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feat(AlgebraicGeometry/Birational/RationalMap): add RationalMap.representative #39656

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feat(AlgebraicGeometry/Birational/RationalMap): add RationalMap.representative #39656

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14 changes: 14 additions & 0 deletions Mathlib/AlgebraicGeometry/Birational/RationalMap.lean
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Expand Up @@ -334,6 +334,20 @@ lemma PartialMap.toRationalMap_eq_iff {f g : X.PartialMap Y} :
f.toRationalMap = g.toRationalMap ↔ f.equiv g :=
Quotient.eq

/-- An arbitrarily chosen partial map representing `f`. Use `RationalMap.toPartialMap` instead
if `X` is reduced and `Y` is separated. -/
noncomputable def RationalMap.representative (f : X ⤏ Y) : X.PartialMap Y :=
f.exists_rep.choose

@[simp]
lemma RationalMap.toRationalMap_representative (f : X ⤏ Y) :
f.representative.toRationalMap = f :=
f.exists_rep.choose_spec

lemma PartialMap.representative_toRationalMap_equiv (f : X.PartialMap Y) :
f.toRationalMap.representative.equiv f := by
rw [← PartialMap.toRationalMap_eq_iff, f.toRationalMap.toRationalMap_representative]

@[simp]
lemma PartialMap.restrict_toRationalMap (f : X.PartialMap Y) (U : X.Opens)
(hU : Dense (U : Set X)) (hU' : U ≤ f.domain) :
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