coprodSpec_coprodMk — Mathlib

English

The left inclusion composed with the coproduct-to-Sigma inverse equals the left injection.

Русский

Левое включение, композиция с обратной стороны от coproduct-to-Sigma равна левому введению.

LaTeX

$$$\\Sigma.\\iota_{\\mathrm{left}} \\circ (coprodIsoSigma X Y)^{-1} = coprod.inl$$$

Lean4

theorem coprodSpec_coprodMk (x) :
    (coprodSpec R S).base (coprodMk _ _ x) = (PrimeSpectrum.primeSpectrumProd R S).symm x :=
  by
  apply PrimeSpectrum.ext
  obtain (x | x) := x <;>
    simp only [coprodMk_inl, coprodMk_inr, ← Scheme.comp_base_apply, coprodSpec, coprod.inl_desc, coprod.inr_desc]
  · change Ideal.comap _ _ = x.asIdeal.prod ⊤
    ext; simp [Ideal.prod, CommRingCat.ofHom]
  · change Ideal.comap _ _ = Ideal.prod ⊤ x.asIdeal
    ext; simp [Ideal.prod, CommRingCat.ofHom]

Похожие утверждения