The characteristic predicate of tensor product

Main definitions

• IsTensorProduct: A predicate on f : M₁ →ₗ[R] M₂ →ₗ[R] M expressing that f realizes M as the tensor product of M₁ ⊗[R] M₂. This is defined by requiring the lift M₁ ⊗[R] M₂ → M to be bijective.
• IsBaseChange: A predicate on an R-algebra S and a map f : M →ₗ[R] N with N being an S-module, expressing that f realizes N as the base change of M along R → S.
• Algebra.IsPushout: A predicate on the following diagram of scalar towers

    R  →  S
    ↓     ↓
    R' →  S'
  

  asserting that is a pushout diagram (i.e. S' = S ⊗[R] R')

Main results

• TensorProduct.isBaseChange: S ⊗[R] M is the base change of M along R → S.

def IsTensorProduct {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] (f : M₁ →ₗ[R] M₂ →ₗ[R] M) : Prop

Given a bilinear map f : M₁ →ₗ[R] M₂ →ₗ[R] M, IsTensorProduct f means that M is the tensor product of M₁ and M₂ via f. This is defined by requiring the lift M₁ ⊗[R] M₂ → M to be bijective.

Equations

• IsTensorProduct f = Function.Bijective ⇑(TensorProduct.lift f)

theorem TensorProduct.isTensorProduct (R : Type u_1) [CommSemiring R] (M : Type u_4) [AddCommMonoid M] [Module R M] (N : Type u_8) [AddCommMonoid N] [Module R N] : IsTensorProduct (TensorProduct.mk R M N)

noncomputable def IsTensorProduct.equiv {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) : TensorProduct R M₁ M₂ ≃ₗ[R] M

If M is the tensor product of M₁ and M₂, it is linearly equivalent to M₁ ⊗[R] M₂.

Equations

• h.equiv = LinearEquiv.ofBijective (TensorProduct.lift f) h

@[simp]

theorem IsTensorProduct.equiv_apply {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) (a✝ : TensorProduct R M₁ M₂) : h.equiv a✝ = (TensorProduct.lift f) a✝

@[simp]

theorem IsTensorProduct.equiv_toLinearMap {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) : ↑h.equiv = TensorProduct.lift f

@[simp]

theorem IsTensorProduct.equiv_symm_apply {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) (x₁ : M₁) (x₂ : M₂) : h.equiv.symm ((f x₁) x₂) = x₁ ⊗ₜ[R] x₂

noncomputable def IsTensorProduct.lift {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} {M' : Type u_5} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [AddCommMonoid M'] [Module R M₁] [Module R M₂] [Module R M] [Module R M'] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) (f' : M₁ →ₗ[R] M₂ →ₗ[R] M') : M →ₗ[R] M'

If M is the tensor product of M₁ and M₂, we may lift a bilinear map M₁ →ₗ[R] M₂ →ₗ[R] M' to a M →ₗ[R] M'.

Equations

• h.lift f' = TensorProduct.lift f' ∘ₗ ↑h.equiv.symm

theorem IsTensorProduct.lift_eq {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} {M' : Type u_5} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [AddCommMonoid M'] [Module R M₁] [Module R M₂] [Module R M] [Module R M'] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) (f' : M₁ →ₗ[R] M₂ →ₗ[R] M') (x₁ : M₁) (x₂ : M₂) : (h.lift f') ((f x₁) x₂) = (f' x₁) x₂

noncomputable def IsTensorProduct.map {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} {N₁ : Type u_6} {N₂ : Type u_7} {N : Type u_8} [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N] [Module R N₁] [Module R N₂] [Module R N] {g : N₁ →ₗ[R] N₂ →ₗ[R] N} (hf : IsTensorProduct f) (hg : IsTensorProduct g) (i₁ : M₁ →ₗ[R] N₁) (i₂ : M₂ →ₗ[R] N₂) : M →ₗ[R] N

The tensor product of a pair of linear maps between modules.

Equations

• hf.map hg i₁ i₂ = ↑hg.equiv ∘ₗ TensorProduct.map i₁ i₂ ∘ₗ ↑hf.equiv.symm

theorem IsTensorProduct.map_eq {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} {N₁ : Type u_6} {N₂ : Type u_7} {N : Type u_8} [AddCommMonoid N₁] [AddCommMonoid N₂] [AddCommMonoid N] [Module R N₁] [Module R N₂] [Module R N] {g : N₁ →ₗ[R] N₂ →ₗ[R] N} (hf : IsTensorProduct f) (hg : IsTensorProduct g) (i₁ : M₁ →ₗ[R] N₁) (i₂ : M₂ →ₗ[R] N₂) (x₁ : M₁) (x₂ : M₂) : (hf.map hg i₁ i₂) ((f x₁) x₂) = (g (i₁ x₁)) (i₂ x₂)

