In algebra, the ring of restricted power series is the subring of a formal power series ring that consists of power series whose coefficients approach zero as degree goes to infinity. Over a non-archimedean complete field, the ring is also called a Tate algebra. Quotient rings of the ring are used in the study of a formal algebraic space as well as rigid analysis, the latter over non-archimedean complete fields. Over a discrete topological ring, the ring of restricted power series coincides with a polynomial ring; thus, in this sense, the notion of "restricted power series" is a generalization of a polynomial.
Definition Let A be a linearly topologized ring, separated and complete and { I λ } {\displaystyle \{I_{\lambda }\}} the fundamental system of open ideals. Then the ring of restricted power series is defined as the projective limit of the polynomial rings over A / I λ {\displaystyle A/I_{\lambda }} :
A ⟨ x 1 , … , x n ⟩ = lim ← λ A / I λ [ x 1 , … , x n ] {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle =\varprojlim _{\lambda }A/I_{\lambda }[x_{1},\dots ,x_{n}]} . In other words, it is the completion of the polynomial ring A [ x 1 , … , x n ] {\displaystyle A[x_{1},\dots ,x_{n}]} with respect to the filtration { I λ [ x 1 , … , x n ] } {\displaystyle \{I_{\lambda }[x_{1},\dots ,x_{n}]\}} . Sometimes this ring of restricted power series is also denoted by A { x 1 , … , x n } {\displaystyle A\{x_{1},\dots ,x_{n}\}} . Clearly, the ring A ⟨ x 1 , … , x n ⟩ {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle } can be identified with the subring of the formal power series ring A [ [ x 1 , … , x n ] ] {\displaystyle A[[x_{1},\dots ,x_{n}]]} that consists of series ∑ c α x α {\displaystyle \sum c_{\alpha }x^{\alpha }} with coefficients c α → 0 {\displaystyle c_{\alpha }\to 0} ; i.e., each I λ {\displaystyle I_{\lambda }} contains all but finitely many coefficients c α {\displaystyle c_{\alpha }} . Also, the ring satisfies (and in fact is characterized by) the universal property: for (1) each continuous ring homomorphism A → B {\displaystyle A\to B} to a linearly topologized ring B {\displaystyle B} , separated and complete and (2) each elements b 1 , … , b n {\displaystyle b_{1},\dots ,b_{n}} in B {\displaystyle B} , there exists a unique continuous ring homomorphism
A ⟨ x 1 , … , x n ⟩ → B , x i ↦ b i {\displaystyle A\langle x_{1},\dots ,x_{n}\rangle \to B,\,x_{i}\mapsto b_{i}}
extending A → B {\displaystyle A\to B} .
Tate algebra In rigid analysis, when the base ring A is the valuation ring of a complete non-archimedean field ( K , | ⋅ | ) {\displaystyle (K,|\cdot |)} , the ring of restricted power series tensored with K {\displaystyle K} ,
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