Coefficient Ring Research Articles

On the monodromy of complex polynomials

Consider a polynomial function f : ℂn→ℂ with generic fiber F. Let Bf be the bifurcation set of f; hence f induces a smooth locally trivial fibration over ℂ\Bf. Then, for any integer q≥0 and any coefficient ring R, there is an associated monodromy representation $\rho(f)_q : \pi_1(\mathbb {C}\backslash B_f,{\rm pt})\to {\rm Aut}(\tilde {H}_q(F,R))$ in (reduced) homology. Going around a circle in ℂ large enough to contain all of the bifurcation set gives rise to the monodromy operators at infinity, which we denote by M∞(f)q. We show that these monodromy operators at infinity and a certain natural direct sum decomposition of the homology of F in terms of vanishing cycles determine the monodromy representation. The role played by this decomposition is crucial since there are examples of polynomials ℂ2→ℂ having distinct complex monodromy representations but whose monodromy operators at infinity have the same Jordan normal form.

Normed units in abelian group rings

We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas ( J. London Math. Soc. , 1991) and Mollov ( Publ. Math. Debrecen , 1971).

The Green Correspondence for Infinitely Generated Modules

If G is a finite group, then the usual version of the Green correspondence applies to finitely generated kG -modules when k is a field of characteristic p > 0 or a p -adic ring. The paper presents a categorical version of the Green correspondence and a version of the Burry–Carlson–Puig theorem that remain valid for arbitrary modules and coefficient rings. In this generality, however, it is not clear whether the Green correspondent of a finitely generated module is always finitely generated.

The Donald–Flanigan Problem for Finite Reflection Groups

The Donald–Flanigan problem for a finite group H and coefficient ring k asks for a deformation of the group algebra kH to a separable algebra. It is solved here for dihedral groups and Weyl groups of types Bn and Dn (whose rational group algebras are computed), leaving but six finite reflection groups with solutions unknown. We determine the structure of a wreath product of a group with a sum of central separable algebras and show that if there is a solution for H over k which is a sum of central separable algebras and if Sn is the symmetric group then (i) the problem is solvable also for the wreath product H\(\wr\)Sn = H × ··· × H (n times) ⋊ Sn and (ii) given a morphism from a finite Abelian or dihedral group G to Sn it is solvable also for H\(\wr\)G. The theorems suggested by the Donald–Flanigan conjecture and subsequently proven follow, we also show, from a geometric conjecture which although weaker for groups applies to a broader class of algebras than group algebras.

Linear and polylinear recurring sequences over abelian groups and modules

In this paper, the well-known results in the theory of polylinear recurring sequences over fields and their recent generalizations for sequences over modules with commutative rings of coefficients are extended to the class of polylinear sequences over modules with noncommutative rings of coefficients. Possible noncommutativity of the main ring of coefficients requires naturally a study of polylinear recurring sequences over bimodules. To estimate the linear complexity of the considered sequences, we introduce and study polylinear (k-linear) shift registers. A criterion for the theory of polylinear recurring sequences over fields to be adequately generalized in our case is the property of the main bimodule to be quasi-Frobenius with the Artinian (respectively, from the left and the right sides) rings of coefficients (i.e., so-called Artinian duality context).

Periodic flat modules, and flat modules for finite groups

If R is a ring of coefficients and G a finite group, then a flat RG-module which is projective as an R-module is necessarily projective as an RG-module. More generally, if H is a subgroup of finite index in an arbitrary group Γ, then a flat RΓmodule which is projective as an RH-module is necessarily projective as an RΓ-module. This follows from a generalization of the first theorem to modules over strongly G-graded rings. These results are proved using the following theorem about flat modules over an arbitrary ring S: If a flat S-module M sits in a short exact sequence 0 → M → P → M → 0 with P projective, then M is projective. Some other properties of flat and projective modules over group rings of finite groups, involving reduction modulo primes, are also proved.

