Coefficient Ring Research Articles

Explicit factors of some iterated resultants and discriminants

In this paper, the result of applying iterative univariate resultant constructions to multivariate polynomials is analyzed. We consider the input polynomials as generic polynomials of a given degree and exhibit explicit decompositions into irreducible factors of several constructions involving two times iterated univariate resultants and discriminants over the integer universal ring of coefficients of the entry polynomials. Cases involving from two to four generic polynomials and resultants or discriminants in one of their variables are treated. The decompositions into irreducible factors we get are obtained by exploiting fundamental properties of the univariate resultants and discriminants and induction on the degree of the polynomials. As a consequence, each irreducible factor can be separately and explicitly computed in terms of a certain multivariate resultant. With this approach, we also obtain as direct corollaries some results conjectured by Collins and McCallum which correspond to the case of polynomials whose coefficients are themselves generic polynomials in other variables. Finally, a geometric interpretation of the algebraic factorization of the iterated discriminant of a single polynomial is detailled.

Generalized Artin and Brauer induction for compact Lie groups

Let G be a compact Lie group. We present two induction theorems for certain generalized G-equivariant cohomology theories. The theory applies to G-equivariant K-theory K G , and to the Borel cohomology associated with any complex oriented cohomology theory. The coefficient ring of K G is the representation ring R(G) of G. When G is a finite group the induction theorems for K G coincide with the classical Artin and Brauer induction theorems for R(G).

The flow and tension spaces and lattices of signed graphs

This paper is to introduce circuit, bond, flow, and tension spaces and lattices for signed graphs, and to study the relations among these spaces and lattices. The key ingredient is to introduce circuit and bond characteristic vectors so that the desired spaces and lattices can be defined such that their dimensions and ranks match well to that of matroids of signed graphs. The main results can be stated as follows: (1) the classification of minimal directed cuts; (2) the circuit space (lattice) equals flow space (lattice), and the bond space equals the tension space; (3) the bond lattice equals the row lattice of the incidence matrix, and the reduced bond lattice equals the tension lattice; and (4) for unbalanced signed graphs, the module of potentials is isomorphic to the module of tensions if the coefficient ring is 2-torsion free.

Rational BV-algebra in string topology

Let M be a 1-connected closed manifold of dimension m and LM be the space of free loops on M. M.Chas and D.Sullivan defined a structure of BV-algebra on the singular homology of LM, H * (LM;k). When the ring of coefficients is a field of characteristic zero, we prove that there exists a BV-algebra structure on the Hochschild cohomology HH * (C * (M);C * (M)) which extends the canonical structure of Gerstenhaber algebra. We construct then an isomorphism of BV-algebras between HH * (C * (M);C * (M)) and the shifted homology H *+m (LM;k). We also prove that the Chas-Sullivan product and the BV-operator behave well with a Hodge decomposition of H * (LM).

Commutativity and ideals in algebraic crossed products

We investigate properties of commutative subrings and ideals in non-commutative algebraic crossed products for actions by arbitrary groups. A description of the commutant of the coefficient subring in the crossed product ring is given. Conditions for commutativity and maximal commutativity of the commutant of the coefficient subring are provided in terms of the action as well as in terms of the intersection of ideals in the crossed product ring with the coefficient subring, specially taking into account both the case of coefficient rings without non-trivial zero-divisors and the case of coefficient rings with non-trivial zero-divisors.

Kernels, inflations, evaluations, and imprimitivity of Mackey functors

Let M be a Mackey functor for a finite group G. By the kernel of M we mean the largest normal subgroup N of G such that M can be inflated from a Mackey functor for G / N . We first study kernels of Mackey functors, and (relative) projectivity of inflated Mackey functors. For a normal subgroup N of G, denoting by P H , V G the projective cover of a simple Mackey functor for G of the form S H , V G we next try to answer the question: how are the Mackey functors P H / N , V G / N and P H , V G related? We then study imprimitive Mackey functors by which we mean Mackey functors for G induced from Mackey functors for proper subgroups of G. We obtain some results about imprimitive Mackey functors of the form P H , V G , including a Mackey functor version of Fong's theorem on induced modules of modular group algebras of p-solvable groups. Aiming to characterize subgroups H of G for which the module P H , V G ( H ) is the projective cover of the simple K N ¯ G ( H ) -module V where the coefficient ring K is a field, we finally study evaluations of Mackey functors.

