Theorems · Theorem · algebraic geometry

Algebra.QuasiFiniteAt.of_quasiFiniteAt_residueField

∀ {R : Type u_1} {S : Type u_2} [inst : CommRing R] [inst_1 : CommRing S] [inst_2 : Algebra R S]
  [Algebra.FiniteType R S] (p : Ideal R) (q : Ideal S) [inst_4 : q.IsPrime] [inst_5 : p.IsPrime] [q.LiesOver p]
  (Q : Ideal (p.Fiber S)) [inst_7 : Q.IsPrime],
  Ideal.comap Algebra.TensorProduct.includeRight.toRingHom Q = q →
    ∀ [Algebra.QuasiFiniteAt p.ResidueField Q], Algebra.QuasiFiniteAt R q

Defined in: Mathlib.RingTheory.ZariskisMainTheorem
Cited by: 1 results in Mathlib
Foundations: Depth 167 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing CommRing Algebra Algebra.FiniteType Ideal.IsPrime Ideal.IsPrime Ideal.LiesOver Ideal.IsPrime Algebra.QuasiFiniteAt

Mathlib docs Source Loogle LeanSearch

Around this declaration

Dashed lines are statement dependencies; solid lines are citations in proofs.

Cites20

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
RingHomstatement · cited by 10,189
Idealstatement and proof · cited by 4,748
TensorProductstatement · cited by 2,545
Ideal.IsPrimestatement and proof · cited by 827
AlgHom.toRingHomstatement and proof · cited by 490
Ideal.primeComplstatement · cited by 462
Ideal.comapstatement and proof · cited by 443
Localization.AtPrimestatement · cited by 299
Ideal.LiesOverstatement and proof · cited by 272
Algebra.TensorProduct.includeRightstatement and proof · cited by 165

Cited by1

Results whose statement or proof uses this declaration.

Algebra.QuasiFiniteAt.of_isOpen_singleton_fiberproof · cited by 1