Theorems · Definition · commutative algebra

Algebra.pushoutDesc

{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) → (∀ (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.

Defined in

Mathlib.RingTheory.IsTensorProduct

Cited by

6 results in Mathlib

Foundations

Depth 79 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommSemiringCommSemiringAlgebraCommSemiringCommSemiringAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTowerAlgebra.IsPushoutSemiringAlgebra