Tags: #todo #NT/algebraic Refs: DVR

Valuation

Motivation:

For divisors in schemes

See divisor. Order of vanishing:

Valuations (Definitions)

• Definition of a valuation:

  • Start with any group morphism \begin{align*}v^{\times}: {\mathbf{G}}_a(K^{\times})\to {\mathbf{G}}_a({\mathbf{R}}),\end{align*} so \(v(x + y) = v(x) + v(y)\), satisfying an ultrametric-type inequality \begin{align*} v(x+y) \geq \min (v(x), v(y)) \end{align*}

• Extend to \begin{align*} v: {\mathbf{G}}_a(K)\to {\mathbf{G}}_a({\mathbf{R}}) \cup\left\{{\infty}\right\} \qquad v(x) \coloneqq\begin{cases}v^{\times}(x) & x\neq 0 \\ \infty & x=0 \end{cases} \end{align*} so that \(v(x) = \infty \iff x=0\).

• Can then define an associated absolute value \begin{align*}{\left\lvert {x} \right\rvert}_v := \exp(v(x))\end{align*} where \(e\) can be replaced with \(c\) another constant.

• See also valuation ring, unit groups, and the integral subring at ring of integers of a nonarchimedean field.

• The \(p{\hbox{-}}\)adic valuation \(v_p: {\mathbf{Q}}\to {\mathbf{Z}}\) is defined using unique factorization in \({\mathbf{Q}}^{\times}\): \begin{align*} v_p(x) = v_p\left( \pm 1 \prod_{p_i\in \operatorname{Spec}{\mathbf{Z}}} p_i^{e_i} \right) \coloneqq e_i, \quad v_p(0) \coloneqq\infty \end{align*} with an associated nonarchimedean absolute value \begin{align*} {\left\lvert {x} \right\rvert}_p \coloneqq C^{-v_p(x)}, \quad c \in (0, 1),\quad {\left\lvert {0} \right\rvert}_p \coloneqq p^{-\infty}\coloneqq 0 \end{align*}

• A place is an equivalence class of valuations.

• The value group of \(v\) is the image \(v(K) \leq {\mathbf{R}}\).

• \(v\) is a discrete valuation if \(v(K) \cong {\mathbf{Z}}\leq {\mathbf{R}}\). See also DVR.

• If \(x\in K^{\times}\), use that \(v(x^{-1}) = v(1) - v(x) = -v(x)\), so either \(v(x)\geq 0\) or \(v(x^{-1})\geq 0\). This forces the unit group of \(A \coloneqq\left\{{v(x)\leq 1}\right\}\) to be \(A^{\times}= \left\{{v(x) = 0}\right\}\). Use this to partition \(A\):

  • Positive valuation: non-units in \(A\)
  • Zero valuation: units in \(A\)
  • Negative valuation: \(x\not \in A\) but \(x^{-1}\in A\)

Places

• For function fields, maximal ideals \(p\) of some valuations rings \({\mathcal{O}}\). If \(p = \left\langle{t}\right\rangle = t{\mathcal{O}}\), then \(t\) is a uniformizer.

• Equivalence classes of valuations.

Infinite places

Infinite places are those not arising from a prime in \({\mathcal{O}}_K\).

Embeddings

• For \(L/K\), real places of \(K\) are embeddings \(K\hookrightarrow{\mathbf{C}}\) with image in \({\mathbf{R}}\), complex places are embeddings which intersect \({\mathbf{C}}\setminus{\mathbf{R}}\).
• A real place \(v\) of \(K\) is unramified in this situation if all of the extensions of \(v\) to \(L\) are again totally real, and is ramified otherwise. Complex places are always unramified.
• Example: in the extension \(\mathbb{Q}\left(\zeta_{p}\right) / \mathbb{Q}\left(\zeta_{p}+\zeta_{p}^{-1}\right)\), where \(\zeta_{p}\) is a primitive \(p\)-th root of unity for some odd prime \(p\), the base field is totally real (all its Archimedean places are real) while the top field is totally imaginary, so all real places ramify in the extension. # Uniformizers
• Any element \(\pi\) for which \(v(\pi) = 1\) is a uniformizer, so \(\pi \in v_p({{\partial}}{\mathbb{D}})\).

Norms

See Adeles.

Degrees and rational places

Results

• Ostrowski’s Theorem:
• Valuation rings are integrally closed:

See adic completion