Theorems · Definition · commutative algebra

HomogeneousLocalization.localRingHom

{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] →
                                  (f : 𝒜 →+*ᵍ ℬ) →
                                    (I : Ideal A) →
                                      [inst_10 : I.IsPrime] →
                                        (J : Ideal B) →
                                          [inst_11 : J.IsPrime] →
                                            I = Ideal.comap f J →
                                              HomogeneousLocalization.AtPrime 𝒜 I →+*
                                                HomogeneousLocalization.AtPrime ℬ J

If f : 𝒜 →+*ᵍ ℬ is a graded ring homomorphism and I is a prime ideal of A and J is a prime ideal of B and f⁻¹ J = I then we have a map AtPrime 𝒜 I →+* AtPrime ℬ J.

Defined in

Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Cited by

5 results in Mathlib

Foundations

Depth 71 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassAddCommMonoidDecidableEqGradedRingCommRingSetLikeAddSubgroupClassGradedRingIdeal.IsPrimeIdeal.IsPrime