Theorems · Theorem · commutative algebra
Localization.exists_awayMap_injective_of_localRingHom_injective
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] {p : Ideal R}
[inst_3 : p.IsPrime] {q : Ideal S} [inst_4 : q.IsPrime],
(RingHom.ker (algebraMap R S)).FG →
∀ [inst_5 : q.LiesOver p],
Function.Injective ⇑(Localization.localRingHom p q (algebraMap R S) ⋯) →
∃ r ∉ p, ∀ (r' : R), r ∣ r' → Function.Injective ⇑(Localization.awayMap (algebraMap R S) r')- Defined in
- Mathlib.RingTheory.Unramified.LocalRing
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 86 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites42
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coestatement and proof · cited by 62,936
- CommRingstatement and proof · cited by 17,173
- Finsetproof · cited by 13,712
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- SetLike.coeproof · cited by 8,199
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Finset.univproof · cited by 3,473
- Compl.complproof · cited by 2,925
- Finset.prodproof · cited by 2,356
- MulZeroClass.mul_zeroproof · cited by 2,091
Cited by1
Results whose statement or proof uses this declaration.
- Localization.exists_awayMap_bijective_of_localRingHom_bijectiveproof · cited by 1