Theorems · Definition · algebraic geometry
PrimeSpectrum.preimageHomeomorphFiber
(R : Type u_3) →
(S : Type u_4) →
[inst : CommRing R] →
[inst_1 : CommRing S] →
[inst_2 : Algebra R S] →
(p : PrimeSpectrum R) → ↑(PrimeSpectrum.comap (algebraMap R S) ⁻¹' {p}) ≃ₜ PrimeSpectrum (p.asIdeal.Fiber S)The Homeomorph between the fiber of PrimeSpectrum S → PrimeSpectrum R
at a prime ideal p : PrimeSpectrum R and the prime spectrum of κ(p) ⊗[R] S.
- Cited by
- 6 results in Mathlib
- Foundations
- Depth 110 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites17
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
- Set.Elemstatement and proof · cited by 7,166
- Set.preimagestatement and proof · cited by 4,946
- Algebra.algebraMapstatement and proof · cited by 4,706
- OrderIsoproof · cited by 874
- Homeomorphstatement · cited by 725
- PrimeSpectrumstatement and proof · cited by 625
- Ideal.primeComplstatement · cited by 462
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- Localization.AtPrimestatement · cited by 299
Cited by6
Results whose statement or proof uses this declaration.
- Algebra.QuasiFinite.isDiscrete_comap_preimage_singletonproof · cited by 2
- Algebra.quasiFiniteAt_iff_isOpen_singleton_fiberproof · cited by 1
- Algebra.QuasiFiniteAt.of_isOpen_singleton_fiberproof · cited by 1
- Ideal.exists_not_mem_forall_mem_of_ne_of_liesOverproof · cited by 1
- PrimeSpectrum.coe_preimageHomeomorphFiber_apply_asIdealstatement and proof · cited by 0
- PrimeSpectrum.coe_preimageHomeomorphFiber_symm_apply_coe_asIdealstatement and proof · cited by 0