isLocallyConstant_rankAtStalk — Mathlib

English

If M is a subsingleton, then rankAtStalk(M) = 0 for all base rings.

Русский

Если модуль одиничный, тогда rankAtStalk(M) равно нулю для всех оснований.

LaTeX

$$$[Subsingleton\ M] \Rightarrow \mathrm{rankAtStalk}(M) = 0$$$

Lean4

theorem isLocallyConstant_rankAtStalk [Module.FinitePresentation R M] [Module.Flat R M] :
    IsLocallyConstant (rankAtStalk (R := R) M) :=
  by
  let e : freeLocus R M ≃ₜ PrimeSpectrum R :=
    (Homeomorph.setCongr freeLocus_eq_univ).trans (Homeomorph.Set.univ (PrimeSpectrum R))
  convert isLocallyConstant_rankAtStalk_freeLocus.comp_continuous e.symm.continuous

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