Theorems · Definition · algebraic geometry
AlgebraicGeometry.LocallyRingedSpace.evaluation
(X : AlgebraicGeometry.LocallyRingedSpace) →
{U : TopologicalSpace.Opens ↑X.toTopCat} → (x : ↥U) → X.presheaf.obj (Opposite.op U) ⟶ X.residueField ↑xIf U is an open of X containing x, we have a canonical ring map from the sections
over U to the residue field of x.
If we interpret sections over U as functions of X defined on U, then this ring map
corresponds to evaluation at x.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 96 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- CategoryTheory.CategoryStruct.compproof · cited by 17,999
- Oppositestatement · cited by 8,081
- TopCat.carrierstatement and proof · cited by 3,184
- CommRingCatstatement · cited by 2,333
- TopologicalSpace.Opensstatement and proof · cited by 2,040
- AlgebraicGeometry.PresheafedSpace.carrierstatement · cited by 2,020
- AlgebraicGeometry.SheafedSpace.toPresheafedSpacestatement and proof · cited by 1,988
- AlgebraicGeometry.LocallyRingedSpace.toSheafedSpacestatement and proof · cited by 1,892
- AlgebraicGeometry.PresheafedSpace.presheafstatement and proof · cited by 1,104
- TopCat.Presheaf.germproof · cited by 208
Cited by7
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.LocallyRingedSpace.Γevaluationproof · cited by 5
- AlgebraicGeometry.LocallyRingedSpace.evaluation_naturalitystatement · cited by 4
- AlgebraicGeometry.LocallyRingedSpace.evaluation_eq_zero_iff_notMem_basicOpenstatement and proof · cited by 2
- AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_applystatement · cited by 2
- AlgebraicGeometry.LocallyRingedSpace.evaluation_ne_zero_iff_mem_basicOpenstatement · cited by 1
- AlgebraicGeometry.LocallyRingedSpace.basicOpen_eq_bot_iff_forall_evaluation_eq_zerostatement · cited by 1
- AlgebraicGeometry.LocallyRingedSpace.evaluation_naturality_assocstatement and proof · cited by 0