tensorProductTo_surjective — Mathlib

English

The map tensorProductTo of Kaehler differential is surjective onto the cotangent, i.e., every cotangent element arises from tensoring with differential.

Русский

Карта tensorProductTo каехлерова дифференциала сюръективна на касательную.

LaTeX

$$$\\mathrm{tensorProductTo}: \\mathrm{D}R S \\to \\mathrm{Cotangent}$ is surjective$$

Lean4

theorem tensorProductTo_surjective : Function.Surjective (KaehlerDifferential.D R S).tensorProductTo :=
  by
  intro x; obtain ⟨x, rfl⟩ := (KaehlerDifferential.ideal R S).toCotangent_surjective x
  exact ⟨x, KaehlerDifferential.D_tensorProductTo x⟩

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