Theorems · Definition · algebraic geometry
AlgebraicGeometry.StructureSheaf.stalkIso
(R : Type u) →
[inst : CommRing R] →
(x : PrimeSpectrum R) →
Localization.AtPrime x.asIdeal ≃ₐ[R] ↑((AlgebraicGeometry.structurePresheafInCommRingCat R).stalk x)The stalk of Spec R at x is isomorphic to Rₚ,
where p is the prime corresponding to x.
- Defined in
- Mathlib.AlgebraicGeometry.StructureSheaf
- Cited by
- 5 results in Mathlib
- Foundations
- Depth 117 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- CommRing
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites12
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- CommRingCatstatement · cited by 2,333
- AlgEquivstatement · cited by 1,681
- CommRingCat.carrierstatement and proof · cited by 1,096
- PrimeSpectrumstatement and proof · cited by 625
- Ideal.primeComplstatement and proof · cited by 462
- TopCat.Presheaf.stalkstatement and proof · cited by 407
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- Localization.AtPrimestatement and proof · cited by 299
- AlgebraicGeometry.PrimeSpectrum.Topstatement · cited by 104
- IsLocalization.algEquivproof · cited by 45
- AlgebraicGeometry.structurePresheafInCommRingCatstatement and proof · cited by 23
Cited by6
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Spec.stalkIsoproof · cited by 8
- AlgebraicGeometry.basicOpen_eq_of_affineproof · cited by 5
- AlgebraicGeometry.localRingHom_comp_stalkIsostatement and proof · cited by 3
- AlgebraicGeometry.stalkClosedPointIso_invproof · cited by 3
- AlgebraicGeometry.localRingHom_comp_stalkIso_applystatement and proof · cited by 0
- AlgebraicGeometry.isIso_SpecMap_stakMap_localizationproof · cited by 0