Ring Of Formal Power Series Research Articles

Non-Cohen-Macaulay symbolic blow-ups for space monomial curves

Let p = p ( n 1 , n 2 , n 3 ) \mathfrak {p} = \mathfrak {p}({n_1},{n_2},{n_3}) denote the prime ideal in the formal power series ring A = k [ [ X , Y , Z ] ] A = k[[X,Y,Z]] over a field k k defining the space monomial curve X = T n 1 , Y = T n 2 X = {T^{{n_1}}},Y = {T^{{n_2}}} , and Z = T n 3 Z = {T^{{n_3}}} with GCD ⁡ ( n 1 , n 2 , n 3 ) = 1 \operatorname {GCD} ({n_1},{n_2},{n_3}) = 1 . Then the symbolic Rees algebra R s ( p ) = ⊕ n ≥ 0 p ( n ) {R_s}(\mathfrak {p}) = { \oplus _{n \geq 0}}{\mathfrak {p}^{(n)}} for p = p ( n 2 + 2 n + 2 , n 2 + 2 n + 1 , n 2 + n + 1 ) \mathfrak {p} = \mathfrak {p}({n^2} + 2n + 2,{n^2} + 2n + 1,{n^2} + n + 1) is Noetherian but not Cohen-Macaulay if ch k = p > 0 {\text {ch}}k = p > 0 and n = p e n = {p^e} with e ≥ 1 e \geq 1 . The same is true for p = p ( n 2 , n 2 + 1 , n 2 + n + 1 ) \mathfrak {p} = \mathfrak {p}({n^2},{n^2} + 1,{n^2} + n + 1) if ch k = p > 0 {\text {ch}}k = p > 0 and n = p e ≥ 3 n = {p^e} \geq 3 .

Universal deformations of simple Lie algebras

This Letter examines the question of the structure of the Hopf algebra deformations of the universal enveloping algebras of the simple Lie algebras. Deformations of a complex algebra A are viewed as algebras defined over formal power series rings that specialize to A when the parameters go to 0. Only the case of U(sl(2,C)) is treated but the methods are general. Under the Ansatz that the two Borel subalgebras are deformed as Hopf algebras but possibly differently, we construct a universal two-parameter deformation.

On the maximal condition in formal power series rings

On the maximal condition in formal power series rings

Topics on symbolic Rees algebras for space monomial curves

Let A be a regular local ring of dim A = 3 and p a prime ideal in A of dim A/p = 1. We put Rs(p) = (here t denotes an indeterminate over A) and call it the symbolic Rees algebra of p. With this notation the authors [5, 6] investigated the condition under which the A-algebra Rs(p) is Cohen-Macaulay and gave a criterion for Rs(p) to be a Gorenstein ring in terms of the elements f and g in Huneke’s condition [11, Theorem 3.1] of Rs(p) being Noetherian. They furthermore explored the prime ideals p = p(n1, n2, n3) in the formal power series ring A = k[X, Y, Z] over a field k defining space monomial curves and Z = with GCD(n1, n2, nz) = 1 and proved that Rs(p) are Gorenstein rings for certain prime ideals p = p(n1 n2, n3). In the present research, similarly as in [5, 6], we are interested in the ring-theoretic properties of Rs(p) mainly for p = p(n1 n2) nz) and the results of [5, 6] will play key roles in this paper.

Normal matrices over hermitian discrete valuation rings

A generalization of the classical spectral theorem for normal complex matrices is proved for matrices with entries from a class of discrete valuation rings which we have called hermitian. The hermitian discrete valuation rings include the convergent complex power series in one variable and the complex formal power series in one variable. Thus one obtains an algebraic proof of Rellich's theorem on diagnolizability of hermitian analytic matrices which is simultaneously valid for both the rings of convergent and formal power series.

La queste del saint Gr a(AL): A computational approach to local algebra

We show how, by means of the Tangent Cone Algorithm, the basic functions related to the maximal ideal topology of a local ring can be effectively computed in the situations of geometrical significance, i.e.: (1) localizations of coordinate rings of a variety at the prime ideal defining a subvariety, (2) rings of algebraic formal power series rings. In particular we show how the method of “associated graded rings” can be turned into an effective tool to compute local algebraic invariants of varieties.

