Ring Of Formal Power Series Research Articles

Colength of Derivation Ideals

In this paper, $D$ is a derivation acting on the formal power series ring $K[[{t_1}, \cdots ,{t_r}]]$ over a field $K$ of characteristic $p \ne 0$. We conjecture that the colengths of the ideals ($({D^{{p^n}}}{t_1}, \cdots ,{D^{{p^n}}}{t_r})$ and $({D^{{p^{n - 1}}}}{t_1}, \cdots ,{D^{{p^{n - 1}}}}{t_r})$ are congruent modulo ${p^n}$, provided they are finite. We give a proof for the case $r = 1$ and any $n \geqq 1$, and for the case $n = 1$ and any $r \geqq 1$.

On Germs of DIfferentiable Functions In Two Variables

In Q4H we have shown some sufficient conditions for a germ (of a differentiable function in two variables) to be transformed into an analytic one or a polynomial through a change of coordinates. Here we shall refine the result above and find a necessary and sufficient condition. We denote respectively by 0, $ the rings of germs at 0 in R of real analytic and C°°-functions, and by J the ring of formal power series in 2 indeterminates over R. If A is one of the above rings, let m(A) denote the maximal ideal of A. Let Ta denote the Taylor expansion at a. Elements f , g o f & are called equivalent if there exists a local diffeomorphism (of class C°°) r of R around 0 such that f°t=g. If an element / of m(«f) can be factorized into the following form

PROJECTIVE RESOLUTIONS AND COHOMOLOGICAL TRIVIALITY OFp-PERIODIC BIMODULES OVER FROBENIUS ORDERS

Let Λ be a Frobenius order in a simple algebra over the field of p-adic numbers, . For a finitely-generated p-periodic Λ-bimodule, we establish the existence of a Λe/p-free resolution whose generating function is an associate of the Poincare series in the ring of formal power series with integral coefficients. Our subsequent investigations are restricted to orders of the form described which in addition satisfy a certain disjointness condition modulo p, which is formulated in terms of constraints on the Cartan matrix of the ring Λe/p. We find conditions sufficient for the existence of a p-periodic module with trivial homology (in the sense of Hochschild) and having infinite projective dimension over the ring Λe/p. We prove a Nakayama-type theorem on the triviality of the cohomology groups ot Λ with coefficients in irreducible Λ-bimodules. Bibliography. 12 items.

�ber die Cohomologie von BSO(2k+1) und Vn,2

If h* is a multiplicative cohomology theory on the category of CW-pairs, then it is shown that h*(BSO (2k+1)) is under certain conditions isomorphic to the ring of formal power series over h*(*) in universal Pontrjagin-classes p1,...,pk. In the second part of the paper one finds a calculation of the cohomology of the Stiefel manifolds Vn,2=SO(n)/SO(n−2).

Rings of Formal Power Series

In this brief exposition we collect several results on rings of formal power series with coefficients from a field or a ring with some special properties. The results that are catalogued below are mostly algebraic in nature.

How changing 𝐷[[𝑥]] changes its quotient field

Let D [ [ x ] ] D[[x]] be the ring of formal power series over the commutative integral domain D D . It is shown that changing D [ [ x ] ] D[[x]] to D [ [ x / a ] ] D[[x/a]] changes (i.e. increases) the quotient field by an infinite transcendence degree over the original field whenever ∩ i = 1 ∞ a i D = 0 \cap _{i = 1}^\infty {a^i}D = 0 . From this it follows that if D 1 {D_1} and D 2 {D_2} are two distinct rings between the integers and the rational numbers, with D 1 {D_1} contained in D 2 {D_2} , then the change in the ring of coefficients from D 1 [ [ x ] ] {D_1}[[x]] to D 2 [ [ x ] ] D_{2}[[x]] again yields a change in the quotient fields by an infinite transcendence degree. More generally, it is shown that D D is completely integrally closed iff any increase in the ring of coefficients yields an increase in the quotient field of D [ [ x ] ] D[[x]] . Moreover, D D is a one-dimensional Prüfer domain iff any change in the ring of coefficients from one overring of D D to another overring of D D yields a change in the quotient field of the respective power series rings. Finally it is shown that many of the domain properties of interest are really properties of their divisibility groups, and some examples are constructed by first constructing the required divisibility groups.

Some remarks on the formal power series ring

Some remarks on the formal power series ring

RINGS WITH A DISCRETE GROUP OF DIVISOR CLASSES

We shall say that a ring A has a DGC (discrete group of classes) if the group of divisor classes is preserved in going to the ring of formal power series, i.e. C(A) → C(A[[T]]) is an isomorphism. We prove the localness and faithfully flat descent of the DGC property. We establish a connection between the DGC property of a ring and its depth. We also give a characterization of two-dimensional rings with DGC and characteristic zero rings with DGC. Finally, it is shown that the discreteness of the group of divisor classes is preserved under regular extensions of rings such as A[T1,⋯, Tn], A[[T1 ,⋯, Tn ]], completions, etc. Bibliography: 13 items.

Finite automorphism groups of restricted formal power series rings

Finite automorphism groups of restricted formal power series rings

A Theorem on Henselian Rings

It is known that if K is a field, then the ring of formal power series in one or more variables, with coefficients in K, is Henselian at its maximal ideal. In this note we show that if R is a ring (commutative and with identity element) which is Henselian at the maximal ideals M1, M2, … then R[[x]] - the ring of formal power series in x with coefficients from R - is also Henselian at the maximal ideals M1 ⋅ R[[x]] + x⋅ R[[x]], etc.

