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Section 7.5 Rigid analytic spaces (Aash)

References.
  1. Several approaches to non-archimidean geomeetry - Conrad
  2. Lectures on formal and rigid geometry - Bosch
  3. Non-Archimidean geometry - Matt Baker
  4. Rigid geometry and applications - Fresnel, van der Put

Usual geomeetry involves polynomial rings over fields, we switsch to tate algebras. Fix a non-archimidean field \(k\text{,}\) \(R\) a valuation ring and \(\tilde k\) its residue field.

Tate algebras over \(k\text{.}\)

\begin{equation*} T_n = T_n(k) = \left\{ \sum a_J x^J : |a_J| \to 0 \text{ as } |J| \to \infty \right\} \end{equation*}
\begin{equation*} J = \{j_1, \ldots, j_n\} \end{equation*}
\begin{equation*} x^J = \prod x_i ^{j_i} \end{equation*}

\(f\) converges on

\begin{equation*} \mathbf B^n(\overline k) \iff f \in k \langle x_1, \ldots, x_n \rangle = T_n(k) \end{equation*}

The Gauss norm / sup norm

\begin{equation*} |\sum a_J x^J | = \max_J |a_J| \ge 0 \end{equation*}

properties

  1. \begin{equation*} |f| = 0 \iff f = 0 \end{equation*}
  2. \begin{equation*} |cf| = |c|_k |f| \end{equation*}
  3. \begin{equation*} |f + g| \le \max\{|f|, |g|\} \end{equation*}
  4. \begin{equation*} |fg| = |f||g| \end{equation*}
\begin{equation*} |f(x)| \le |f|,\,\forall x \in \mathbf B^n(\overline k) \end{equation*}

consider

\begin{equation*} \pi \colon R\langle x_1, \ldots, x_n \rangle \to \tilde k \lb x_1,\ldots, x_n\rb \end{equation*}

let \(\tilde f = \pi(f)\) be non-trivial (\(|f| = 1\)), then there exists

\begin{equation*} \tilde x \in \overline{\tilde k}^n \end{equation*}

s.t.

\begin{equation*} \tilde f (\tilde x) \ne 0\text{.} \end{equation*}

Have

\begin{equation*} \xymatrix{ \overline R \lb x_1, \ldots, x_n \rb \ar[r] \ar[d] & \overline{\tilde k} \lb x_1, \ldots, x_n\rb \ar[d] \\ \overline R \ar[r] & \overline{\tilde k} } \end{equation*}

so lift \(\tilde x\) to \(x \in \overline R^n\text{.}\) Since

\begin{equation*} f(x)\mapsto \tilde f( \tilde x) \end{equation*}

and \(\tilde f(\tilde x) \ne 0\) and \(|f(x)| = 1\text{.}\)

Algebraic properties of \(T_n\text{.}\)

  1. \(T_n\) is noetherian, regular and a UFD, for every maximal ideal \(\ideal m\) of \(T_n\text{,}\) \(T_n /\ideal m\) has finite degree over \(k\text{.}\)
  2. \(T_n\) is Jacobson: \(x\in T_n / I\) is nilpotent iff \(x\) lies in all maximal ideals of \(T_n/I\text{.}\)
  3. \(I\) is closed w.r.t. the Gauss norm for all ideals.
Definition 7.5.2. Affinoid algebras.

A \(k\)-affinoid algebra is a \(k\)-algebra \(A\) admitting an isomorphism \(A \simeq T_n/I\) as \(k\)-algebras \(I\subseteq T_n\text{.}\) The set \(\Max(A)\) for maximal ideals is denoted \(M(A)\text{.}\)

