Theorems · Theorem · commutative algebra

Algebra.pushoutDesc_apply

∀ {R : Type u_1} {S : Type v₃} [inst : CommSemiring R] [inst_1 : CommSemiring S] [inst_2 : Algebra R S] {R' : Type u_6}
  (S' : Type u_7) [inst_3 : CommSemiring R'] [inst_4 : CommSemiring S'] [inst_5 : Algebra R R'] [inst_6 : Algebra S S']
  [inst_7 : Algebra R' S'] [inst_8 : Algebra R S'] [inst_9 : IsScalarTower R R' S'] [inst_10 : IsScalarTower R S S']
  [H : Algebra.IsPushout R S R' S'] {A : Type u_8} [inst_11 : Semiring A] [inst_12 : 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) (x : S'),
  (Algebra.pushoutDesc S' f g hf) x = (Algebra.TensorProduct.lift f g hf) ((Algebra.IsPushout.equiv R S R' S').symm x)

Defined in

Mathlib.RingTheory.IsTensorProduct

Cited by

2 results in Mathlib

Foundations

Depth 80 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringCommSemiringAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerAlgebra.IsPushoutSemiringAlgebra