Exact and coexact matrices

(1) f: R” + R” is a C” map. Here k is the real numbers R and R is the ring of C” maps from R” to R. (2) f: C” + C” is an analytic map. Here k is the complex numbers C and R is the ring of analytic maps from C” to C. (3) f: k” -+ k” is a polynomial map. Here k is a commutative ring and R is the polynomial ring kc”‘. (4) f:k[[I',k[[nll is a k-algebra homomorphism. Here k is a commutative ring and R is the formal power series ring kccnl’.

The relative form of Gersten’s conjecture for power series over a complete discrete valuation ring

A relative form of Gersten’s Conjecture is established for a ring of formal power series over a complete discrete valuation ring. The main corollaries are that the absolute version of Gersten’s Conjecture is valid for such a ring if it is valid for arbitrary discrete valuation rings, and, consequently, that the conjecture is true for such a ring if we use K K -theory with finite coefficients of order prime to the characteristic of the residue field.

The Burnside ring of the infinite cyclic group and its relations to the necklace algebra, λ-rings, and the universal ring of Witt vectors

It is shown that well-known product decompositions of formal power series arise from combinatorially defined canonical isomorphisms between the Burnside ring of the infinite cyclic group on the one hand and Grothendieck's ring of formal power series with constant term 1 as well as the universal ring of Witt vectors on the other hand.

OnF-integrable actions of the restricted Lie algebra of a formal groupFin characteristicp> 0

Letkbe an integral domain, letF= (F1X, Y),…,Fn(X, Y)),X= (X1,…,Xn),Y= (Y1,…,Yn), be ann-dimensional formal group overk, and letL(F) be the Lie algebra of allF-invariantk-derivations of the ring of formal power seriesk[X] (cf. § 2). If A is a (commutative)k-algebra and Derk(A) denotes the Lie algebra of allk-derivationsd:A→A, then by an action ofL(F) onAwe mean a morphism of Lie algebrasφ:L(F) → Derk(A) such thatφ(dp) = φ(d)p, provided char (k) =p> 0. An action of the formal groupFonAis a morphism ofk-algebrasD:A-→A[X] such thatD(a)≡a mod (X) fora ∊ A, andFA∘D=DY∘D, whereFA:A[X] →A[X, Y],DF:A[X] →A[X, Y] are morphisms ofk-algebras given byFA(g(X))=g(F), DY(ΣaaaXa) =ΣaD(aa)Y;for a motivation of this notion, see [15]. LetD:A→A[X] be such an action.

Differential Completions and Differentially Simple Algebras

AbstractDifferentially simple local noetherian Q -algebras are shown to be always (a certain type of) subrings of formal power series rings. The result is established as an illustration of a general theory of differential filtrations and differential completions.

Über ein Problem von J. Schwaiger und Z. Moszner betreffend die Vertauschbarkeit von einparametrigen Automorphismengruppen formaler Potenzreihenringe

Let ℂ〚x〛 be the ring of formal power series in one indeterminate over ℂ, and let Γ be the group of automorphisms of ℂ〚x〛 which are continuous in the order topology and leave ℂ elementwise fixed. Assume that(F t ) t ∈ ℂ and(G t ) t ∈ ℂ are iteration groups, i.e. one-parameter subgroups of Γ which are solutions of the translation equationF t ∘F s =F t + s ,G t ∘G s =G t + s . Suppose moreover that the following weak commutativity condition holds: (1) $$F_t \circ G_t = G_t \circ F_t forallt \in \mathbb{C}.$$ Does (1) imply the stronger condition (2) $$F_t \circ G_s = G_s \circ F_t foralls,t \in \mathbb{C}?$$ (This problem had been posed by J. Schwaiger. Similar problems for homomorphisms of (ℂ, + ) into groups of matrices have been dealt with by Z. Moszner and Z. Leszczynska.) We give an affirmative answer to this question by characterizing all pairs (Ft)t∈ℂ, (Gt)t∈ℂ of iteration groups which satisfy (1). For such pairs of iteration groups exactly two cases occur: We do not assume that the iteration groups under consideration are analytic. Indeed, no assumption on the regularity of the dependence on the group parameter is made.

