map_over — Mathlib

English

For morphism f: S → T of schemes, the square formed by map n f and the affine-space projections is commutative, i.e., map n f followed by 𝔸(n; T) ↓ T equals 𝔸(n; S) ↓ S followed by f.

Русский

Для морфизма f: S → T график диаграммы: map n f ∘ projection = projection ∘ map n f; квадрат коммутирует.

LaTeX

$$$\\mathrm{map}\\ n\\ f \\circ (\\mathbb{A}(n; T) \\downarrow T) = (\\mathbb{A}(n; S) \\downarrow S) \\circ f$.$$

Lean4

@[reassoc (attr := simp)]
theorem map_over {S T : Scheme.{max u v}} (f : S ⟶ T) : map n f ≫ 𝔸(n; T) ↘ T = 𝔸(n; S) ↘ S ≫ f :=
  pullback.lift_fst _ _ _

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