power-bounded element

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A power-bounded element is an element of a topological ring whose powers are bounded. These elements are used in the theory of adic spaces.

Definition

Let $A$ be a topological ring. A subset $T \subset A$ is called bounded, if, for every neighbourhood $U$ of zero, there exists an open neighbourhood $V$ of zero such that $T \cdot V := \{t \cdot v \mid t \in T, v \in V\} \subset U$ holds. An element $a \in A$ is called power-bounded, if the set $\{a^n \mid n \in \mathbb N\}$ is bounded.Wedhorn: Def. 5.27

Examples

An element $x \in \mathbb R$ is power-bounded if and only if $|x| \leq 1$ hold.
More generally, if $A$ is a topological commutative ring whose topology is induced by an absolute value, then an element $x \in A$ is power-bounded if and only if $|x| \leq 1$ holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring, denoted by $A^{\circ}$ . This follows from the ultrametric inequality.
The ring of power-bounded elements in $\mathbb Q_p$ is $\mathbb Q_p^{\circ} = \mathbb Z_p$ .
Every topological nilpotent element is power-bounded.Wedhorn: Rem. 5.28 (4)

Literature

Morel: [https://web.math.princeton.edu/~smorel/adic_notes.pdf Adic spaces]

Wedhorn: [https://arxiv.org/pdf/1910.05934.pdf Adic spaces]

References

Category:Topological algebra