Cohomology of discrete groups and their finite subgroups

We investigate the cohomology of a group having finite virtual cohomological dimension in terms of the contributions from finite subgroups. As a result, we prove a variant of Quillen's F-isomorphism theorem which remains valid for an arbitrary commutative ring of coefficients and for suitable families of finite subgroups.

Intermediate rings between matrix rings and Ornstein dual pairs

We show that isomorphism of intermediate rings between row and column finite matrix rings and row finite matrix rings implies Morita equivalence of the coefficient rings and equality of the cardinality of the set of indices. Among the applications we extend the Isomorphism Theorem for Dual Pairs over Division Rings to Ornstein dual pairs over any class of rings for which Morita equivalence implies isomorphism.

Thom Spectra of Generalized Braid Groups

It is proved that Thom spectra of generalized braid groups are the wedges of suspensions over the Eilenberg–MacLane spectrum for ℤ/2. The precise structure of the Thom spectra of the generalized braid groups of the types C and D is obtained. For the generalized braid groups of type C the natural pairing analogous to the pairing of the classical braids is studied. This pairing generates the multiplicative structure of the Thom spectrum such that the corresponding bordism theory has the coefficient ring isomorphic to the polynomial ring over ℤ/2 on one generator of dimension 1[ratio ]ℤ/2[ s ].

The Picard group of a structural matrix algebra

We investigate the Picard group of a structural matrix (or incidence) algebra A of a finite preordered set P over a field and we consider five interrelated problems. Our main technique involves establishing a connection between the group of outer automorphisms Out( A) of A and the group of outer automorphisms of the basic algebra A ̃ which is an incidence algebra of the associated partially ordered set P ̃ of P. We discuss necessary and sufficient conditions for Out( A) to be a natural invariant for the Morita equivalence class of A, and necessary and sufficient conditions for M n ( K) to be strongly graded by a group G and coefficient ring A containing n primitive, orthogonal idempotents.

Skew Group Rings which are Semi-Hereditary Orders and Prüfer Orders in Simple Artinian Rings

Let R be a commutative ring, let G be a finite group acting on R as automorphisms of R and let R * G be the skew group ring. By using the decomposition subgroups of G, the inertial subgroups of G, the properties of the coefficient ring R and the properties of the fixed subring RG, some necessary and sufficient conditions for R * G to be a prime Goldie ring, a semi-hereditary order in a simple Artinian ring, or a Prufer order in a simple Artinian ring are given.

Actions of Picard Groups on Graded Rings

Throughout this paper, G denotes a group with identity e. If R is a G-graded ring, we write R = ⊕g∈GRg and we refer to Re as the coefficient ring. We denote the category of graded (left) R-modules (i.e., those left R-modules M with a G-grading M = ⊕g∈GMg such that RhMg ⊆ Mhg) by R-gr. The aim of this paper is to present a method to reduce the study of one strongly graded ring R to the study of another strongly graded ring R′ that is more tractable. This reduction process has two aspects: reducing the coefficient ring or reducing the grading. Since we apply this method in our study of module-theoretic properties such as semisimplicity for graded rings, we require that this reduction process preserve the category of modules R-mod and the category of graded modules R-gr. This leads us to the notion of graded equivalence. In this paper, a Morita context is the usual tuple (A,B,P,Q, τ, μ) where we assume τ and μ are isomorphisms. A G-graded context is a Morita context (R,R′, P,Q, τ, μ), where R and R′ are G-graded rings, RPR′ and R′QR are graded bimodules (i.e., they have a grading which makes them into both graded R-modules and graded R′-modules on the corresponding side) and τ and μ are graded bimodule homomorphisms. See [14] for details about graded modules and homomorphisms. Given a functor F : R-gr → R′-gr, we say that F is a graded equivalence of categories if any of the following equivalent conditions hold (see for example [8]):

On the uniqueness of rings of coefficients in skew polynomial rings

On the uniqueness of rings of coefficients in skew polynomial rings

Strong convergence properties of sft rings

This paper investigates a strong convergence property of rings of fbrmal power series where the coefficient ring has the strong finite type (SFT) property. In particular, we show that if R[[x]] is SFT, then given two primes whose variable annihilation ideals agree, then x∈ Γ1.We also show that this strong convergence property completely characterizes Noetherian rings.

