Restricted power series - Misplaced Pages

(Redirected from Gauss norm) Formal power series with coefficients tending to 0

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 $\{I_{\lambda }\}$ 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 }

In other words, it is the completion of the polynomial ring $A$ with respect to the filtration $\{I_{\lambda }\}$ . 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 $A]$ 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: 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. It is equivalently the subring of formal power series $k]$ 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 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 $\nu >s$ . Then for each $f\in T_{n}$ , there exist a unique $q\in T_{n}$ and a unique polynomial $r\in T_{n-1}$ of degree $<s$ such that
$f=qg+r.$
(Weierstrass preparation) As above, let $g$ be a $\xi _{n}$ -distinguished series of order s. Then there exist a unique monic polynomial $f\in T_{n-1}$ of degree $s$ and a unit element $u\in T_{n}$ such that $g=fu$ .
(Noether normalization) If ${\mathfrak {a}}\subset T_{n}$ is an ideal, then there is a finite homomorphism $T_{d}\hookrightarrow T_{n}/{\mathfrak {a}}$ .

As consequence of the division, preparation theorems and Noether normalization, $T_{n}$ is a Noetherian unique factorization domain of Krull dimension n. 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).

Results

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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 $g\in k$ 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 $A$ and $H$ in $A\langle \xi \rangle$ such that
$F=GH,\,\varphi (G)=g,\varphi (H)=h$ .