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.

Let A be a linearly topologized ring, separated and complete and ${\displaystyle \{I_{\lambda }\}}$ the fundamental system of open ideals. Then the ring of restricted power series is defined as the projective limit of the polynomial rings over ${\displaystyle A/I_{\lambda }}$:

${\displaystyle A\langle x_{1},\dots ,x_{n}\rangle =\varprojlim _{\lambda }A/I_{\lambda }[x_{1},\dots ,x_{n}]}$.^[2][3]

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

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

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

extending ${\displaystyle A\to B}$.

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

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

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

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

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

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

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

Some key results are:

• (Weierstrass division) Let ${\displaystyle g\in T_{n}}$ be a ${\displaystyle \xi _{n}}$-distinguished series of order s; i.e., ${\displaystyle g=\sum _{\nu =0}^{\infty }g_{\nu }\xi _{n}^{\nu }}$ where ${\displaystyle g_{\nu }\in T_{n-1}}$, ${\displaystyle g_{s}}$ is a unit element and ${\displaystyle |g_{s}|=\|g\|>|g_{v}|}$ for ${\displaystyle \nu >s}$.^[7] Then for each ${\displaystyle f\in T_{n}}$, there exist a unique ${\displaystyle q\in T_{n}}$ and a unique polynomial ${\displaystyle r\in T_{n-1}[\xi _{n}]}$ of degree ${\displaystyle <s}$ such that

  ${\displaystyle f=qg+r.}$^[8]

• (Weierstrass preparation) As above, let ${\displaystyle g}$ be a ${\displaystyle \xi _{n}}$-distinguished series of order s. Then there exist a unique monic polynomial ${\displaystyle f\in T_{n-1}[\xi _{n}]}$ of degree ${\displaystyle s}$ and a unit element ${\displaystyle u\in T_{n}}$ such that ${\displaystyle g=fu}$.^[9]
• (Noether normalization) If ${\displaystyle {\mathfrak {a}}\subset T_{n}}$ is an ideal, then there is a finite homomorphism ${\displaystyle T_{d}\hookrightarrow T_{n}/{\mathfrak {a}}}$.^[10]

As consequence of the division, preparation theorems and Noether normalization, ${\displaystyle T_{n}}$ 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 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 ${\displaystyle {\mathfrak {m}}\subset A}$ be a maximal ideal and ${\displaystyle \varphi :A\to k:=A/{\mathfrak {m}}}$ the quotient map. Given an ${\displaystyle F}$ in ${\displaystyle A\langle \xi \rangle }$, if ${\displaystyle \varphi (F)=gh}$ for some monic polynomial ${\displaystyle g\in k[\xi ]}$ and a restricted power series ${\displaystyle h\in k\langle \xi \rangle }$ such that ${\displaystyle g,h}$ generate the unit ideal of ${\displaystyle k\langle \xi \rangle }$, then there exist ${\displaystyle G}$ in ${\displaystyle A[\xi ]}$ and ${\displaystyle H}$ in ${\displaystyle A\langle \xi \rangle }$ such that

  ${\displaystyle F=GH,\,\varphi (G)=g,\varphi (H)=h}$.^[13]

• Bourbaki, N. (2006). Algèbre commutative: Chapitres 1 à 4. Springer Berlin Heidelberg. ISBN 9783540339373.
• Grothendieck, Alexandre; Dieudonné, Jean (1960). "Éléments de géométrie algébrique: I. Le langage des schémas". Publications Mathématiques de l'IHÉS. 4. doi:10.1007/bf02684778. MR 0217083.
• Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold (1984), "Chapter 5", Non-archimedean analysis, Springer
• Bosch, Siegfried (2014), Lectures on Formal and Rigid Geometry, ISBN 9783319044170
• Fujiwara, Kazuhiro; Kato, Fumiharu (2018), Foundations of Rigid Geometry I

• Weierstrass preparation theorem

• https://ncatlab.org/nlab/show/restricted+formal+power+series
• http://math.stanford.edu/~conrad/papers/aws.pdf
• https://web.archive.org/web/20060916051553/http://www-math.mit.edu/~kedlaya//18.727/tate-algebras.pdf