Theorems · Theorem · algebraic geometry
AlgebraicGeometry.IsImmersion.isImmersion_iff_exists
∀ {X Y : AlgebraicGeometry.Scheme} {f : X ⟶ Y},
AlgebraicGeometry.IsImmersion f ↔
∃ Z g₁ g₂,
AlgebraicGeometry.IsClosedImmersion g₁ ∧
AlgebraicGeometry.IsOpenImmersion g₂ ∧ CategoryTheory.CategoryStruct.comp g₁ g₂ = fA morphism is a (locally-closed) immersion if and only if it can be factored into a closed immersion followed by an open immersion.
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- Foundations
- Depth 144 from the axioms · uses propext, Classical.choice, Quot.sound
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Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Quiver.Homstatement and proof · cited by 32,603
- CategoryTheory.CategoryStruct.compstatement and proof · cited by 17,999
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- AlgebraicGeometry.IsOpenImmersionstatement and proof · cited by 476
- AlgebraicGeometry.Scheme.Opens.toSchemeproof · cited by 433
- AlgebraicGeometry.Scheme.Opens.ιproof · cited by 275
- AlgebraicGeometry.IsClosedImmersionstatement and proof · cited by 49
- AlgebraicGeometry.IsImmersionstatement and proof · cited by 17
- AlgebraicGeometry.Scheme.Hom.coborderRangeproof · cited by 8
- AlgebraicGeometry.Scheme.Hom.liftCoborderproof · cited by 6
- AlgebraicGeometry.Scheme.Hom.liftCoborder_ιproof · cited by 6
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