Theorems · Definition · algebraic geometry

AlgebraicGeometry.AffineSpace.isoOfIsAffine

(n : Type u) →
  (S : AlgebraicGeometry.Scheme) →
    [AlgebraicGeometry.IsAffine S] →
      AlgebraicGeometry.AffineSpace n S ≅
        AlgebraicGeometry.Spec (CommRingCat.of (MvPolynomial n ↑(S.presheaf.obj (Opposite.op ⊤))))

The affine space over an affine base is isomorphic to the spectrum of the polynomial ring. Also see AffineSpace.SpecIso.

Defined in: Mathlib.AlgebraicGeometry.AffineSpace
Cited by: 10 results in Mathlib
Foundations: Depth 175 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: AlgebraicGeometry.IsAffine

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Cites37

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

DFunLike.coeproof · cited by 62,936
CategoryTheory.Functor.objstatement and proof · cited by 19,642
CategoryTheory.CategoryStruct.compproof · cited by 17,999
Top.topstatement and proof · cited by 9,680
Oppositestatement · cited by 8,081
CategoryTheory.Iso.invproof · cited by 6,514
Finsuppstatement · cited by 5,255
CategoryTheory.ConcreteCategory.homproof · cited by 4,022
CategoryTheory.Isostatement · cited by 3,963
TopCat.carrierstatement · cited by 3,184
AlgebraicGeometry.Schemestatement and proof · cited by 2,540
CommRingCatstatement · cited by 2,333

Cited by11

Results whose statement or proof uses this declaration.

AlgebraicGeometry.AffineSpace.SpecIsoproof · cited by 7
AlgebraicGeometry.AffineSpace.SpecIso_inv_overproof · cited by 4
AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_overstatement · cited by 2
AlgebraicGeometry.AffineSpace.SpecIso_inv_appTop_coordproof · cited by 1
AlgebraicGeometry.AffineSpace.isoOfIsAffine_homstatement and proof · cited by 1
AlgebraicGeometry.AffineSpace.isoOfIsAffine_hom_appTopstatement · cited by 1
AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_appTop_coordstatement · cited by 1
AlgebraicGeometry.AffineSpace.SpecIso_hom_appTopproof · cited by 0
AlgebraicGeometry.AffineSpace.isoOfIsAffine_invstatement and proof · cited by 0
AlgebraicGeometry.AffineSpace.isoOfIsAffine_inv_over_assocstatement and proof · cited by 0
AlgebraicGeometry.AffineSpace.isoOfIsAffine.congr_simpstatement and proof · cited by 0