Theorems · Theorem · algebraic geometry

IsArtinianRing.finrank_eq_sum_primeSpectrum

∀ (R : Type u_1) [inst : CommRing R] [IsArtinianRing R] (F : Type u_2) [inst_2 : Field F] [inst_3 : Algebra F R]
  [Module.Finite F R] [inst_5 : Fintype (PrimeSpectrum R)],
  Module.finrank F R = ∑ p, Module.finrank F (Localization.AtPrime p.asIdeal)

Defined in: Mathlib.RingTheory.Spectrum.Prime.Noetherian
Cited by: 1 results in Mathlib
Foundations: Depth 113 from the axioms · uses propext, Classical.choice, Quot.sound
Assumes: CommRing IsArtinianRing Field Algebra Module.Finite Fintype

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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
Fintypestatement and proof · cited by 7,736
Fieldstatement and proof · cited by 7,404
Finset.sumstatement · cited by 5,195
Idealproof · cited by 4,748
Finset.univstatement · cited by 3,473
Module.finrankstatement · cited by 1,770
Module.Finitestatement and proof · cited by 1,032
Ideal.IsPrimeproof · cited by 827
PrimeSpectrumstatement and proof · cited by 625
Ideal.primeComplstatement and proof · cited by 462

Cited by1

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Ideal.sum_ramification_inertia_eq_finrank_fiberproof · cited by 1