English
For a prime p, rankAtStalk M equals the finrank of the localized tensor product with M over R, taken at p.
Русский
Для точки p рангообразование M равно размерности локализованного тензорного произведения: ранг равен finrank локализованного тензорного произведения.
LaTeX
$$$\\operatorname{rankAtStalk}(M)(p) = \\operatorname{finrank}_{\\operatorname{Localization.AtPrime}(p)} (\\operatorname{Localization.AtPrime}(p) \\otimes_{R} M)$$$
Lean4
/-- Purely inseparable field extensions are universal homeomorphisms. -/
@[stacks 0BRA "Special case for purely inseparable field extensions"]
theorem isHomeomorph_comap_of_isPurelyInseparable [IsPurelyInseparable k K] :
IsHomeomorph (comap <| algebraMap R (R ⊗[k] K)) :=
by
let q := ringExpChar k
refine isHomeomorph_comap _ (IsPurelyInseparable.exists_pow_mem_range_tensorProduct) ?_
convert bot_le
rw [← RingHom.injective_iff_ker_eq_bot]
exact Algebra.TensorProduct.includeLeft_injective (S := R) (algebraMap k K).injective
