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

Examples

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

Literature

• Morel: Adic spaces

• Wedhorn: Adic spaces

References