Theorems · Theorem · commutative algebra
Algebra.Smooth.exists_span_eq_top_isStandardSmooth
∀ (R : Type u_1) (S : Type u_2) [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Algebra.Smooth R S], ∃ s, Ideal.span s = ⊤ ∧ ∀ x ∈ s, Algebra.IsStandardSmooth R (Localization.Away x)
If S is R-smooth, there exists a cover by basic opens D(sᵢ) such that
S[1/sᵢ] is R-standard-smooth.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 156 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites27
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Top.topstatement and proof · cited by 9,680
- Idealstatement and proof · cited by 4,748
- Set.rangeproof · cited by 4,705
- Compl.complproof · cited by 2,925
- Set.iUnionproof · cited by 2,483
- Ideal.spanstatement · cited by 948
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumproof · cited by 625
- Ideal.primeComplproof · cited by 462
Cited by1
Results whose statement or proof uses this declaration.
- RingHom.Smooth.locally_isStandardSmoothproof · cited by 1