Theorems · Definition · algebraic geometry
AlgebraicGeometry.Scheme.functionField
(X : AlgebraicGeometry.Scheme) → [IrreducibleSpace ↥X] → CommRingCat
The function field of an irreducible scheme is the local ring at its generic point. Despite the name, this is a field only when the scheme is integral.
- Defined in
- Mathlib.AlgebraicGeometry.FunctionField
- Cited by
- 23 results in Mathlib
- Foundations
- Depth 141 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- IrreducibleSpace
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites11
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- TopCat.carrierstatement and proof · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- CommRingCatstatement · cited by 2,333
- AlgebraicGeometry.PresheafedSpace.carrierstatement and proof · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
- AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement and proof · cited by 1,892
- AlgebraicGeometry.Scheme.toLocallyRingedSpacestatement and proof · cited by 1,734
- AlgebraicGeometry.PresheafedSpace.presheafproof · cited by 1,104
- TopCat.Presheaf.stalkproof · cited by 407
- IrreducibleSpacestatement and proof · cited by 40
- genericPointproof · cited by 15
Cited by31
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Scheme.ordstatement and proof · cited by 12
- AlgebraicGeometry.Scheme.ordHomstatement · cited by 10
- AlgebraicGeometry.Scheme.germToFunctionFieldstatement · cited by 7
- AlgebraicGeometry.Scheme.ord_eq_zero_of_coheight_neq_onestatement and proof · cited by 5
- AlgebraicGeometry.Scheme.RationalMap.fromFunctionFieldstatement · cited by 3
- AlgebraicGeometry.Scheme.ord_eq_unzero_ordHomstatement and proof · cited by 3
- AlgebraicGeometry.Scheme.ord.congr_simpstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.ord_eq_iffstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.ord_eq_ordHom_of_coheight_eq_onestatement and proof · cited by 2
- AlgebraicGeometry.Scheme.ord_le_ord_iffstatement and proof · cited by 2
- AlgebraicGeometry.Scheme.ord_zerostatement and proof · cited by 2
- AlgebraicGeometry.Scheme.PartialMap.fromFunctionFieldstatement · cited by 2