Theorems · Inductive type · algebraic geometry
AlgebraicGeometry.Flat
{X Y : AlgebraicGeometry.Scheme} → (X ⟶ Y) → PropA morphism of schemes f : X ⟶ Y is flat if for each affine U ⊆ Y and
V ⊆ f ⁻¹' U, the induced map Γ(Y, U) ⟶ Γ(X, V) is flat. This is equivalent to
asking that all stalk maps are flat (see AlgebraicGeometry.Flat.iff_flat_stalkMap).
- Defined in
- Mathlib.AlgebraicGeometry.Morphisms.Flat
- Cited by
- 49 results in Mathlib
- Foundations
- Depth 99 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites2
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Quiver.Homstatement · cited by 32,603
- AlgebraicGeometry.Schemestatement · cited by 2,540
Cited by55
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.Hom.finrank_comp_left_of_isIsostatement and proof · cited by 7
- AlgebraicGeometry.Scheme.Hom.finrank_pullback_sndstatement and proof · cited by 6
- AlgebraicGeometry.Scheme.fppfPrecoverageproof · cited by 4
- AlgebraicGeometry.Scheme.fpqcPrecoverageproof · cited by 4
- AlgebraicGeometry.Scheme.Hom.flat_appLEstatement · cited by 3
- AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_left_of_ringHomFlatstatement and proof · cited by 2
- AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_rightstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.fppfPrecoverage_le_fpqcPrecoverageproof · cited by 2
- AlgebraicGeometry.GeometricallyReduced.isReduced_of_flat_of_isLocallyNoetherianstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.Hom.one_le_finrank_mapstatement and proof · cited by 1
- AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_leftstatement and proof · cited by 1
- AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_rightstatement and proof · cited by 1