Contents

Power Series RingEdit

Power series rings are a fundamental construction in algebra and algebraic geometry, providing a precise way to study local behavior and infinitesimal deformations. Given a ring R, the power series ring Rt consists of all formal sums sum_{n≥0} a_n t^n with coefficients a_n in R. Unlike the polynomial ring R[t], these sums are not required to terminate, and the algebraic structure is designed to handle infinite tails in a controlled way. The ring Rt is the completion of the polynomial ring R[t] with respect to the (t)-adic topology, and it comes equipped with a natural surjection to R given by evaluation at t = 0, f(t) ↦ a_0, whose kernel is the ideal generated by t. For a field k, the prototypical example is kt, the ring of formal power series over a field. ring polynomial ring inverse limit t-adic topology completion.

Definition and construction

Elements of Rt are formal series Σ{n≥0} a_n t^n with a_n ∈ R. Addition is termwise, and multiplication is given by the Cauchy product: (sum a_n t^n) · (sum b_n t^n) = sum c_n t^n where c_n = sum{i=0}^n a_i b_{n−i}. A convenient way to view Rt is as the inverse limit of the quotient rings R[t]/(t^N) as N → ∞: Rt ≅ lim← R[t]/(t^N). This perspective underlines its role as a completion of R[t] with respect to the (t)-adic topology. The natural projection f(t) ↦ a_0 exhibits a surjection Rt → R with kernel the ideal (t). If R is local with maximal ideal m, the maximal ideal of Rt is m + (t). RW Cauchy product inverse limit.

Units and ideals

An element f(t) = Σ a_n t^n is a unit in Rt if and only if its constant term a_0 is a unit in R. When R is local with maximal ideal m, the units are exactly those series with a_0 ∉ m. The ideal generated by t, denoted (t), consists of all series whose constant term is zero. More generally, if R has an ideal I, one can consider the I-adic topology and related completions, with Rt playing the role of a basic one-variable example. unit maximal ideal.

Algebraic and topological structure

Rt is a topological ring, complete with respect to the (t)-adic topology. If R is Noetherian, then Rt is also Noetherian. The ring is a domain whenever R is a domain, and its dimension satisfies dim Rt ≥ dim R; for many nice cases (for instance, when R is a Noetherian local ring), one has dim Rt = dim R + 1. When R is a field k, kt is a complete discrete valuation ring (DVR) with uniformizing parameter t and residue field k. These structural properties make Rt a central object in studies of local behavior and deformations. Noetherian ring local ring dimension theory discrete valuation ring field.

Examples

  • The canonical example is kt for a field k, the ring of formal power series in one variable. This ring serves as the local model for smooth one-dimensional objects in algebraic geometry and for local fields in number theory. field k[[t]].
  • More generally, Rt with R any ring (such as Z, a finite product of rings, or a local ring) provides a basic one-variable local-like structure in which coefficients come from R. For instance, Zt appears in contexts modeling p-adic-like phenomena without committing to a particular completion of Z; it acts as a testing ground for formal deformation ideas. Z ring.
  • In several variables one can form Rt_1, ..., t_n, the ring of formal power series in n indeterminates, which plays a central role in the study of local properties of schemes and in deformation theory. power series multivariable power series ring.

Connections and applications

  • Local geometry: For a scheme X and a closed point x, the local ring O_{X,x} captures infinitesimal structure around x. When X is a smooth curve over a field k, the completed local ring can be identified with kt in many standard coordinates, illustrating how power series rings model neighborhoods of points. scheme local ring curve.
  • Deformation theory: Deformations of objects are often encoded by algebras over a base ring with a formal parameter t, and Rt-algebras provide natural receptacles for families parametrized by infinitesimal thickenings. deformation theory.
  • Analogy with p-adics: The (t)-adic topology in Rt parallels the p-adic topology on Z_p in spirit; both arise from completing a ring with respect to a chosen ideal and both provide contexts in which formal variable behavior encodes infinitesimal information. p-adic.
  • Non-archimedean and formal geometry: Power series rings serve as algebraic scaffolding for formal schemes and for certain non-archimedean analytic frameworks where convergence is formal rather than analytic. formal scheme.
  • Variants and generalizations: One can consider Rx_1, ..., x_n or noncommutative power series rings, expanding the utility to multivariate or noncommutative settings. These variants underpin a wide range of results in algebraic geometry, representation theory, and beyond. multivariable power series ring.

Properties in practice

  • Calculations often exploit the fact that evaluation at t = 0 recovers the constant term, while higher coefficients control infinitesimal data. The t-adic filtration { (t)^N } gives a natural way to approximate elements and to study convergence in a purely algebraic sense. valuation filtration.
  • When R is Noetherian, standard algebraic tools (noetherian approximation, dimension theory, flatness considerations) behave well under passage to Rt. This makes Rt a reliable base for constructing and analyzing families of algebraic objects. flatness.

See also