Theorems · Theorem · commutative algebra

HomogeneousLocalization.map_comp

∀ {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] {C : Type u_6} {ψ : Type u_7}
  [inst_10 : CommRing C] [inst_11 : SetLike ψ C] [inst_12 : AddSubgroupClass ψ C] {𝒞 : ι → ψ} [inst_13 : GradedRing 𝒞]
  {f : 𝒜 →+*ᵍ ℬ} {g : ℬ →+*ᵍ 𝒞} {P : Submonoid A} {Q : Submonoid B} {R : Submonoid C} (hpq : P ≤ Submonoid.comap f Q)
  (hqr : Q ≤ Submonoid.comap g R),
  HomogeneousLocalization.map (g.comp f) ⋯ =
    (HomogeneousLocalization.map g hqr).comp (HomogeneousLocalization.map f hpq)

Defined in

Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Cited by

2 results in Mathlib

Foundations

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

Assumes

CommRingSetLikeAddSubgroupClassAddCommMonoidDecidableEqGradedRingCommRingSetLikeAddSubgroupClassGradedRingCommRingSetLikeAddSubgroupClassGradedRing