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.[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.

## Contents

* 1 Definition
* 2 Tate algebra
* 3 Results
* 4 Notes
* 5 References
* 6 See also
* 7 External links

## Definition[edit]

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}]} .[2][3]

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:[4] 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[edit]

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} ,

T n = K ⟨ ξ 1 , … ξ n ⟩ = A ⟨ ξ 1 , … , ξ n ⟩ ⊗ A 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  k [ [ ξ 1 , … , ξ n ] ] {\displaystyle k[[\xi _{1},\dots ,\xi _{n}]]} which consists of series convergent on  o k ¯ n {\displaystyle {\mathfrak {o}}_{\overline {k}}^{n}} , where  o k ¯ := { x ∈ k ¯ : | x | ≤ 1 } {\displaystyle {\mathfrak {o}}_{\overline {k}}:=\\{x\in {\overline {k}}:|x|\leq 1\\}} is the valuation ring in the algebraic closure k ¯ {\displaystyle {\overline {k}}} .

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

Define the Gauss norm of  f = ∑ a α ξ α {\displaystyle f=\sum a_{\alpha }\xi ^{\alpha }} in  T n {\displaystyle T_{n}} by

‖ f ‖ = max α | a α | . {\displaystyle \|f\|=\max _{\alpha }|a_{\alpha }|.}

This makes  T n {\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 I {\displaystyle I} of  T n {\displaystyle T_{n}} is closed[6] and thus, if I is radical, the quotient  T n / I {\displaystyle T_{n}/I} is also a (reduced) Banach algebra called an affinoid algebra.

Some key results are:

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

f = q g + r . {\displaystyle f=qg+r.} [8]
* (Weierstrass preparation) As above, let  g {\displaystyle g} be a  ξ n {\displaystyle \xi _{n}} -distinguished series of order s. Then there exist a unique monic polynomial f ∈ T n − 1 [ ξ n ] {\displaystyle f\in T_{n-1}[\xi _{n}]} of degree  s {\displaystyle s} and a unit element  u ∈ T n {\displaystyle u\in T_{n}} such that  g = f u {\displaystyle g=fu} .[9]
* (Noether normalization) If  a ⊂ T n {\displaystyle {\mathfrak {a}}\subset T_{n}} is an ideal, then there is a finite homomorphism  T d ↪ T n / a {\displaystyle T_{d}\hookrightarrow T_{n}/{\mathfrak {a}}} .[10]

As consequence of the division, preparation theorems and Noether normalization,  T n {\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[edit]

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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  m ⊂ A {\displaystyle {\mathfrak {m}}\subset A} a maximal ideal and  φ : A -> k := A / m {\displaystyle \varphi :A\to k:=A/{\mathfrak {m}}} the quotient map. Given a  F {\displaystyle F} in  A ⟨ ξ ⟩ {\displaystyle A\langle \xi \rangle } , if  φ ( F ) = g h {\displaystyle \varphi (F)=gh} for some monic polynomial  g ∈ k [ ξ ] {\displaystyle g\in k[\xi ]} and a restricted power series  h ∈ k ⟨ ξ ⟩ {\displaystyle h\in k\langle \xi \rangle } such that  g , h {\displaystyle g,h} generate the unit ideal of  k ⟨ ξ ⟩ {\displaystyle k\langle \xi \rangle } , then there exist  G {\displaystyle G} in  A [ ξ ] {\displaystyle A[\xi ]} and  H {\displaystyle H} in  A ⟨ ξ ⟩ {\displaystyle A\langle \xi \rangle } such that

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

## Notes[edit]

1. ^ Stacks Project, Tag 0AKZ.
2. ^ Grothendieck & Dieudonné 1960, Ch. 0, § 7.5.1.
3. ^ Bourbaki 2006, Ch. III, § 4. Definition 2 and Proposition 3.
4. ^ Grothendieck & Dieudonné 1960, Ch. 0, § 7.5.3.
5. ^ Fujiwara & Kato 2018, Ch 0, just after Proposition 9.3.
6. ^ Bosch 2014, § 2.3. Corollary 8
7. ^ Bosch 2014, § 2.2. Definition 6.
8. ^ Bosch 2014, § 2.2. Theorem 8.
9. ^ Bosch 2014, § 2.2. Corollary 9.
10. ^ Bosch 2014, § 2.2. Corollary 11.
11. ^ Bosch 2014, § 2.2. Proposition 14, Proposition 15, Proposition 17.
12. ^ Bosch 2014, § 2.2. Proposition 16.
13. ^ Bourbaki 2006, Ch. III, § 4. Theorem 1.

## References[edit]

* 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

## See also[edit]

* Weierstrass preparation theorem

## External links[edit]

* 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