Theorems · Definition · commutative algebra

HomogeneousIdeal.irrelevant

{ι : Type u_1} →
  {σ : Type u_2} →
    {A : Type u_3} →
      [inst : Semiring A] →
        [inst_1 : DecidableEq ι] →
          [inst_2 : AddCommMonoid ι] →
            [inst_3 : PartialOrder ι] →
              [CanonicallyOrderedAdd ι] →
                [inst_5 : SetLike σ A] →
                  [inst_6 : AddSubmonoidClass σ A] → (𝒜 : ι → σ) → [inst_7 : GradedRing 𝒜] → HomogeneousIdeal 𝒜

For a graded ring ⨁ᵢ 𝒜ᵢ graded by [AddCommMonoid ι] [PartialOrder ι] [CanonicallyOrderedAdd ι], the irrelevant ideal refers to ⨁_{i>0} 𝒜ᵢ, or equivalently {a | a₀ = 0}. This definition is used in Proj construction where ι is always ℕ so the irrelevant ideal is simply elements with 0 as 0-th coordinate.

Defined in

Mathlib.RingTheory.GradedAlgebra.Homogeneous.Ideal

Cited by

46 results in Mathlib

Foundations

Depth 95 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

SemiringDecidableEqAddCommMonoidPartialOrderCanonicallyOrderedAddSetLikeAddSubmonoidClassGradedRing