Theorems · Definition · algebraic geometry
PrimeSpectrum.preimageEquivFiber
(R : Type u_1) →
(S : Type u_2) →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
(p : PrimeSpectrum R) → ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {p}) ≃ PrimeSpectrum (p.asIdeal.Fiber S)The fiber PrimeSpectrum S → PrimeSpectrum R at a prime ideal
p : PrimeSpectrum R is in bijection with the prime spectrum of κ(p) ⊗[R] S.
- Cited by
- 8 results in Mathlib
- Foundations
- Depth 104 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites21
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- Setstatement · cited by 53,352
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Equivstatement · cited by 8,337
- Set.Elemstatement and proof · cited by 7,166
- Set.preimagestatement and proof · cited by 4,946
- Algebra.algebraMapstatement and proof · cited by 4,706
- PrimeSpectrumstatement and proof · cited by 625
- AlgHom.toRingHomproof · cited by 490
- Ideal.primeComplstatement · cited by 462
- RingHom.kerproof · cited by 363
- PrimeSpectrum.asIdealstatement and proof · cited by 333
Cited by9
Results whose statement or proof uses this declaration.
- Algebra.QuasiFinite.iff_finite_comap_preimage_singletonproof · cited by 4
- PrimeSpectrum.preimageOrderIsoFiberproof · cited by 3
- Algebra.QuasiFinite.finite_comap_preimage_singletonproof · cited by 3
- PrimeSpectrum.preimageEquivFiber_apply_asIdealstatement and proof · cited by 2
- Polynomial.not_weaklyQuasiFiniteAtproof · cited by 2
- Polynomial.not_ker_le_map_C_of_surjective_of_weaklyQuasiFiniteAtproof · cited by 1
- Localization.exists_finite_awayMapₐ_of_surjective_awayMapₐproof · cited by 1
- Ideal.exists_not_mem_forall_mem_of_ne_of_liesOverproof · cited by 1
- PrimeSpectrum.preimageEquivFiber_symm_apply_coestatement and proof · cited by 0