Tags: #todo #todo/learning/definitions

valuation

ultrametric triangle inequality

• Integral subring: \(\left\{{x\in X {~\mathrel{\Big\vert}~}|x| \leq 1}\right\}\) where \({\left\lvert {{-}} \right\rvert}\) is the valuation.

• Definition of a valuation: a group morphism \({\mathbb{F}}^{\times}\to {\mathbb{R}}\) such that \begin{align*}v(x+y) \geq \min (v(x), v(y))\end{align*}

• Extend to \({\mathbb{F}}\to {\mathbb{R}}\cup\left\{{\infty}\right\}\) by defining \({\left\lvert {x} \right\rvert}_v := e^{v(x)}\) yields an nonarchimedean absolute value

• The value group is \(v({\mathbb{F}})\), and \(v\) is a discrete valuation if \(v({\mathbb{F}}) \cong {\mathbb{Z}}\leq {\mathbb{R}}\).

• The valuation ring is \(A := \left\{{x\in {\mathbb{F}}{~\mathrel{\Big\vert}~}v(x) \geq 0}\right\}\in \mathsf{CRing}\), and its unit group is \(A^{\times}:= \left\{{x\in {\mathbb{F}}{~\mathrel{\Big\vert}~}v(x) = 0}\right\}\).

  • Note that \(x\in A\) is invertible in \(A\) iff \(v(x) = 0\).

• Any element \(\pi\) for which \(v(\pi) = 1\) is a uniformizer.