Theorems · Theorem · commutative algebra
Algebra.QuasiFinite.iff_finite_comap_preimage_singleton
∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
[Algebra.FiniteType R S],
Algebra.QuasiFinite R S ↔ ∀ (x : PrimeSpectrum R), (PrimeSpectrum.comap (algebraMap R S) ⁻¹' {x}).Finite- Defined in
- Mathlib.RingTheory.QuasiFinite.Basic
- Cited by
- 4 results in Mathlib
- Foundations
- Depth 151 from the axioms · uses propext, Classical.choice, Quot.sound
Around this declaration
Dashed lines are statement dependencies; solid lines are citations in proofs.
Cites24
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.preimagestatement and proof · cited by 4,946
- Idealproof · cited by 4,748
- Algebra.algebraMapstatement and proof · cited by 4,706
- Finiteproof · cited by 3,029
- Set.Finitestatement and proof · cited by 1,814
- Ideal.IsPrimeproof · cited by 827
- PrimeSpectrumstatement and proof · cited by 625
- PrimeSpectrum.comapstatement and proof · cited by 199
- Ideal.ResidueFieldproof · cited by 119
Cited by4
Results whose statement or proof uses this declaration.
- AlgebraicGeometry.LocallyQuasiFinite.of_finite_preimage_singletonproof · cited by 2
- AlgebraicGeometry.locallyQuasiFinite_iff_isDiscrete_preimage_singletonproof · cited by 1
- Algebra.QuasiFiniteAt.of_isOpen_singletonproof · cited by 1
- Algebra.QuasiFinite.iff_finite_primesOverproof · cited by 0