Restricted power series

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.^[1] 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 $\{I_{\lambda }\}$ the fundamental system of open ideals. Then the ring of restricted power series is defined as the projective limit of the polynomial rings over $A/I_{\lambda }$ :

A\langle x_{1},\dots ,x_{n}\rangle =\varprojlim _{\lambda }A/I_{\lambda }[x_{1},\dots ,x_{n}]

.^[2]^[3]

In other words, it is the completion of the polynomial ring $A[x_{1},\dots ,x_{n}]$ with respect to the filtration $\{I_{\lambda }[x_{1},\dots ,x_{n}]\}$ . Sometimes this ring of restricted power series is also denoted by $A\{x_{1},\dots ,x_{n}\}$ .

Clearly, the ring $A\langle x_{1},\dots ,x_{n}\rangle$ can be identified with the subring of the formal power series ring $A[[x_{1},\dots ,x_{n}]]$ that consists of series $\sum c_{\alpha }x^{\alpha }$ with coefficients $c_{\alpha }\to 0$ ; i.e., each $I_{\lambda }$ contains all but finitely many coefficients $c_{\alpha }$ . Also, the ring satisfies (and in fact is characterized by) the universal property:^[4] for (1) each continuous ring homomorphism $A\to B$ to a linearly topologized ring $B$ , separated and complete and (2) each elements $b_{1},\dots ,b_{n}$ in $B$ , there exists a unique continuous ring homomorphism

A\langle x_{1},\dots ,x_{n}\rangle \to B,\,x_{i}\mapsto b_{i}

extending $A\to B$ .

Tate algebra

In rigid analysis, when the base ring A is the valuation ring of a complete non-archimedean field $(K,|\cdot |)$ , the ring of restricted power series tensored with $K$ ,

T_{n}=K\langle \xi _{1},\dots \xi _{n}\rangle =A\langle \xi _{1},\dots ,\xi _{n}\rangle \otimes _{A}K

is called a Tate algebra, named for John Tate.^[5] It is equivalently the subring of formal power series $k[[\xi _{1},\dots ,\xi _{n}]]$ which consists of series convergent on ${\mathfrak {o}}_{\overline {k}}^{n}$ , where ${\mathfrak {o}}_{\overline {k}}:=\{x\in {\overline {k}}:|x|\leq 1\}$ is the valuation ring in the algebraic closure ${\overline {k}}$ .

The maximal spectrum of $T_{n}$ is then a rigid-analytic space that models an affine space in rigid geometry.

Define the Gauss norm of $f=\sum a_{\alpha }\xi ^{\alpha }$ in $T_{n}$ by

\|f\|=\max _{\alpha }|a_{\alpha }|.

This makes $T_{n}$ a Banach algebra over k; i.e., a normed algebra that is complete as a metric space. With this norm, any ideal $I$ of $T_{n}$ is closed^[6] and thus, if I is radical, the quotient $T_{n}/I$ is also a (reduced) Banach algebra called an affinoid algebra.

Some key results are:

(Weierstrass division) Let $g\in T_{n}$ be a $\xi _{n}$ -distinguished series of order s; i.e., $g=\sum _{\nu =0}^{\infty }g_{\nu }\xi _{n}^{\nu }$ where $g_{\nu }\in T_{n-1}$ , $g_{s}$ is a unit element and $|g_{s}|=\|g\|>|g_{v}|$ for $\nu >s$ .^[7] Then for each $f\in T_{n}$ , there exist a unique $q\in T_{n}$ and a unique polynomial $r\in T_{n-1}[\xi _{n}]$ of degree $<s$ such that
$f=qg+r.$ ^[8]
(Weierstrass preparation) As above, let $g$ be a $\xi _{n}$ -distinguished series of order s. Then there exist a unique monic polynomial $f\in T_{n-1}[\xi _{n}]$ of degree $s$ and a unit element $u\in T_{n}$ such that $g=fu$ .^[9]
(Noether normalization) If ${\mathfrak {a}}\subset T_{n}$ is an ideal, then there is a finite homomorphism $T_{d}\hookrightarrow T_{n}/{\mathfrak {a}}$ .^[10]

As consequence of the division, preparation theorems and Noether normalization, $T_{n}$ is a Noetherian unique factorization domain of Krull dimension n.^[11] An analog of Hilbert's Nullstellensatz is valid: the radical of an ideal is the intersection of all maximal ideals containing the ideal (we say the ring is Jacobson).^[12]

Results

Results for polynomial rings such as Hensel's lemma, division algorithms (or the theory of Gröbner bases) are also true for the ring of restricted power series. Throughout the section, let A denote a linearly topologized ring, separated and complete.

(Hensel) Let ${\mathfrak {m}}\subset A$ be a maximal ideal and $\varphi :A\to k:=A/{\mathfrak {m}}$ the quotient map. Given an $F$ in $A\langle \xi \rangle$ , if $\varphi (F)=gh$ for some monic polynomial $g\in k[\xi ]$ and a restricted power series $h\in k\langle \xi \rangle$ such that $g,h$ generate the unit ideal of $k\langle \xi \rangle$ , then there exist $G$ in $A[\xi ]$ and $H$ in $A\langle \xi \rangle$ such that
$F=GH,\,\varphi (G)=g,\varphi (H)=h$ .^[13]