rTensor M h.toLinearMap)
PolynomialLaw.isCompat' 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] (self : M →ₚₗ[R] N) {S : Type u} [CommSemiring S] [Algebra R S] {S' : Type u} [CommSemiring S'] [Algebra R S'] (φ : S →ₐ[R] S') : ⇑(LinearMap.rTensor N φ.toLinearMap) ∘ self.toFun' S = self.toFun' S' ∘ ⇑(LinearMap.rTensor M φ.toLinearMap)
PolynomialLaw.mk 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] (toFun' : (S : Type u) → [inst : CommSemiring S] → [inst_1 : Algebra R S] → TensorProduct R S M → TensorProduct R S N) (isCompat' : ∀ {S : Type u} [inst : CommSemiring S] [inst_1 : Algebra R S] {S' : Type u} [inst_2 : CommSemiring S'] [inst_3 : Algebra R S'] (φ : S →ₐ[R] S'), ⇑(LinearMap.rTensor N φ.toLinearMap) ∘ toFun' S = toFun' S' ∘ ⇑(LinearMap.rTensor M φ.toLinearMap) := by aesop) : M →ₚₗ[R] N
PolynomialLaw.exists_lift 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {S : Type v} [CommSemiring S] [Algebra R S] (t : TensorProduct R S M) : ∃ n ψ p, (LinearMap.rTensor M ψ.toLinearMap) p = t
PolynomialLaw.exists_lift' 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {S : Type v} [CommSemiring S] [Algebra R S] (t : TensorProduct R S M) (s : S) : ∃ n ψ p q, (LinearMap.rTensor M ψ.toLinearMap) p = t ∧ ψ q = s
PolynomialLaw.toFun_eq_rTensor_φ_toFun' 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {S : Type v} [CommSemiring S] [Algebra R S] (f : M →ₚₗ[R] N) {t : TensorProduct R S M} {s : Finset S} {p : TensorProduct R (MvPolynomial (Fin s.card) R) M} (ha : PolynomialLaw.π R M S ⟨s, p⟩ = t) : PolynomialLaw.toFun S f t = (LinearMap.rTensor N (PolynomialLaw.φ R s).toLinearMap) (f.toFun' (MvPolynomial (Fin s.card) R) p)
PolynomialLaw.exists_lift_of_mem_range_rTensor 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {S : Type v} [CommSemiring S] [Algebra R S] {T : Type u_3} [CommSemiring T] [Algebra R T] (A : Subalgebra R T) {φ : S →ₐ[R] T} (hφ : A ≤ φ.range) {t : TensorProduct R T M} (ht : t ∈ (LinearMap.rTensor M A.val.toLinearMap).range) : ∃ s, (LinearMap.rTensor M φ.toLinearMap) s = t
PolynomialLaw.toFun'_eq_of_inclusion 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {S : Type v} [CommSemiring S] [Algebra R S] (f : M →ₚₗ[R] N) {A : Type u} [CommSemiring A] [Algebra R A] {φ : A →ₐ[R] S} (p : TensorProduct R A M) {B : Type u} [CommSemiring B] [Algebra R B] (q : TensorProduct R B M) {ψ : B →ₐ[R] S} (h : φ.range ≤ ψ.range) (hpq : (LinearMap.rTensor M ((Subalgebra.inclusion h).comp φ.rangeRestrict).toLinearMap) p = (LinearMap.rTensor M ψ.rangeRestrict.toLinearMap) q) : (LinearMap.rTensor N φ.toLinearMap) (f.toFun' A p) = (LinearMap.rTensor N ψ.toLinearMap) (f.toFun' B q)
PolynomialLaw.toFun'_eq_of_diagram 📋 Mathlib.RingTheory.PolynomialLaw.Basic
{R : Type u} [CommSemiring R] {M : Type u_1} [AddCommMonoid M] [Module R M] {N : Type u_2} [AddCommMonoid N] [Module R N] {S : Type v} [CommSemiring S] [Algebra R S] (f : M →ₚₗ[R] N) {A : Type u} [CommSemiring A] [Algebra R A] {φ : A →ₐ[R] S} (p : TensorProduct R A M) {T : Type w} [CommSemiring T] [Algebra R T] {B : Type u} [CommSemiring B] [Algebra R B] {ψ : B →ₐ[R] T} (q : TensorProduct R B M) (h : S →ₐ[R] T) (h' : ↥φ.range →ₐ[R] ↥ψ.range) (hh' : ψ.range.val.comp h' = h.comp φ.range.val) (hpq : (LinearMap.rTensor M (h'.comp φ.rangeRestrict).toLinearMap) p = (LinearMap.rTensor M ψ.rangeRestrict.toLinearMap) q) : (LinearMap.rTensor N (h.comp φ).toLinearMap) (f.toFun' A p) = (LinearMap.rTensor N ψ.toLinearMap) (f.toFun' B q)

About

Loogle searches Lean and Mathlib definitions and theorems.

You can use Loogle from within the Lean4 VSCode language extension using the Loogle command from the command palette. You can also try the #loogle command from LeanSearchClient, the CLI version, the Loogle VS Code extension, the lean.nvim integration or the Zulip bot.

Usage

Loogle finds definitions and lemmas in various ways:

By constant:
🔍 Real.sin
finds all lemmas whose statement somehow mentions the sine function.
By lemma name substring:
🔍 "differ"
finds all lemmas that have "differ" somewhere in their lemma name.
By subexpression:
🔍 _ * (_ ^ _)
finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in
🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map:
🔍 (?a -> ?b) -> List ?a -> List ?b
🔍 List ?a -> (?a -> ?b) -> List ?b
By main conclusion:
🔍 |- tsum _ = _ * tsum _
finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example,
🔍 |- _ < _ → tsum _ < tsum _
will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.
You can filter for definitions vs theorems: Using ⊢ (_ : Type _) finds all definitions which provide data while ⊢ (_ : Prop) finds all theorems (and definitions of proofs).

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search
🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _
would find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

The #lucky button will directly send you to the documentation of the first hit.

Source code

You can find the source code for this service at https://github.com/nomeata/loogle. The https://loogle.lean-lang.org/ service is provided by the Lean FRO. Please review the Lean FRO Terms of Use and Privacy Policy.

This is Loogle revision 9f11169 serving mathlib revision 69fae59