Theorems · Theorem · algebraic geometry
Algebra.basicOpen_subset_smoothLocus_iff
∀ {R : Type u_1} {A : Type u_2} [inst : CommRing R] [inst_1 : CommRing A] [inst_2 : Algebra R A]
[Algebra.FinitePresentation R A] {f : A},
↑(PrimeSpectrum.basicOpen f) ⊆ Algebra.smoothLocus R A ↔ Algebra.FormallySmooth R (Localization.Away f)- Defined in
- Mathlib.RingTheory.Smooth.Locus
- Cited by
- 3 results in Mathlib
- Foundations
- Depth 137 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites40
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement and proof · cited by 53,352
- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- SetLike.coestatement and proof · cited by 8,199
- LinearEquivproof · cited by 3,317
- Compl.complproof · cited by 2,925
- TopologicalSpace.Opensstatement · cited by 2,040
- LinearEquiv.symmproof · cited by 1,461
- PrimeSpectrumstatement and proof · cited by 625
- Submonoid.powersstatement and proof · cited by 408
- compl_complproof · cited by 229
Cited by3
Results whose statement or proof uses this declaration.
- Algebra.smoothLocus_eq_univ_iffproof · cited by 3
- Algebra.IsSmoothAt.exists_notMem_smoothproof · cited by 2
- Algebra.basicOpen_subset_smoothLocus_iff_smoothproof · cited by 0