fromSpec_toSpec — Mathlib

English

The image of the carrier under the projection is a proper subset, not equal to the whole ring.

Русский

Образ носителя под проекцией не равен 전체 кольцу.

LaTeX

$$$fdeg hm q \neq ⊤ \Rightarrow \dots$$$

Lean4

theorem fromSpec_toSpec {f : A} {m : ℕ} (f_deg : f ∈ 𝒜 m) (hm : 0 < m) (x : Proj.T| pbo f) :
    FromSpec.toFun f_deg hm (toSpec 𝒜 f x) = x :=
  by
  refine Subtype.ext <| ProjectiveSpectrum.ext <| HomogeneousIdeal.ext' ?_
  intro i z hzi
  refine (FromSpec.mem_carrier_iff_of_mem f_deg hm _ _ hzi).trans ?_
  exact (ToSpec.mk_mem_carrier _ _).trans (x.1.2.pow_mem_iff_mem m hm)

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