ker_of_isAffine — Mathlib

English

If f: X ⟶ Y is an isomorphism, then ker f = ⊤.

Русский

Если f: X ⟶ Y — изоморфизм, тогда ker f = ⊤.

LaTeX

$$(f).ker = \\top$$

Lean4

theorem ker_of_isAffine {X Y : Scheme} (f : X ⟶ Y) [IsAffine Y] : f.ker = ofIdealTop (RingHom.ker f.appTop.hom) :=
  by
  refine (le_of_isAffine ((f.ideal_ker_le _).trans (by simp))).antisymm (le_ofIdeals_iff.mpr fun U ↦ ?_)
  simp only [ofIdealTop_ideal, homOfLE_leOfHom, Ideal.map_le_iff_le_comap, RingHom.comap_ker, ← CommRingCat.hom_comp,
    f.naturality]
  intro x
  simp +contextual

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