Theorems · Definition · algebraic geometry
PrimeSpectrum.preimageOrderIsoFiber
(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}) ≃o PrimeSpectrum (p.asIdeal.Fiber S)The OrderIso 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
- 3 results in Mathlib
- Foundations
- Depth 107 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites16
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
- Equivproof · 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
- OrderIsostatement · cited by 874
- PrimeSpectrumstatement and proof · cited by 625
- Ideal.primeComplstatement · cited by 462
- PrimeSpectrum.asIdealstatement and proof · cited by 333
- Localization.AtPrimestatement · cited by 299
Cited by5
Results whose statement or proof uses this declaration.
- PrimeSpectrum.primesOverOrderIsoFiberproof · cited by 8
- PrimeSpectrum.preimageHomeomorphFiberproof · cited by 6
- Polynomial.ringKrullDim_leproof · cited by 0
- PrimeSpectrum.coe_preimageOrderIsoFiber_apply_asIdealstatement and proof · cited by 0
- PrimeSpectrum.coe_preimageOrderIsoFiber_symm_apply_coe_asIdealstatement and proof · cited by 0