Theorems · Theorem · algebraic geometry

Module.rankAtStalk_eq

∀ {R : Type uR} {M : Type uM} [inst : CommRing R] [inst_1 : AddCommGroup M] [inst_2 : Module R M] [Module.Flat R M]
  [Module.Finite R M] (p : PrimeSpectrum R),
  Module.rankAtStalk M p = Module.finrank p.asIdeal.ResidueField (p.asIdeal.Fiber M)

The rank of a module M at a prime p is equal to the dimension of κ(p) ⊗[R] M as a κ(p)-module.

Defined in

Mathlib.RingTheory.Spectrum.Prime.FreeLocus

Cited by

2 results in Mathlib

Foundations

Depth 120 from the axioms · uses propext, Classical.choice, Quot.sound

Assumes

CommRingAddCommGroupModuleModule.FlatModule.Finite