Global coefficient ring in the Nilpotence Conjecture

In this note we show that the nilpotence conjecture for toric varieties is true over any regular coefficient ring containing Q.

Khovanov homology for virtual knots with arbitrary coefficients

The Khovanov homology theory over an arbitrary coefficient ring is extended to the case of virtual knots. We introduce a complex which is well-defined in the virtual case and is homotopy equivalent to the original Khovanov complex in the classical case. Unlike Khovanov's original construction, our definition of the complex does not use any additional prescription of signs to the edges of a cube. Moreover, our method enables us to construct a Khovanov homology theory for `twisted virtual knots' in the sense of Bourgoin and Viro (including knots in three-dimensional projective space). We generalize a number of results of Khovanov homology theory (the Wehrli complex, minimality problems, Frobenius extensions) to virtual knots with non-orientable atoms.

On the Farrell-Jones Conjecture and its applications

We present the status of the Farrell–Jones Conjecture for algebraic K-theory for a group G and arbitrary coefficient rings R. We add new groups for which the conjecture is known to be true, and we study inheritance properties. We discuss new applications, focussing on the Bass Conjecture, the Kaplansky Conjecture, and conjectures generalizing Moody’s Induction Theorem. Thus, we considerably extend the class of groups for which these conjectures are known.

Morphic and principal-ideal group rings

For R an artinian ring and G a group, we characterize when RG is a principal ideal ring. In the case when G is finite (and R artinian), this yields a characterization of when RG is a left and right morphic ring. This extends work done by Passman, Sehgal and Fisher on principal ideal group rings when the coefficient ring is a field, and work of Chen, Li, and Zhou on morphic group rings.

HECKE OPERATORS ON HILBERT–SIEGEL MODULAR FORMS

We define Hilbert–Siegel modular forms and Hecke "operators" acting on them. As with Hilbert modular forms (i.e. with Siegel degree 1), these linear transformations are not linear operators until we consider a direct product of spaces of modular forms (with varying groups), modulo natural identifications we can make between certain spaces. With Hilbert–Siegel forms (i.e. with arbitrary Siegel degree) we identify several families of natural identifications between certain spaces of modular forms. We associate the Fourier coefficients of a form in our product space to even integral lattices, independent of basis and choice of coefficient rings. We then determine the action of the Hecke operators on these Fourier coefficients, paralleling the result of Hafner and Walling for Siegel modular forms (where the number field is the field of rationals).

Factoring the adjoint and maximal Cohen-Macaulay modules over the generic determinant

A question of Bergman asks whether the adjoint of the generic square matrix over a field can be factored nontrivially as a product of square matrices. We show that such factorizations indeed exist over any coefficient ring when the matrix has even size. Establishing a correspondence between such factorizations and extensions of maximal Cohen–Macaulay modules over the generic determinant, we exhibit all factorizations where one of the factors has determinant equal to the generic determinant. The classification shows not only that the Cohen–Macaulay representation theory of the generic determinant is wild in the tame-wild dichotomy, but that it is quite wild: even in rank two, the isomorphism classes cannot be parametrized by a finite-dimensional variety over the coefficients. We further relate the factorization problem to the multiplicative structure of the Ext–algebra of the two nontrivial rank-one maximal Cohen–Macaulay modules and determine it completely.

The Fast Fourier Transform

Fast Fourier transforms (FFTs) are fast algorithms, i.e., of low complexity, for the computation of the discrete Fourier transform (DFT) on a finite abelian group. They are among the most important algorithms in applied and engineering mathematics and in computer science, in particular for one‐ and multidimensional systems theory and signal processing. We give a relatively short survey of the FFT for arbitrary finite abelian groups, cyclic or not, with complete and partially novel proofs, the main distinction being explicit induction formulas for the FFT in all cases which generalize the original FFT‐algorithm due to Cooley and Tukey and, much earlier, to Gauß. We believe that our approach has didactic advantages over the usual ones. We also present the application of the FFT to fast convolution algorithms, and the so‐called number theoretic transforms over finite coefficient rings. We do not treat those algorithms which decrease the multiplicative complexity at the expense of many more rational linear combinations, which in this context are considered costless, nor do we discuss the DFT for nonabelian finite groups.

