ofIdealTop — Mathlib

English

Auxiliary proof piece about the mkOfMemSupportIff constructor.

Русский

В auxiliar части доказательства конструктора mkOfMemSupportIff.

LaTeX

$$I.e. auxiliary lemma in mkOfMemSupportIff$$

Lean4

/-- The ideal sheaf induced by an ideal of the global sections. -/
@[simps! ideal coe_support]
def ofIdealTop (I : Ideal Γ(X, ⊤)) : IdealSheafData X :=
  mkOfMemSupportIff (fun U ↦ I.map (X.presheaf.map (homOfLE le_top).op).hom)
    (fun U f ↦ by rw [Ideal.map_map, ← CommRingCat.hom_comp, ← Functor.map_comp]; rfl) (X.zeroLocus (U := ⊤) I)
    (fun U x hxU ↦ by
      simp only [Ideal.map, zeroLocus_span, zeroLocus_map, Set.mem_union, Set.mem_compl_iff, SetLike.mem_coe, hxU,
        not_true_eq_false, iff_self_or, IsEmpty.forall_iff])

Похожие утверждения