Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Hom.flat_appLE

∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [self : AlgebraicGeometry.Flat f] {U : Y.Opens},
  AlgebraicGeometry.IsAffineOpen U →
    ∀ {V : X.Opens},
      AlgebraicGeometry.IsAffineOpen V →
        ∀ (e : V ≤ (TopologicalSpace.Opens.map f.base).obj U),
          (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.appLE f U V e)).Flat

Alias of AlgebraicGeometry.Flat.flat_appLE.

Defined in: Mathlib.AlgebraicGeometry.Morphisms.Flat
Cited by: 3 results in Mathlib
Foundations: Depth 139 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.Flat

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Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites24

Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.

Quiver.Homstatement · cited by 32,603
CategoryTheory.Functor.objstatement · cited by 19,642
Oppositestatement · cited by 8,081
TopCat.carrierstatement · cited by 3,184
AlgebraicGeometry.Schemestatement · cited by 2,540
CommRingCatstatement · cited by 2,333
TopologicalSpace.Opensstatement · cited by 2,040
AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement · cited by 1,988
AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement · cited by 1,892
AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement · cited by 1,734
AlgebraicGeometry.Scheme.Opensstatement · cited by 1,149

Cited by3

Results whose statement or proof uses this declaration.

AlgebraicGeometry.mono_pushoutSection_of_isCompact_of_flat_rightproof · cited by 2
AlgebraicGeometry.isIso_pushoutSection_of_isQuasiSeparated_of_flat_rightproof · cited by 1
AlgebraicGeometry.Flat.flat_of_affine_subsetproof · cited by 0