Flat Modules over Group Rings of Finite Groups

Let k be a commutative ring of coefficients and G be a finite group. Does there exist a flat k G-module which is projective as a k-module but not as a k G-module? We relate this question to the question of existence of a k-module which is flat and periodic but not projective. For either question to have a positive answer, it is at least necessary to have |k| ≥ ℵω. There can be no such example if k is Noetherian of finite Krull dimension, or if k is perfect.

A Realization Algorithm for SL2(R[x1 , . . . ,xm]) over the Euclidean Domain

Let R be an arbitrary Euclidean domain and A be an n x n matrix of determinant 1 whose entries are elements of R. Gaussian elimination process based on Euclidean division algorithm (on R) allows one to write A as a product of elementary matrices over R. Suslin's stability theorem states that any n x n multivariate polynomial matrix of determinant 1 with $n\geq 3$ can be written as a product of elementary matrices. This result fails when n=2, and a counter-example was constructed by P. M. Cohn [ Inst. Hautes Études Sci. Publ. Math. 30 (1996), pp. 365--413]. In this paper, an algorithm is developed that determines precisely when a given matrix in SL2(R[x1 , . . . ,xm]) allows a factorization into elementary matrices and, if it does, expresses it as a product of elementary matrices. This algorithm has potential applications to signal processing, in which case the coefficient ring R is usually taken as a field or a ring $\Bbb Z$ of integers. Extending the idea used in this algorithm, we disprove a conjecture of Tolhuizen--Hollmann--Kalker regarding the realizability of a certain FIR filter.

Augmentation ideals of equivariant cohomology rings

The purpose of this note is to establish a number of useful results about the augmentation ideal J for the coefficient ring F ∗ G of a Noetherian complex stable equivariant cohomology theory. The results show that various naturally occurring substitutes for the ideal have the same radical, and can therefore be used instead of the augmentation ideal in all geometric constructions.

Tate cohomology of theories with one-dimensional coefficient ring

For a finite group G we calculate the E-Tate cohomology t(E) G ∗ and the E-homology E ∗(BG +) as functors of the augmented commutative ring E ∗(BG +) when E ∗(.) is a complex oriented, v n -periodic cohomology theory with one-dimensional graded coefficient ring E ∗ .

Dubrovin valuation properties of skew group rings and crossed products

In this paper some conditions for a skew group ring or a crossed product to have finite weak global dimension are given.Using these results we obtain some necessary conditions and some sufficient conditions for a skew group ring or a crossed product to be a Dubrovin valuation ring.If R*G is a skew group ring, where the coefficient ring R is a commutative ring and G is a finite group, then we prove that the conditions we obtained become necessary and sufficient conditions.In particular, if R is a commutative valuation ring, then R*G is a Dubrovin valuation ring if and only if G T=<1>,where G T is the inertial group of R.

Localization and Completion Theorems for MU-Module Spectra

Let G be a finite extension of a torus. Working with highly structured ring and module spectra, let M be any module over MU ; examples include all of the standard homotopical MU -modules, such as the Brown-Peterson and Morava K-theory spectra. We shall prove localization and completion theorems for the computation of M∗(BG) and M∗(BG). The G-spectrum MUG that represents stabilized equivariant complex cobordism is an algebra over the equivariant sphere spectrum SG, and there is an MUG-module MG whose underlying MU -module is M . This allows the use of topological analogues of constructions in commutative algebra. The computation ofM∗(BG) andM∗(BG) is expressed in terms of spectral sequences whose respective E2 terms are computable in terms of local cohomology and local homology groups that are constructed from the coefficient ring MU ∗ and its module M ∗ . The central feature of the proof is a new norm map in equivariant stable homotopy theory, the construction of which involves the new concept of a global I∗-functor with smash product.