Theorems · Definition · commutative algebra

HomogeneousLocalization.map

{ι : Type u_1} →
  {A : Type u_2} →
    {σ : Type u_3} →
      [inst : CommRing A] →
        [inst_1 : SetLike σ A] →
          [inst_2 : AddSubgroupClass σ A] →
            [inst_3 : AddCommMonoid ι] →
              [inst_4 : DecidableEq ι] →
                {𝒜 : ι → σ} →
                  [inst_5 : GradedRing 𝒜] →
                    {B : Type u_4} →
                      {τ : Type u_5} →
                        [inst_6 : CommRing B] →
                          [inst_7 : SetLike τ B] →
                            [inst_8 : AddSubgroupClass τ B] →
                              {ℬ : ι → τ} →
                                [inst_9 : GradedRing ℬ] →
                                  {P : Submonoid A} →
                                    {Q : Submonoid B} →
                                      (g : 𝒜 →+*ᵍ ℬ) →
                                        P ≤ Submonoid.comap g Q →
                                          HomogeneousLocalization 𝒜 P →+* HomogeneousLocalization ℬ Q

Let A, B be two graded rings with the same indexing set and g : 𝒜 →+*ᵍ ℬ be a graded ring homomorphism. Let P ≤ A be a submonoid and Q ≤ B be a submonoid such that P ≤ g⁻¹ Q, then g induces a map from the homogeneous localization A⁰_P to the homogeneous localization B⁰_Q.

Defined in

Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Cited by

5 results in Mathlib

Foundations

Depth 70 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassAddCommMonoidDecidableEqGradedRingCommRingSetLikeAddSubgroupClassGradedRing