theorem IsTensorProduct.inductionOn {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (h : IsTensorProduct f) {C : M → Prop} (m : M) (h0 : C 0) (htmul : ∀ (x : M₁) (y : M₂), C ((f x) y)) (hadd : ∀ (x y : M), C x → C y → C (x + y)) : C m

theorem IsTensorProduct.of_equiv {R : Type u_1} [CommSemiring R] {M₁ : Type u_2} {M₂ : Type u_3} {M : Type u_4} [AddCommMonoid M₁] [AddCommMonoid M₂] [AddCommMonoid M] [Module R M₁] [Module R M₂] [Module R M] {f : M₁ →ₗ[R] M₂ →ₗ[R] M} (e : TensorProduct R M₁ M₂ ≃ₗ[R] M) (he : ∀ (x : M₁) (y : M₂), e (x ⊗ₜ[R] y) = (f x) y) : IsTensorProduct f

def IsBaseChange {R : Type u_1} {M : Type v₁} {N : Type v₂} (S : Type v₃) [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] (f : M →ₗ[R] N) : Prop

Given an R-algebra S and an R-module M, an S-module N together with a map f : M →ₗ[R] N is the base change of M to S if the map S × M → N, (s, m) ↦ s • f m is the tensor product.

Equations

• IsBaseChange S f = IsTensorProduct (↑R ((Algebra.linearMap S (Module.End S (M →ₗ[R] N))).flip f))

noncomputable def IsBaseChange.lift {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) {Q : Type u_3} [AddCommMonoid Q] [Module S Q] [Module R Q] [IsScalarTower R S Q] (g : M →ₗ[R] Q) : N →ₗ[S] Q

Suppose f : M →ₗ[R] N is the base change of M along R → S. Then any R-linear map from M to an S-module factors through f.

Equations

• h.lift g = { toAddHom := (IsTensorProduct.lift h (↑R ((Algebra.linearMap S (Module.End S (M →ₗ[R] Q))).flip g))).toAddHom, map_smul' := ⋯ }

theorem IsBaseChange.lift_eq {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) {Q : Type u_3} [AddCommMonoid Q] [Module S Q] [Module R Q] [IsScalarTower R S Q] (g : M →ₗ[R] Q) (x : M) : (h.lift g) (f x) = g x

theorem IsBaseChange.lift_comp {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) {Q : Type u_3} [AddCommMonoid Q] [Module S Q] [Module R Q] [IsScalarTower R S Q] (g : M →ₗ[R] Q) : ↑R (h.lift g) ∘ₗ f = g

theorem IsBaseChange.inductionOn {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) (x : N) (P : N → Prop) (h₁ : P 0) (h₂ : ∀ (m : M), P (f m)) (h₃ : ∀ (s : S) (n : N), P n → P (s • n)) (h₄ : ∀ (n₁ n₂ : N), P n₁ → P n₂ → P (n₁ + n₂)) : P x

theorem IsBaseChange.algHom_ext {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) {Q : Type u_3} [AddCommMonoid Q] [Module S Q] (g₁ g₂ : N →ₗ[S] Q) (e : ∀ (x : M), g₁ (f x) = g₂ (f x)) : g₁ = g₂

theorem IsBaseChange.algHom_ext' {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) {Q : Type u_3} [AddCommMonoid Q] [Module S Q] [Module R Q] [IsScalarTower R S Q] (g₁ g₂ : N →ₗ[S] Q) (e : ↑R g₁ ∘ₗ f = ↑R g₂ ∘ₗ f) : g₁ = g₂

theorem TensorProduct.isBaseChange (R : Type u_1) (M : Type v₁) (S : Type v₃) [AddCommMonoid M] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] : IsBaseChange S ((TensorProduct.mk R S M) 1)

noncomputable def IsBaseChange.equiv {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) : TensorProduct R S M ≃ₗ[S] N

The base change of M along R → S is linearly equivalent to S ⊗[R] M.

