Theorems · Theorem · commutative algebra
PrimeSpectrum.comap_surjective_iff_injective_of_finite
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S] [Module.Flat R S]
[Module.Finite R S], Function.Surjective (PrimeSpectrum.comap (algebraMap R S)) ↔ Function.Injective ⇑(algebraMap R S)- Defined in
- Mathlib.RingTheory.Flat.Rank
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 132 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites29
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
- Algebrastatement and proof · cited by 11,388
- RingHomstatement · cited by 10,189
- Idealproof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- AlgHomproof · cited by 3,236
- TensorProductproof · cited by 2,545
- Nontrivialproof · cited by 2,416
- Module.Finitestatement and proof · cited by 1,032
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumstatement and proof · cited by 625
Cited by1
Results whose statement or proof uses this declaration.
- Module.rankAtStalk_pos_iff_algebraMap_injectiveproof · cited by 1