Theorems · Theorem · commutative algebra

HomogeneousLocalization.Away.span_mk_prod_pow_eq_top

∀ {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] {f : A} {d : ι} (hf : f ∈ 𝒜 d) {ι' : Type u_4} [inst_6 : Fintype ι'] (v : ι' → A),
  Algebra.adjoin (↥(𝒜 0)) (Set.range v) = ⊤ →
    ∀ (dv : ι' → ι) (hxd : ∀ (i : ι'), v i ∈ 𝒜 (dv i)),
      Submodule.span ↥(𝒜 0)
          {x |
            ∃ a ai,
              ∃ (hai : ∑ i, ai i • dv i = a • d), HomogeneousLocalization.Away.mk 𝒜 hf a (∏ i, v i ^ ai i) ⋯ = x} =
        ⊤

Let 𝒜 be a graded ring, finitely generated (as an algebra) over 𝒜₀ by { vᵢ }, where vᵢ has degree dvᵢ. If f : A has degree d, then 𝒜_(f) is generated (as a module) over 𝒜₀ by elements of the form (∏ i, vᵢ ^ aᵢ) / fᵃ such that ∑ aᵢ • dvᵢ = a • d.

Defined in

Mathlib.RingTheory.GradedAlgebra.HomogeneousLocalization

Cited by

1 results in Mathlib

Foundations

Depth 100 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingSetLikeAddSubgroupClassAddCommMonoidDecidableEqGradedRingFintype