Reduction of Singularities of the Differential Equation Ady = Bdx

Let k [ [X, Y]] be the ring of formal power series in two letters over an algebraically closed base field k of characteristic zero, let A, B C k [ [X, Y]] be elements without non-unit common divisor, and consider the differential equation A (x, y) dy B (x, y) dx. By a solution (at the origin) of the equation we mean an analytic branch (centered at the origin) which satisfies the equation; such a branch is represented by a pair of formal power series x cit + c2t2? + * * , y d,t + d2t2 + * , with x, y not both =0, such that A(x,y) (Vic.ti-1) -B(x,y) (Ejdjtj-1). Let r=min{subdA,subdB}. The number r is a measure of the complexity of the equation at the origin, and in particular the origin is called singular for Ady = Bdx if r ? 1. In Theorem 12 below we show how, after a finite number of transformations of the form X' = X, Y'Y/X, translations tacitly included, the solutions of Ady -Bdx can be made to go over into solutions of equations with r ? 1. In Theorems 8 and 10 the case r = 1 is analyzed; and we thus have, in one direction at any rate, a complete theory. In Theorem 13 we show that the set S of solutions can be divided into a finite number of subsets Si, S = U Si, such that any pair in a given Si are equisingular in the sense of [1]. The other theorems are for the most part preparatory. Undoubtedly some of them (in particular Theorems 2, 5, 7, 14) are known, but they are included for the sake of coherence.

Rings of formal power series over a Krull domain

Rings of formal power series over a Krull domain

Diophantine Problems Over Local Fields: III. Decidable Fields

In [3] we gave a complete set of elementary axioms for the valued field of p-adic numbers. In this paper we show how the valued field F((t)) of formal power series over a field F (of characteristic 0) may be similarly characterized, if a complete set of axioms for F is known. This enables us to answer affirmatively (for the characteristic 0 case) the question of A. Robinson [14] whether the elementary equivalence of two fields F and F' (written F _ F') implies the elementary equivalence of the formal power series rings F[[t]] and F'[[t]]. At the same time we obtain the result that if F is a decidable field of characteristic 0, then F((t)) and F[[t]] are decidable. In particular, the rings R[[t]] and C[[t]] of formal power series over the fields R of reals and C of complexes are decidable. This answers a question raised in R. Robinson [17]. The decidability of C[[t]] may be contrasted with the undecidability of the ring of entire functions over C proved in R. Robinson [17]. To offset partially the impression to which this might give rise, namely, that analysis is intrinsically more difficult than algebra, we mention the result proved in ? 2 that the ring of germs of analytic functions is elementarily equivalent to C[[t]], and hence is also decidable. In ? 1, we prove the algebraic results necessary for ? 2. It is first shown that ultraproducts of abelian groups have a certain analytic property called Z-completeness. As a corollary, we then identify denumerable ultraproducts of Z-groups of cardinality ? 2o0. The Z-completeness property also enables us to prove that ultraproducts of L-fields have cross-sections in the sense defined in [2]. This result allows us to apply Theorem 2 of [2] to show that any two L-fields of cardinality < 2Ro with elementarily equivalent residue class fields have isomorphic ultrapowers. The applications mentioned earlier are all consequences of this result. In ? 3 we show that, for each of the theories we have previously shown decidable, there is an elimination of quantifiers. We also give a different

A group ring method for finitely generated groups

It is an essential principle employed in this paper that properties of a group which has only a multiplication can be studied in a ring which has both additive and multiplicative operations. We also achieve, to a certain extent, a unified theory to explain some facts concerning finitely generated groups which are contained in the works of J. W. Alexander, W. Magnus, K. Reidemeister, and R. H. Fox. A group may be defined by generators and relations; i.e., it may be taken as a factor group F/R, where R is a normal subgroup of a free group F. H. Hopf [5] has shown that [F, F]/[F, R] is invariantly defined for F/R; to be explicit, F/R-F'/R' implies [F, F]/[F, R]-[F', F']/[F', R']. After some preliminaries about group rings in ?1, we proceed to establish a theory of presentations of groups and to generalize the above mentioned factor group in ?3. A family of invariantly defined quotient rings are obtained from the group ring of JF of F over J. In ?5, we extract invariants from these quotient rings by mapping each of them homomorphically into another ring of a suitable structure. The homomorphisms we use for this purpose are those induced by a homomorphism of JF into a noncommutative formal power series ring due to Magnus. A family of groups under our consideration for equivalence by isomorphism often have some common properties. For example, in the family of knot groups, each group made abelian is infinite cyclic. Therefore stronger invariants may be expected. Having this in mind, we develop the B-presentation theory in ?6. In order to derive invariants from the invariantly defined quotient rings just mentioned, Fox free differentiation becomes our main tool. Invariants resembling Alexander polynomials are produced; the computation for these invariants bears close relation with the Jacobian matrix theory in [4 ]. ?4 is mainly concerned with properties of noncommutative formal power series rings. The following notations will be universal in this paper: For elements u1, u2, of a group, [u1, u2] =ulu2u' 1uj, and [u1, * * , uP1]= [[u1, * u, up], up+i], p>2. Let A and B be subgroups of