Theorems · Theorem · commutative algebra
RingHom.smooth_iff_locally_isStandardSmooth
∀ {R S : Type u} [inst : CommRing R] [inst_1 : CommRing S] {f : R →+* S},
f.Smooth ↔ RingHom.Locally (fun {R S} [CommRing R] [CommRing S] => RingHom.IsStandardSmooth) fA ring homomorphism is smooth if and only if it is locally standard smooth.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 158 from the axioms · uses propext, Classical.choice, Quot.sound
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
- RingHomstatement and proof · cited by 10,189
- RingHom.Smoothstatement and proof · cited by 30
- RingHom.Locallystatement and proof · cited by 28
- RingHom.IsStandardSmoothstatement and proof · cited by 16
- RingHom.PropertyIsLocal.respectsIsoproof · cited by 16
- RingHom.Smooth.propertyIsLocalproof · cited by 5
- RingHom.locally_iff_of_localizationSpanTargetproof · cited by 3
- RingHom.Smooth.ofLocalizationSpanTargetproof · cited by 2
- RingHom.locally_of_locallyproof · cited by 1
- RingHom.IsStandardSmooth.smoothproof · cited by 1
- RingHom.Smooth.locally_isStandardSmoothproof · cited by 1
Cited by1
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.Smooth.iff_forall_exists_isStandardSmoothproof · cited by 2