Theorems · Theorem · algebraic geometry
AlgebraicGeometry.isField_stalk_of_closure_mem_irreducibleComponents
∀ (X : AlgebraicGeometry.Scheme) (x : ↥X),
closure {x} ∈ irreducibleComponents ↥X → ∀ [AlgebraicGeometry.IsReduced X], IsField ↑(X.presheaf.stalk x)If X is reduced and has finitely many irreducible components, then the stalks at the generic
points of the irreducible components are fields.
- Defined in
- Mathlib.AlgebraicGeometry.Properties
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 140 from the axioms · uses propext, Classical.choice, Quot.sound
- Assumes
- AlgebraicGeometry.IsReduced
Around this declaration
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Cites50
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- CategoryTheory.ConcreteCategory.homproof · cited by 4,022
- TopCat.carrierstatement and proof · cited by 3,184
- AlgebraicGeometry.Schemestatement and proof · cited by 2,540
- CommRingCatstatement and proof · 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
Cited by1
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