A.5. The hybrid circle [018J]

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A.5. The hybrid circle

Now consider the hybrid circle of radius $r\in(0,1)$ , that is, $C_{\mathrm{hyb}}(r):=\{|{t}|=r\}\subset{\mathbb{A}}^{1,\mathrm{hyb}}=(\operatorname{Spec}{\mathbb{C}}[{t}])^{\mathrm{hyb}}$ . By [Poi10, Prop 2.1.1], this is compact and realized as the Berkovich spectrum of the Banach ring

A_{r}:=\left\{f=\sum_{\alpha\in{\mathbb{Z}}}c_{\alpha}{t}^{\alpha}\in{\mathbb{C}}(\!({t})\!)\ \bigg|\ \|f\|_{\mathrm{hyb}}:=\sum_{\alpha\in{\mathbb{Z}}}\|c_{\alpha}\|_{\mathrm{hyb}}r^{\alpha}<+\infty\right\}.

Since $\|c_{\alpha}\|_{\mathrm{hyb}}\geq|c_{\alpha}|_{\infty}$ , every $f\in A_{r}$ defines a continuous function $f^{\operatorname{hol}}$ on the punctured closed disc $\overline{{\mathbb{D}}}^{*}_{r}$ that is holomorphic on ${\mathbb{D}}^{*}_{r}$ and meromorphic at 0.

Proposition A.4.

There is a homeomorphism $\overline{{\mathbb{D}}}_{r}\overset{\sim}{\to}{\mathcal{M}}(A_{r})\simeq C_{\mathrm{hyb}}(r)$ , that maps $z\in\overline{{\mathbb{D}}}_{r}\subset{\mathbb{C}}$ to the seminorm on $A_{r}$ defined by

|f|=\begin{cases}r^{\operatorname{ord}_{0}(f)}&\ \text{if $z=0$}\\ r^{\frac{\log|f^{\operatorname{hol}}(z)|_{\infty}}{\log|z|_{\infty}}}\ &\text{otherwise},\end{cases}

(A.1)

and via which the map $\lambda\colon C_{\mathrm{hyb}}(r)\to[0,1]$ is given by $\lambda(z)=\frac{\log r}{\log|z|_{\infty}}$ .

Proof.

The map $\tau\colon\overline{{\mathbb{D}}}_{r}\to{\mathcal{M}}(A_{r})$ given by (A.1) is clearly well defined. It is also continuous on $\overline{{\mathbb{D}}}^{*}_{r}$ . To prove continuity at $0$ , we note that for each $f\in A_{r}$ , we can write $f^{\operatorname{hol}}=z^{\operatorname{ord}_{0}(f)}u$ , where $u$ is a continuous function on $\overline{{\mathbb{D}}}_{r}$ that is holomorphic on ${\mathbb{D}}_{r}$ with $u(0)\neq 0$ . As a consequence, we get $\lim_{z\to 0}\frac{\log|f^{\operatorname{hol}}(z)|_{\infty}}{\log|z|_{\infty}}=\operatorname{ord}_{0}(f)$ .

Now, for each $\rho\in(0,1]$ , $\lambda^{-1}(\rho)\subset C_{\mathrm{hyb}}(r)$ can be identified with the circle of radius $r$ with respect to the absolute value $|\cdot|_{\infty}^{\rho}$ , while $\lambda^{-1}(0)$ is the non-Archimedean absolute value $r^{-\operatorname{ord}_{0}}$ on ${\mathbb{C}}(\!({t})\!)$ . This proves that the map $\tau$ above is bijective, and hence a homeomorphism by compactness. ∎

Remark A.5.

When $r<s$ , the identity gives a bounded map from $A_{s}$ to $A_{r}$ , and $\varinjlim_{r\to 0}A_{r}$ is the fraction field of ${\mathcal{O}}_{{\mathbb{C}},0}$ , i.e. the ring of meromorphic germs at the origin of ${\mathbb{C}}$ .