English
Quotient maps commute with comaps along ramification indices under a prime/prime setting, preserving the inertia degree.
Русский
Квотиентные отображения commuting с comap через ramificationIdx, сохраняют инерийную степень.
LaTeX
$$inertiaDeg_comap_eq p P e$$
Lean4
/-- Quotienting `P^i / P^e` by its subspace `P^(i+1) ⧸ P^e` is
`R ⧸ p`-linearly isomorphic to `S ⧸ P`. -/
noncomputable def quotientRangePowQuotSuccInclusionEquiv [IsDedekindDomain S] [P.IsPrime] (hP : P ≠ ⊥) {i : ℕ}
(hi : i < e) :
((P ^ i).map (Ideal.Quotient.mk (P ^ e)) ⧸ LinearMap.range (powQuotSuccInclusion p P i)) ≃ₗ[R ⧸ p] S ⧸ P :=
by
choose a a_mem a_notMem using
SetLike.exists_of_lt (Ideal.pow_right_strictAnti P hP (Ideal.IsPrime.ne_top inferInstance) (le_refl i.succ))
refine (LinearEquiv.ofBijective ?_ ⟨?_, ?_⟩).symm
· exact quotientToQuotientRangePowQuotSucc p P a_mem
· exact quotientToQuotientRangePowQuotSucc_injective p P hi a_mem a_notMem
· exact quotientToQuotientRangePowQuotSucc_surjective p P hP hi a_mem a_notMem
