discr_eq_discr_of_algEquiv — Mathlib

English

For an irreducible ϖ in the valuation ring, the nth power of the maximal ideal corresponds to {y | v(y) ≤ v(ϖ)^n} in the subtype.

Русский

Для неприводимого ϖ в локальном кольце, n-я степень максимального идеала соответствует {y | v(y) ≤ v(ϖ)^n}.

LaTeX

$$Irreducible ϖ → ∀ n, (maximalIdeal v.integer)^n = { y | v(y) ≤ v(ϖ)^n }$$

Lean4

/-- Mapping a family of vectors along an `AlgEquiv` preserves the discriminant. -/
theorem discr_eq_discr_of_algEquiv [Fintype ι] (b : ι → B) (f : B ≃ₐ[A] C) :
    Algebra.discr A b = Algebra.discr A (f ∘ b) := by
  rw [discr_def]; congr; ext
  simp_rw [traceMatrix_apply, traceForm_apply, Function.comp, ← map_mul f, trace_eq_of_algEquiv]

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