Theorems · Definition · commutative algebra

GradedAlgHom.liftEquiv

{ι : Type u_1} →
  {R : Type u_2} →
    {S : Type u_3} →
      {A : Type u_4} →
        {B : Type u_5} →
          [inst : DecidableEq ι] →
            [inst_1 : AddMonoid ι] →
              [inst_2 : CommSemiring R] →
                [inst_3 : CommSemiring S] →
                  [inst_4 : Semiring A] →
                    [inst_5 : Semiring B] →
                      [inst_6 : Algebra R A] →
                        [inst_7 : Algebra S B] →
                          (𝒜 : ι → Submodule R A) →
                            (ℬ : ι → Submodule S B) →
                              [inst_8 : GradedAlgebra 𝒜] →
                                [inst_9 : GradedAlgebra ℬ] →
                                  [inst_10 : Algebra R S] →
                                    [inst_11 : Algebra R B] →
                                      [inst_12 : IsScalarTower R S B] →
                                        (𝒜 →ₐᵍ[R] fun x => Submodule.restrictScalars R (ℬ x)) ≃
                                          ((fun x => Submodule.baseChange S (𝒜 x)) →ₐᵍ[S] ℬ)

A map from the base change of a graded algebra is the same as a map to the scalar restriction. In category-theoretical terms, this is an adjunction between: 1. 𝒜 ↦ (𝒜 · |>.baseChange S), a functor from Graded R-Algebra to Graded S-Algebra; and: 2. ℬ ↦ (ℬ · |>.restrictScalars R), a functor from Graded S-Algebra to Graded R-Algebra.

Defined in

Mathlib.RingTheory.GradedAlgebra.TensorProduct

Cited by

2 results in Mathlib

Foundations

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

Assumes

DecidableEqAddMonoidCommSemiringCommSemiringSemiringSemiringAlgebraAlgebraGradedAlgebraGradedAlgebraAlgebraAlgebraIsScalarTower