There are some basic terminologies that resembles the ones used in algebraic geometry. Let \(K\) be a complete valued (non-archimedean) field with ring \(\mathcal{O}_K\) and prime \(\mathfrak{p}\), \(k=\mathcal{O}_K/\mathfrak{p}\).
| AG terms | Analytic terms |
|---|---|
| Polynomial algebra \(K[X_{1\dots n}]\) | Tate algebra \(T_n = K\langle X_{1\dots n}\rangle = \{\sum c_\alpha x^\alpha | |c_\alpha| \to 0 \}\) |
| Gauss norm\(\left\|\sum c_\alpha x^\alpha \right\| = \max |c_\alpha|\) | |
| Affine algebra \(A=R/I\) | Affinoid algebra (quotient of Tate)\(T_n/J\) |
| Affine Space \(\mathrm{Spec}(A)\) | Affinoid space (maximalideals) \(\mathrm{Sp}(T_n/J)\) |
| Zariski Topology \(V(\mathfrak{a}), \mathfrak{a}\subset A\) | Zariski Topology \(V(\mathfrak{a}),\mathfrak{a}\subset A\) |
| Canonical Topology whose topological base generated by (finite intersections and arbitrary unions) \(X(f,\varepsilon)= \{x| |f(x)|\le \varepsilon\}\) | |
| Ringed space \((X,\mathcal{O}_X)\) | Ringed space with Grothendieck topology \((X,\mathcal{O}_X)\) |
The projective line \(\mathbb{P}^1\) over an algebraically closed non-archimedean field \(K\) for example \(\mathbb{C}_p\) is intuitively depicted as \(K\cup \{\infty\}\). It serves as a basic and useful example before heavy mechanisms come into place.
The linear group \(\mathrm{PGL}_2(K)\) acts on \(\mathbb{P}^1\) by coordinates, and in affine coordinate it is just a fraction linear transform \(z\mapsto \frac{az+b}{cz+d}\).
The ’open’ disks are \[\{|z-a|<r\},\quad \{|z-a|>r\}\] and closed disks where \(<,>\) are replaced by \(\le,\ge\). They are both open and closed. The \(\mathrm{PGL}_2(K)\) group acts transitively on all open disks, and all closed disks.
A connected affinoid subset of \(\mathbb{P}\) is the complement of a non-empty finite union of open disks. i.e. discarding finite many open disks, or equivalently, finite intersection of non-full closed disks, whose complement is of the form \[F^c=\bigcup B_i\] where \(B_i\) are open disks. Thus \(\mathrm{PGL}\) transforms connected affinoid subsets to connected affinoid subsets. Connected affinoid subset can be empty, but not full.
An affinoid subset of \(\mathbb{P}\) is a finite union of connected affinoid subsets.
Example 1. Think through the following interesting examples to get a feeling of open disks in \(\mathbb{P}^1\),
Let \(B_1, B_2\) be two open disks s.t. \(B_1\cup B_2\neq \mathbb{P}\), then \(B_1\cap B_2\) is either empty or an open disk. Because we can use group action to move them into affine line, and the intersection is either empty or one of them is contained in another.
If \(B_1\cup B_2=\mathbb{P}\), what is \(B_1\cap B_2\)? They are strips, you can see this by moving their centers to \(0\) and \(\infty\).
Lemma 1. Let \(F_1, F_2\) denote two connected affinoid subsets of \(\mathbb{P}^1\)
\(F_1\cap F_2\) is always a connected affinoid subset.
if \(F_1\) and \(F_2\) have both non-trivial intersection and union, \(F_1\cup F_2\) is a connected affinoid subset.
Proof.
Finite intersections of connected affinoid subsets are obviously connected affinoid subsets, since \((F_1\cap F_2)^c = F_1^c\cup F_2^c\).
The condition \(F_1\cap F_2\neq \varnothing\) is crucial, it means \(\mathbb{P}\neq (F_1^c\cup F_2^c)=\bigcup B_i\cup \bigcup B_i'\). Therefore by lemmas \(B_i\cap B_j'\) is either empty or an open disk, so \[(F_1\cup F_2)^c = \bigcup_{i,j}(B_i\cap B_j')\] is (the complement of) a connected affinoid subset.
\(F_1\cup F_2\neq \mathbb{P}\) ensures at least one of the above \(B_i\cap B_j'\) is non-empty.
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Every affinoid subset \(F\) is in a unique way a finite union of connected affinoid subsets, called the connectedcomponents of \(F\). This is similar to writing things in disjunctive normal form.
Affinoids are closed under finite union and intersection.
Let’s explore affinoids defined by a rational function. Let \(f\in K(z)\) be any rational function on \(\mathbb{P}^1\) and \(r\in |K^*|\) we consider \[F:=\{a\in \mathbb{P}^1 | |f(a)|\le r\}\] then \(F\) is either an affinoid subset or empty.
Proof. One may write \(f=\prod_{i=1}^s (z-a_i)^{n_i}\) with distinct \(a_i\in K\) and integers \(n_i\in\mathbb{Z}\). We can prove by induction on the number of factors of \(f\).
The case of \(s=1\) is trivial, regardless whether \(n_1\ge 0\) or \(n_1 < 0\) (we are talking about affinoids so \(\mathbb{P}^1\) is accepted as an affinoid subset).
Assume now \(a_1=0, a_2=1\). We know that \(\mathbb{P}^1=U_1\cup U_2\cup U_3\) where \[U_1 = \{a\in\mathbb{P}^1 : |a|\ge 1, |a-1|\ge 1\}\] \[U_2 = \{a\in\mathbb{P}^1 : |a|\ge 1, |a-1|\le 1\}\] \[U_3 = \{a\in\mathbb{P}^1 : |a|\le 1, |a-1|\ge 1\}.\] (the other case degenerates to their intersections \(\{|a|=1, |a-1|=1\}\))
We can see from \(|a-1|\ge 1, |a|\ge 1\Rightarrow |a|=|a-1|\), this means \(F\) is locally \(F_{z^{n_1+n_2}g_2(z)}\), \[U_1\cap F=\{a\in U_1: |z^{n_1+n_2} g_2(z)|\le r\}= U_1 \cap F_{z^{n_1+n_2}g_2(z)}.\] Therefore by induction hypothesis, \(F_{z^{n_1+n_2}g_2(z)}\) is affinoid, \(U_1\) is also affinoid, so \(U_1\cap F\) is affinoid since finite intersection of affinoids is affinoid.
In fact in \(U_2\) we also have \(|a|=|a-1|\). Since if \(|a|=1\) we would have \(1\le |a-1|\le 1\), and when \(|a|>1\) we must have \(|a-1|=\max(|a|,|a-1|)=|a|\). Therefore \(U_2\cap F\) is also an affinoid subset by the same arguments. Similarly for \(U_3\).
We can conclude by using the fact that the union of affinoid subsets with non-trivial intersection and union is an affinoid subset.
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Proposition 1. A rational function viewed as \(f:\mathbb{P}^1\to \mathbb{P}^1\) has the property that any preimage of affinoid subset is again an affinoid subset.
Proof. Consider first the case of a non-full closed disk \(D\subset \mathbb{P}^1\). Then its preimage is an affinoid subset as we have proved in previous lemma. This means the preimage of connected affinoid subsets are affinoid subsets, as a result the preimage of affinoid subsets are affinoid subsets. ◻
Connected affinoid subsets are characterized by the fact that the analytic function rings on it does not have zero divisors.