Theorems · Theorem · algebraic geometry
PrimeSpectrum.nontrivial_iff_mem_rangeComap
∀ {R : Type u} [inst : CommRing R] {S : Type u_1} [inst_1 : CommRing S] [inst_2 : Algebra R S] (p : PrimeSpectrum R),
Nontrivial (TensorProduct R p.asIdeal.ResidueField S) ↔ p ∈ Set.range (PrimeSpectrum.comap (algebraMap R S))A prime p is in the range of Spec S → Spec R if the fiber over p is nontrivial.
- Cited by
- 1 results in Mathlib
- Foundations
- Depth 102 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites36
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- DFunLike.coeproof · cited by 62,936
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- RingHomproof · cited by 10,189
- Idealproof · cited by 4,748
- Bot.botproof · cited by 4,720
- Algebra.algebraMapstatement and proof · cited by 4,706
- Set.rangestatement and proof · cited by 4,705
- AlgHomproof · cited by 3,236
- TensorProductstatement and proof · cited by 2,545
- Nontrivialstatement and proof · cited by 2,416
Cited by1
Results whose statement or proof uses this declaration.
- PrimeSpectrum.rankAtStalk_pos_iff_mem_range_comapproof · cited by 2