Theorems · Theorem · commutative algebra
Localization.localRingHom_injective_of_primesOver_eq_singleton
∀ {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] (hq : p.primesOver S = {q}) [Algebra.IsIntegral R S]
[FaithfulSMul R S], Function.Injective ⇑(Localization.localRingHom p q (algebraMap R S) ⋯)- Defined in
- Mathlib.RingTheory.Unramified.LocalRing
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 119 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites26
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
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Idealstatement and proof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Ideal.IsPrimestatement and proof · cited by 827
- Ideal.primeComplstatement and proof · cited by 462
- Eq.gestatement · cited by 375
- FaithfulSMulstatement and proof · cited by 340
- Localization.AtPrimestatement and proof · cited by 299
Cited by1
Results whose statement or proof uses this declaration.
- Localization.exists_awayMap_bijective_of_residueField_surjectiveproof · cited by 0