Equations

• h.equiv = { toAddHom := (↑(IsTensorProduct.equiv h)).toAddHom, map_smul' := ⋯, invFun := (IsTensorProduct.equiv h).invFun, left_inv := ⋯, right_inv := ⋯ }

theorem IsBaseChange.equiv_tmul {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) (s : S) (m : M) : h.equiv (s ⊗ₜ[R] m) = s • f m

theorem IsBaseChange.equiv_symm_apply {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (h : IsBaseChange S f) (m : M) : h.equiv.symm (f m) = 1 ⊗ₜ[R] m

theorem IsBaseChange.of_equiv {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} (e : TensorProduct R S M ≃ₗ[S] N) (he : ∀ (x : M), e (1 ⊗ₜ[R] x) = f x) : IsBaseChange S f

theorem isBaseChange_tensorProduct_map {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {P : Type u_2} [AddCommMonoid P] [Module R P] (A : Type u_4) [CommSemiring A] [Algebra R A] [Algebra S A] [IsScalarTower R S A] [Module S M] [IsScalarTower R S M] [Module A N] [IsScalarTower S A N] [IsScalarTower R A N] {f : M →ₗ[S] N} (hf : IsBaseChange A f) : IsBaseChange A (TensorProduct.AlgebraTensorModule.map f LinearMap.id)

If N is the base change of M to A, then N ⊗[R] P is the base change of M ⊗[R] P to A. This is simply the isomorphism A ⊗[S] (M ⊗[R] P) ≃ₗ[A] (A ⊗[S] M) ⊗[R] P.

theorem IsBaseChange.of_lift_unique {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] (f : M →ₗ[R] N) (h : ∀ (Q : Type (max v₁ v₂ v₃)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module S Q] [inst_3 : IsScalarTower R S Q] (g : M →ₗ[R] Q), ∃! g' : N →ₗ[S] Q, ↑R g' ∘ₗ f = g) : IsBaseChange S f

theorem IsBaseChange.iff_lift_unique {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {f : M →ₗ[R] N} : IsBaseChange S f ↔ ∀ (Q : Type (max v₁ v₂ v₃)) [inst : AddCommMonoid Q] [inst_1 : Module R Q] [inst_2 : Module S Q] [inst_3 : IsScalarTower R S Q] (g : M →ₗ[R] Q), ∃! g' : N →ₗ[S] Q, ↑R g' ∘ₗ f = g

theorem IsBaseChange.ofEquiv {R : Type u_1} {M : Type v₁} {N : Type v₂} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [Module R M] [Module R N] (e : M ≃ₗ[R] N) : IsBaseChange R ↑e

theorem IsBaseChange.comp {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {T : Type u_4} {O : Type u_5} [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [AddCommMonoid O] [Module R O] [Module S O] [Module T O] [IsScalarTower S T O] [IsScalarTower R S O] [IsScalarTower R T O] {f : M →ₗ[R] N} (hf : IsBaseChange S f) {g : N →ₗ[S] O} (hg : IsBaseChange T g) : IsBaseChange T (↑R g ∘ₗ f)

theorem IsBaseChange.of_comp {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {T : Type u_4} {O : Type u_5} [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [AddCommMonoid O] [Module R O] [Module S O] [Module T O] [IsScalarTower S T O] [IsScalarTower R S O] [IsScalarTower R T O] {f : M →ₗ[R] N} (hf : IsBaseChange S f) {h : N →ₗ[S] O} (hc : IsBaseChange T (↑R h ∘ₗ f)) : IsBaseChange T h

If N is the base change of M to S and O the base change of M to T, then O is the base change of N to T.

theorem IsBaseChange.comp_iff {R : Type u_1} {M : Type v₁} {N : Type v₂} {S : Type v₃} [AddCommMonoid M] [AddCommMonoid N] [CommSemiring R] [CommSemiring S] [Algebra R S] [Module R M] [Module R N] [Module S N] [IsScalarTower R S N] {T : Type u_4} {O : Type u_5} [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] [AddCommMonoid O] [Module R O] [Module S O] [Module T O] [IsScalarTower S T O] [IsScalarTower R S O] [IsScalarTower R T O] {f : M →ₗ[R] N} (hf : IsBaseChange S f) {h : N →ₗ[S] O} : IsBaseChange T (↑R h ∘ₗ f) ↔ IsBaseChange T h

If N is the base change M to S, then O is the base change of M to T if and only if O is the base change of N to T.

class Algebra.IsPushout (R : Type u_1) (S : Type v₃) [CommSemiring R] [CommSemiring S] [Algebra R S] (R' : Type u_6) (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] : Prop

A type-class stating that the following diagram of scalar towers R → S ↓ ↓ R' → S' is a pushout diagram (i.e. S' = S ⊗[R] R')

