Theorems · Theorem · algebraic geometry

AlgebraicGeometry.Scheme.Hom.toNormalization_app_preimage

∀ {X Y : AlgebraicGeometry.Scheme} (f : X ⟶ Y) [inst : AlgebraicGeometry.QuasiCompact f]
  [inst_1 : AlgebraicGeometry.QuasiSeparated f] (U : ↑Y.affineOpens),
  let this := (CommRingCat.Hom.hom (AlgebraicGeometry.Scheme.Hom.app f ↑U)).toAlgebra;
  AlgebraicGeometry.Scheme.Hom.app (AlgebraicGeometry.Scheme.Hom.toNormalization f)
      ((TopologicalSpace.Opens.map (AlgebraicGeometry.Scheme.Hom.fromNormalization f).base).obj ↑U) =
    CategoryTheory.CategoryStruct.comp (AlgebraicGeometry.Scheme.Hom.normalizationObjIso f ⋯).hom
      (CategoryTheory.CategoryStruct.comp
        (CommRingCat.ofHom
          ↑(integralClosure ↑(Y.presheaf.obj (Opposite.op ↑U))
                ↑(X.presheaf.obj (Opposite.op ((TopologicalSpace.Opens.map f.base).obj ↑U)))).val)
        (X.presheaf.map (CategoryTheory.eqToHom ⋯).op))

Defined in

Mathlib.AlgebraicGeometry.Normalization

Cited by

2 results in Mathlib

Foundations

Depth 208 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

AlgebraicGeometry.QuasiCompactAlgebraicGeometry.QuasiSeparated