Theorems · Theorem · commutative algebra
Algebra.HasGoingDown.iff_generalizingMap_primeSpectrumComap
- 1000+ list: Going-up and going-down theorems
∀ {R : Type u_3} {S : Type u_4} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S],
Algebra.HasGoingDown R S ↔ GeneralizingMap (PrimeSpectrum.comap (algebraMap R S))An R-algebra S has the going down property if and only if generalizations lift
along Spec S → Spec R.
- Defined in
- Mathlib.RingTheory.Ideal.GoingDown
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 83 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites18
Mathlib declarations this one mentions in its statement or cites explicitly in its proof. Plumbing is filtered out.
- CommRingstatement and proof · cited by 17,173
- Algebrastatement and proof · cited by 11,388
- Idealproof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- LT.lt.leproof · cited by 2,189
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumstatement and proof · cited by 625
- PrimeSpectrum.asIdealproof · cited by 333
- Ideal.LiesOverproof · cited by 272
- PrimeSpectrum.comapstatement and proof · cited by 199
- Specializesproof · cited by 176
- Ideal.underproof · cited by 170
Cited by4
Results whose statement or proof uses this declaration.
- PrimeSpectrum.isOpenMap_comap_of_hasGoingDown_of_finitePresentationproof · cited by 2
- RingHom.Flat.generalizingMap_comapproof · cited by 1
- AlgebraicGeometry.Flat.generalizingMapproof · cited by 0
- Algebra.HasGoingDown.transproof · cited by 0