valuation — Mathlib

English

Nonzero divisors have nonzero valuation: for x in nonZeroDivisors, v.intValuation x ≠ 0.

Русский

Ненулевые делители имеют ненулевую оценку: для x из nonZeroDivisors, v.intValuation x ≠ 0.

LaTeX

$$x \in \text{nonZeroDivisors} \Rightarrow v(x) \neq 0$$

Lean4

/-- The `v`-adic valuation of `x ∈ K` is the valuation of `r` divided by the valuation of `s`,
where `r` and `s` are chosen so that `x = r/s`. -/
def valuation (v : HeightOneSpectrum R) : Valuation K ℤᵐ⁰ :=
  v.intValuation.extendToLocalization (fun r hr => Set.mem_compl <| v.intValuation_ne_zero' ⟨r, hr⟩) K

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