toΓSpec_preimage_basicOpen_eq — Mathlib

English

A restatement: for X integral and arbitrary r and x, the same equivalence between not being in the prime associated to x and being a unit in the stalk holds, expressed through the germ map.

Русский

Переформулировка: для X интеграл и произвольного r и x справедлива та же эквивалентность между не принадлежностью к соответствующему идеалу и being единицей в стержне, через герм-маршрут.

LaTeX

$$$r \notin (X.toΓSpecFun x).asIdeal \iff IsUnit\left(((X.presheaf.Γgerm x)\right. r)\right).$$$

Lean4

/-- The preimage of a basic open in `Spec Γ(X)` under the unit is the basic
open in `X` defined by the same element (they are equal as sets). -/
theorem toΓSpec_preimage_basicOpen_eq (r : Γ.obj (op X)) :
    X.toΓSpecFun ⁻¹' basicOpen r = SetLike.coe (X.toRingedSpace.basicOpen r) :=
  by
  ext
  dsimp
  simp only [Set.mem_preimage, SetLike.mem_coe]
  rw [X.toRingedSpace.mem_top_basicOpen]
  exact notMem_prime_iff_unit_in_stalk ..

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