• out : IsBaseChange S (IsScalarTower.toAlgHom R R' S').toLinearMap

theorem Algebra.isPushout_iff (R : Type u_1) (S : Type v₃) [CommSemiring R] [CommSemiring S] [Algebra R S] (R' : Type u_6) (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] : Algebra.IsPushout R S R' S' ↔ IsBaseChange S (IsScalarTower.toAlgHom R R' S').toLinearMap

theorem Algebra.IsPushout.symm {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} {S' : Type u_7} [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] (h : Algebra.IsPushout R S R' S') : Algebra.IsPushout R R' S S'

theorem Algebra.IsPushout.comm (R : Type u_1) (S : Type v₃) [CommSemiring R] [CommSemiring S] [Algebra R S] (R' : Type u_6) (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] : Algebra.IsPushout R S R' S' ↔ Algebra.IsPushout R R' S S'

instance TensorProduct.isPushout {R : Type u_8} {S : Type u_9} {T : Type u_10} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] : Algebra.IsPushout R S T (TensorProduct R S T)

instance TensorProduct.isPushout' {R : Type u_8} {S : Type u_9} {T : Type u_10} [CommRing R] [CommRing S] [CommRing T] [Algebra R S] [Algebra R T] : Algebra.IsPushout R T S (TensorProduct R S T)

noncomputable def Algebra.pushoutDesc {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [H : Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (hf : ∀ (x : S) (y : R'), f x * g y = g y * f x) : S' →ₐ[R] A

If S' = S ⊗[R] R', then any pair of R-algebra homomorphisms f : S → A and g : R' → A such that f x and g y commutes for all x, y descends to a (unique) homomorphism S' → A.

Equations

• Algebra.pushoutDesc S' f g hf = AlgHom.ofLinearMap (↑R (⋯.lift g.toLinearMap)) ⋯ ⋯

theorem Algebra.pushoutDesc_apply {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [H : Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (hf : ∀ (x : S) (y : R'), f x * g y = g y * f x) (a : S') : (Algebra.pushoutDesc S' f g hf) a = (⋯.lift g.toLinearMap) a

@[simp]

theorem Algebra.pushoutDesc_left {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H : ∀ (x : S) (y : R'), f x * g y = g y * f x) (x : S) : (Algebra.pushoutDesc S' f g H) ((algebraMap S S') x) = f x

theorem Algebra.lift_algHom_comp_left {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H : ∀ (x : S) (y : R'), f x * g y = g y * f x) : (Algebra.pushoutDesc S' f g H).comp (IsScalarTower.toAlgHom R S S') = f

@[simp]

theorem Algebra.pushoutDesc_right {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H : ∀ (x : S) (y : R'), f x * g y = g y * f x) (x : R') : (Algebra.pushoutDesc S' f g H) ((algebraMap R' S') x) = g x

theorem Algebra.lift_algHom_comp_right {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] (f : S →ₐ[R] A) (g : R' →ₐ[R] A) (H : ∀ (x : S) (y : R'), f x * g y = g y * f x) : (Algebra.pushoutDesc S' f g H).comp (IsScalarTower.toAlgHom R R' S') = g

theorem Algebra.IsPushout.algHom_ext {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] [H : Algebra.IsPushout R S R' S'] {A : Type u_8} [Semiring A] [Algebra R A] {f g : S' →ₐ[R] A} (h₁ : f.comp (IsScalarTower.toAlgHom R R' S') = g.comp (IsScalarTower.toAlgHom R R' S')) (h₂ : f.comp (IsScalarTower.toAlgHom R S S') = g.comp (IsScalarTower.toAlgHom R S S')) : f = g

theorem Algebra.IsPushout.comp_iff {R : Type u_1} {S : Type v₃} [CommSemiring R] [CommSemiring S] [Algebra R S] {T : Type u_4} [CommSemiring T] [Algebra R T] [Algebra S T] [IsScalarTower R S T] {R' : Type u_6} (S' : Type u_7) [CommSemiring R'] [CommSemiring S'] [Algebra R R'] [Algebra S S'] [Algebra R' S'] [Algebra R S'] [IsScalarTower R R' S'] [IsScalarTower R S S'] {T' : Type u_8} [CommRing T'] [Algebra R T'] [Algebra S' T'] [Algebra S T'] [Algebra T T'] [Algebra R' T'] [IsScalarTower R T T'] [IsScalarTower S T T'] [IsScalarTower S S' T'] [IsScalarTower R R' T'] [IsScalarTower R S' T'] [IsScalarTower R' S' T'] [Algebra.IsPushout R S R' S'] : Algebra.IsPushout R T R' T' ↔ Algebra.IsPushout S T S' T'

Let the following be a commutative diagram of rings

  R  →  S  →  T
  ↓     ↓     ↓
  R' →  S' →  T'

where the left-hand square is a pushout. Then the following are equivalent:

• the big rectangle is a pushout.
• the right-hand square is a pushout.

Note that this is essentially the isomorphism T ⊗[S] (S ⊗[R] R') ≃ₐ[T] T ⊗[R] R'.