Theorems · Definition · commutative algebra

Algebra.Generators.Hom.equivAlgHom

{R : Type u} →
  {S : Type v} →
    {ι : Type w} →
      [inst : CommRing R] →
        [inst_1 : CommRing S] →
          [inst_2 : Algebra R S] →
            {P : Algebra.Generators R S ι} →
              {R' : Type u_1} →
                {S' : Type u_2} →
                  {ι' : Type u_3} →
                    [inst_3 : CommRing R'] →
                      [inst_4 : CommRing S'] →
                        [inst_5 : Algebra R' S'] →
                          {P' : Algebra.Generators R' S' ι'} →
                            [inst_6 : Algebra R R'] →
                              [inst_7 : Algebra S S'] →
                                [inst_8 : Algebra R S'] →
                                  [IsScalarTower R R' S'] →
                                    [IsScalarTower R S S'] →
                                      P.Hom P' ≃
                                        { f //
                                          ∀ (x : P.Ring),
                                            (MvPolynomial.aeval P'.val) (f x) =
                                              (algebraMap S S') ((MvPolynomial.aeval P.val) x) }

Giving a hom between two families of generators is equivalent to giving an algebra homomorphism between the polynomial rings.

Defined in

Mathlib.RingTheory.Extension.Generators

Cited by

2 results in Mathlib

Foundations

Depth 101 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingCommRingAlgebraCommRingCommRingAlgebraAlgebraAlgebraAlgebraIsScalarTowerIsScalarTower