Theorems · Theorem · commutative algebra
Algebra.IsSmoothAt.exists_notMem_isStandardSmooth
∀ (R : Type u_1) {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.FinitePresentation R S] (p : Ideal S) [inst_4 : p.IsPrime] [Algebra.IsSmoothAt R p],
∃ f ∉ p, Algebra.IsStandardSmooth R (Localization.Away f)If S is R-smooth at a prime p, then S is R-standard-smooth in a neighbourhood of p:
there exists a basic open p ∈ D(f) of Spec S such that S[1/f] is standard smooth.
- Cited by
- 2 results in Mathlib
- Foundations
- Depth 155 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites73
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
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- RingHom.idproof · cited by 18,349
- CommRingstatement and proof · cited by 17,173
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- LinearMapproof · cited by 10,215
- Finsuppproof · cited by 5,255
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapproof · cited by 4,706
- Set.rangeproof · cited by 4,705
- Finiteproof · cited by 3,029
- AlgEquivproof · cited by 1,681
- Module.Basisproof · cited by 1,477
Cited by2
Results whose statement or proof uses this declaration.
- Algebra.Smooth.exists_span_eq_top_isStandardSmoothproof · cited by 1
- Algebra.IsSmoothAt.exists_isStandardEtale_mvPolynomialproof · cited by 1