Primeness, semiprimeness and localisation in Iwasawa algebras

Necessary and sufficient conditions are given for the completed group algebras of a compactpp-adic analytic group with coefficient ring thepp-adic integers or the field ofppelements to be prime, semiprime and a domain. Necessary and sufficient conditions are found for the localisation at semiprime ideals related to the augmentation ideals of closed normal subgroups. Some information is obtained about the Krull and global dimensions of the localisations. The results extend and complete work of A. Neumann and J. Coateset al.

On Bezout and distributive generalized power series rings

In this paper we give sufficient and necessary conditions on a strongly regular ring of coefficients R and a monoid of nonnegative exponents S such that the generalized power series ring R〚S〛 is right Bezout. It is shown that all such generalized power series rings are right distributive. We also study when a generalized power series ring over a von Neumann regular ring has weak dimension less than or equal to one.

Realizability of algebraic Galois extensions by strictly commutative ring spectra

We discuss some of the basic ideas of Galois theory for commutative S-algebras originally formulated by John Rognes. We restrict our attention to the case of finite Galois groups and to global Galois extensions. We describe parts of the general framework developed by Rognes. Central roles are played by the notion of strong duality and a trace mapping constructed by Greenlees and May in the context of generalized Tate cohomology. We give some examples where algebraic data on coefficient rings ensures strong topological consequences. We consider the issue of passage from algebraic Galois extensions to topological ones by applying obstruction theories of Robinson and Goerss-Hopkins to produce topological models for algebraic Galois extensions and the necessary morphisms of commutative S-algebras. Examples such as the complex $ K-theory spectrum as a KO-algebra indicate that more exotic phenomena occur in the topological setting. We show how in certain cases topological abelian Galois extensions are classified by the same Harrison groups as algebraic ones, and this leads to computable Harrison groups for such spectra. We end by proving an analogue of Hilbert's theorem 90 for the units associated with a Galois extension.

Automorphisms of a chain ring

Let A be a chain ring that is a faithful algebra over a commutative chain ring R, such that \(\overline{A} = A/J(A)\) is a separable, normal, algebraic field extension of \(\overline{R} = R/J(R)\) and \(\overline{A}\) is countably generated over \(\overline{R}\). It has been recently proved by Alkhamees and Singh that A has a coefficient ring R0, and there exists a pair (θ, σ) with θ ∈ A, σ an R-automorphism of R0 such that J (A) = θ A = Aθ, and θa = σ (a) θ, a ∈ R0. The question of the extension of certain R-automorphisms of R0 to R-automorphisms of A is investigated.

Separable Groupoid Rings

We show that groupoid rings are separable over their ring of coefficients if and only if the groupoid is finite and the orders of the associated principal groups are invertible in the ring of coefficients. We use this to show that if we are given a finite groupoid, then the associated groupoid ring is semisimple (or hereditary) if and only if the ring of coefficients is semisimple (or hereditary) and the orders of the principal groups are invertible in the ring of coefficients. To this end, we extend parts of the theory of graded rings and modules from the group graded case to the category graded, and, hence, groupoid graded situation. In particular, we show that strongly groupoid graded rings are separable over their principal components if and only if the image of the trace map contains the identity.

ORE EXTENSIONS SATISFYING A POLYNOMIAL IDENTITY

Necessary and sufficient conditions for an Ore extension S = R[x;σ,δ] to be a PI ring are given in the case σ is an injective endomorphism of a semiprime ring R satisfying the ACC on annihilators. Also, for an arbitrary endomorphism τ of R, a characterization of Ore extensions R[x;τ] which are PI rings is given, provided the coefficient ring R is noetherian.

The joint weight enumerators and Siegel modular forms

The weight enumerator of a binary doubly even self-dual code is an isobaric polynomial in the two generators of the ring of invariants of a certain group of order 192. The aim of this note is to study the ring of coefficients of that polynomial, both for standard and joint weight enumerators.