Algebraic elements in formal power series rings II

We obtain explicit formulae for degrees on diagonals, Hadamard products and Lamperti products.

When is a regular local ring a locality?

Let A be a regular local ring and let R be a polynomial extension of A in a finite number of variables. Several attempts have been made to study the structure of projective modules over R and to find the minimal numbers of generators of ideals of R; to wit, [Z, V.4; 3; 61. The methods employed in these references depend very much on the structure of A; the cases dealt with are formal power series rings over fields and local rings at smooth points of algebraic varieties. Cohen’s structure theorem tells us precisely what regular local rings are formal power series rings. We present here the following characterization for a regular local ring to be a locality over a field :

Algebraic elements in formal power series rings

Algebraic elements in formal power series rings

Roots, iterations and logarithms of formal automorphisms

In this paper it is proved that having a logarithm is equivalent to having roots of arbitrary order in the group of automorphisms of a formal power series ring, and in algebraic subgroups too

Coherence of power series rings over pseudo-Bezout domains

We provide a large class of coherent domains whose rings of formal power series are not coherent, by proving that if R is a pseudo-Bezout domain and R[[ X]] is coherent, then R is completely integrally closed; this generalizes a theorem of Jøndrup and Small's for valuation domains. We also obtain a large class of completely integrally closed pseudo-Bezout domains R for which R[[ X]] is not coherent; in particular, if R is a rank one valuation domain whose group of divisibility is a proper dense subgroup of the reals, then R[[ X]] is not coherent.

Unified proofs of Hilbert's basis theorem and its analogue in formal power series rings

Let R denote a commutative Noetherian ring with an identity element. Hilbert's basis theorem states that the polynomial ring in a finite number of indeterminates over R is also Noetherian. (See Northcott ], theorem 8, p. 26; Zariski and Samuel [4], theorem 1, p. 201). Hilbert's original theorem in [2] is stated for the case when R is a field or the ring of integers. The standard proofs of this fundamental theorem are essentially of a direct type. The analogue of Hilbert's basis theorem in the ring of formal power series in a finite number of indeterminates over R is also true (Chevalley [1]; see also Northcott [3], theorem 3, p. 89; Zariski and Samuel [5], theorem 4, p. 138). In the present note we bring together concise proofs of Hilbert's and Chevalley's theorems by using a single indirect (i.e. reductio ad absurdum) argument. By induction it is sufficient to consider the case of a polynomial ring S and a formal power series ring T in a single indeterminate X over R.

Iterations and logarithms of formal automorphisms

Using the decomposition of an automorphism of the ring of formal power series in several variables, in a semisimple and a unipotent automorphism, I prove in this paper that an automorphism allows a continuous iteration if and only if it is the exponential of a derivation. This result implies a number of results recently obtained by Reich, Schwaiger, and Bucher.

Joseph Becker and Leonard Lipshitz. Remarks on the elementary theories of formal and convergent power series. Fundament a mathematicae, vol. 105 (1980), pp. 229–239. - Françoise Delon. Indécidabilité de la théorie des anneaux de séries formelles à plusiers indéterminées. Fundament a mathematicae, vol. 112 (1981), pp. 215–229. - J. Becker, J. Denef, and L. Lipshitz. Further remarks on the elementary

Joseph Becker and Leonard Lipshitz. Remarks on the elementary theories of formal and convergent power series. Fundament a mathematicae, vol. 105 (1980), pp. 229–239. - Françoise Delon. Indécidabilité de la théorie des anneaux de séries formelles à plusiers indéterminées. Fundament a mathematicae, vol. 112 (1981), pp. 215–229. - J. Becker, J. Denef, and L. Lipshitz. Further remarks on the elementary