Properties

  1. A Noetherian, Jacobson, finite Krull dimension, \(A/\ideal m\) is a finite extension of \(k\text{,}\) where \(\ideal m \in M(A)\text{.}\)
  2. \(k(x) = A/ \ideal m_x\) for \(\ideal m_x \in M(A)\) then \(a\) is nilpotent iff \(a(x) = 0 \) for all \(x\in M(A)\text{.}\)
  3. \(M(A)\) is functorial with pullback. if \(\phi \colon A \to A' \) then
    \begin{equation*} \phi \inv (x) \in M(A) \end{equation*}
    for all \(x\in M(A')\) as
    \begin{equation*} \phi \colon A \to A' \end{equation*}
    \begin{equation*} A/ \phi \inv (x) \hookrightarrow A'/ x \end{equation*}
    \begin{equation*} A/\phi \inv (x) \end{equation*}
    is a finite extension of \(k\) hence a field, so \(\phi \inv(x)\) is maximal.
  4. Noether normalization: For \(A\) affinoid, then \(\exists d = \dim(A)\) s.t.
    \begin{equation*} T_d (k) \hookrightarrow A \end{equation*}
    then \(A/T_d(k)\) is a finite module extension.
  5. Maximum modulus
    \begin{equation*} \|f\|_{\sup} = \max_{x\in M(A)} |f(x)| \lt \infty \text{.} \end{equation*}

Topology on \(M(A)\text{.}\)

Example 7.5.4.
\begin{equation*} M(T_n) \end{equation*}

is disconnected

\begin{equation*} U = \{| x_1|= \cdots = |x_n| = 1\} \end{equation*}
\begin{equation*} V = U^c\text{.} \end{equation*}
Definition 7.5.5.

Tate algebras over \(k\)-Banach algebras \(\mathcal A\)

\begin{equation*} \mathcal A \langle Y_1, \ldots , Y_n \rangle = \{ \sum a_J Y^J : |a_J| \to 0 \text{ as } | J|\to \infty \} \end{equation*}

Universal property

\begin{equation*} \Hom(\mathcal A \langle X_1,\ldots, X_n\rangle, B) \to (B^0)^n \end{equation*}

is bijective.

\begin{equation*} \phi \mapsto (\phi (X_1), \ldots, \phi (X_n)) \end{equation*}
\begin{equation*} B \end{equation*}

is an

\begin{equation*} \mathcal A \end{equation*}

algebra, \(B^0\) are powerbounded elements.

Given \(a', a_1, \ldots, a_n\) with no common zeroes.

\begin{equation*} A \langle a_1/a', \ldots, a_n/a'\rangle = A\langle \underline X \rangle / \langle a' X_1 - a_1, \ldots\rangle\text{.} \end{equation*}

By universal property we have

\begin{equation*} b_1, \ldots , b_n \in B^0 \end{equation*}

s.t. \(\phi (a')b_j = \phi (a_j)\)l, \(\phi (a') \) is a unit otherwise there is \(y\) s.t. \(\phi (a' ) (y) = 0\) so \(\phi (a_j)(y) = 0 \forall j\) so common zero. Hence \(b_j\)'s are unique

\begin{equation*} |\phi (a_j)(y)|/|\phi (a')(y)| = |b_j(y)| \le 1 \end{equation*}

for all \(y\in M(B)\) .

Conversely if \(|\phi (a_j)| \le |\phi (a')|\text{,}\) \(\phi (a')\) a unit else common zero, let

\begin{equation*} b_j = \phi (a_j )/\phi (a') \end{equation*}

want \(|b_j(y) |\le 1\) for all \(y \in M(B)\) but

\begin{equation*} |\phi (a_j)(y)| \le |\phi (a')(y)| \end{equation*}

for all \(y \in M(B)\text{.}\)

Call

\begin{equation*} \{ x \in M(A) : |a_j(x)| \le |a'(x)| \} \end{equation*}

a rational domain: this canonically determines \(A \langle a_1/a', \ldots, a_n/'\rangle\text{.}\) Let \(A\langle \underline a, \underline {a'}\inv\rangle\) a laurent domain, if they are equal a weierstrass domain.

Affinoid subdomains: a \(k\)-affinoid subalgebra \(U \subseteq M(A)\) is called an affinoid subdomain if \(\exists i \colon A \to A'\) such that

\begin{equation*} M(i)\colon M(A') \to M(A) \end{equation*}

lands in \(U\) and is universal. This diagram commutes iff \(M(\phi )(M(B)) \subseteq U\) .

Completed tensor products

\begin{equation*} A \widehat \otimes _k A' \end{equation*}

give us intersection and pullback.

Gerritzen-Grauert