Restricted power series

Summary

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.

DefinitionEdit

Let A be a linearly topologized ring, separated and complete and   the fundamental system of open ideals. Then the ring of restricted power series is defined as the projective limit of the polynomial rings over  :

 .[2][3]

In other words, it is the completion of the polynomial ring   with respect to the filtration  . Sometimes this ring of restricted power series is also denoted by  .

Clearly, the ring   can be identified with the subring of the formal power series ring   that consists of series   with coefficients  ; i.e., each   contains all but finitely many coefficients  . Also, the ring satisfies (and in fact is characterized by) the universal property:[4] for (1) each continuous ring homomorphism   to a linearly topologized ring  , separated and complete and (2) each elements   in  , there exists a unique continuous ring homomorphism

 

extending  .

Tate algebraEdit

In rigid analysis, when the base ring A is the valuation ring of a complete non-archimedean field  , the ring of restricted power series tensored with  ,

 

is called a Tate algebra, named for John Tate.[5] It is equivalently the subring of formal power series   which consists of series convergent on  , where   is the valuation ring in the algebraic closure  .

The maximal spectrum of   is then a rigid-analytic space that models an affine space in rigid geometry.

Define the Gauss norm of   in   by

 

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

Some key results are:

  • (Weierstrass division) Let   be a  -distinguished series of order s; i.e.,   where  ,   is a unit element and   for  .[7] Then for each  , there exist a unique   and a unique polynomial   of degree   such that
     [8]
  • (Weierstrass preparation) As above, let   be a  -distinguished series of order s. Then there exist a unique monic polynomial   of degree   and a unit element   such that  .[9]
  • (Noether normalization) If   is an ideal, then there is a finite homomorphism  .[10]

As consequence of the division, preparation theorems and Noether normalization,   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]

ResultsEdit

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   a maximal ideal and   the quotient map. Given a   in  , if   for some monic polynomial   and a restricted power series   such that   generate the unit ideal of  , then there exist   in   and   in   such that
     .[13]

NotesEdit

  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.

ReferencesEdit

  • 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 alsoEdit

External linksEdit

  • 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