Power-bounded element

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 ${\displaystyle A}$  be a topological ring. A subset ${\displaystyle T\subset A}$  is called bounded, if, for every neighbourhood ${\displaystyle U}$  of zero, there exists an open neighbourhood ${\displaystyle V}$  of zero such that ${\displaystyle T\cdot V:=\{t\cdot v\mid t\in T,v\in V\}\subset U}$  holds. An element ${\displaystyle a\in A}$  is called power-bounded, if the set ${\displaystyle \{a^{n}\mid n\in \mathbb {N} \}}$  is bounded.^[1]

Examples

• An element ${\displaystyle x\in \mathbb {R} }$  is power-bounded if and only if ${\displaystyle |x|\leq 1}$  hold.
• More generally, if ${\displaystyle A}$  is a topological commutative ring whose topology is induced by an absolute value, then an element ${\displaystyle x\in A}$  is power-bounded if and only if ${\displaystyle |x|\leq 1}$  holds. If the absolute value is non-Archimedean, the power-bounded elements form a subring, denoted by ${\displaystyle A^{\circ }}$ . This follows from the ultrametric inequality.
• The ring of power-bounded elements in ${\displaystyle \mathbb {Q} _{p}}$  is ${\displaystyle \mathbb {Q} _{p}^{\circ }=\mathbb {Z} _{p}}$ .
• Every topological nilpotent element is power-bounded.^[2]

Literature

• Morel: Adic spaces

• Wedhorn: Adic spaces

References

1. Wedhorn: Def. 5.27
2. Wedhorn: Rem. 5.28 (4)