Coefficient Ring Research Articles

Symmetry and reversibility properties for quantum algebras and skew Poincaré-Birkhoff-Witt extensions

Our aim in this paper is to investigate symmetry and reversibility pro-perties for quantum algebras and skew PBW extensions. Under certainconditions we prove that these properties transfer from a ring of coeffi-cients to a quantum algebra or skew PBW extension over this ring. In thisway we generalize several results established in the literature and consideralgebras which have not been studied before. We illustrate our results withremarkable examples of theoretical physics

Comparisons between singularity categories and relative stable categories of finite groups

We consider the relationship between the relative stable category of and the usual singularity category for group algebras with coefficients in a commutative noetherian ring. When the coefficient ring is self-injective we show that these categories share a common, relatively large, Verdier quotient. At the other extreme, when the coefficient ring has finite global dimension, there is a semi-orthogonal decomposition, due to Poulton, relating the two categories. We prove that this decomposition is partially compatible with the monoidal structure and study the morphism it induces on spectra.

CHAIN CONDITIONS ON ÉTALE GROUPOID ALGEBRAS WITH APPLICATIONS TO LEAVITT PATH ALGEBRAS AND INVERSE SEMIGROUP ALGEBRAS

The author has previously associated to each commutative ring with unit$R$and étale groupoid$\mathscr{G}$with locally compact, Hausdorff and totally disconnected unit space an$R$-algebra$R\,\mathscr{G}$. In this paper we characterize when$R\,\mathscr{G}$is Noetherian and when it is Artinian. As corollaries, we extend the characterization of Abrams, Aranda Pino and Siles Molina of finite-dimensional and of Noetherian Leavitt path algebras over a field to arbitrary commutative coefficient rings and we recover the characterization of Okniński of Noetherian inverse semigroup algebras and of Zelmanov of Artinian inverse semigroup algebras.

The spectrum for commutative complex K–theory

We study commutative complex $K$-theory, a generalised cohomology theory built from spaces of ordered commuting tuples in the unitary groups. We show that the spectrum for commutative complex $K$-theory is stably equivalent to the $ku$-group ring of $BU(1)$ and thus obtain a splitting of its representing space $B_{com}U$ as a product of all the terms in the Whitehead tower for $BU$, $B_{com}U\simeq BU\times BU\langle 4\rangle \times BU\langle 6\rangle \times \dots .$ As a consequence of the spectrum level identification we obtain the ring of coefficients for this theory. Using the rational Hopf ring for $B_{com}U$ we describe the relationship of our results with a previous computation of the rational cohomology algebra of $B_{com}U$. This gives an essentially complete description of the space $B_{com}U$ introduced by A. Adem and J. G\'omez.

Poincaré duality with cap products in intersection homology

For having a Poincaré duality via a cap product between the intersection homology of a paracompact oriented pseudomanifold and the cohomology given by the dual complex, G. Friedman and J. E. McClure need a coefficient field or an additional hypothesis on the torsion. In this work, by using the classical geometric process of blowing-up, adapted to a simplicial setting, we build a cochain complex which gives a Poincaré duality via a cap product with intersection homology, for any commutative ring of coefficients. We prove also the topological invariance of the blown-up intersection cohomology with compact supports in the case of a paracompact pseudomanifold with no codimension one strata.This work is written with general perversities, defined on each stratum and not only in function of the codimension of strata. It contains also a tame intersection homology, suitable for large perversities.

Strictly commutative complex orientation theory

For a multiplicative cohomology theory E, complex orientations are in bijective correspondence with multiplicative natural transformations to E from complex bordism cohomology MU. If E is represented by a spectrum with a highly structured multiplication, we give an iterative process for lifting an orientation $$MU \rightarrow E$$ to a map respecting this extra structure, based on work of Arone–Lesh. The space of strictly commutative orientations is the limit of an inverse tower of spaces parametrizing partial lifts; stage 1 corresponds to ordinary complex orientations, and lifting from stage $$(m-1)$$ to stage m is governed by the existence of an orientation for a family of E-modules over a fixed base space $$F_m$$ . When E is p-local, we can say more. We find that this tower only changes when m is a power of p, and if E is E(n)-local the tower is constant after stage $$p^n$$ . Moreover, if the coefficient ring $$E^*$$ is p-torsion free, the ability to lift from stage 1 to stage p is equivalent to a condition on the associated formal group law that was shown necessary by Ando.

Wedge operations and doubling operations of real toric manifolds

This paper deals with two things. First, the cohomology of canonical extensions of real topological toric manifolds is computed when coefficient ring G is a commutative ring in which 2 is unit in G. Second, the author focuses on a specific canonical extensions called doublings and presents their various properties. They include existence of infinitely many real topological toric manifolds admitting complex structures, and a way to construct infinitely many real toric manifolds which have an odd torsion in their cohomology groups. Moreover, some questions about real topological toric manifolds related to Halperin’s toral rank conjecture are presented.

Codes over affine algebras with a finite commutative chain coefficient ring

We consider codes defined over an affine algebra A=R[X1,…,Xr]/〈t1(X1),…,tr(Xr)〉, where ti(Xi) is a monic univariate polynomial over a finite commutative chain ring R. Namely, we study the A−submodules of Al (l∈N). These codes generalize both the codes over finite quotients of polynomial rings and the multivariable codes over finite chain rings. Some codes over Frobenius local rings that are not chain rings are also of this type. A canonical generator matrix for these codes is introduced with the help of the Canonical Generating System. Duality of the codes is also considered.

Derived Galois deformation rings

We define a derived version of Mazur's Galois deformation ring. It is a pro-simplicial ring R classifying deformations of a fixed Galois representation to simplicial coefficient rings; its zeroth homotopy group π0R recovers Mazur's deformation ring.We give evidence that these rings R occur in the wild: For suitable Galois representations, the Langlands program predicts that π0R should act on the homology of an arithmetic group. We explain how the Taylor–Wiles method can be used to upgrade such an action to a graded action of π⁎R on the homology.

Note on the injectivity of the Loday assembly map

We show that the Loday assembly map for the algebraic K-theory for a group algebra of a finite group with coefficient ring a finite field is in general not injective.

Simulations of Polarimetric, X-Band Radar Signatures in Supercells. Part I: Description of Experiment and Simulated ρhv Rings

AbstractWith the development of multimoment bulk microphysical schemes and polarimetric radar forward operators, one can better examine convective storms simulated in high-resolution numerical models from a simulated polarimetric radar perspective. Subsequently, relationships between observable and unobservable quantities can be examined that may provide useful information about storm intensity and organization that otherwise would be difficult to obtain. This paper, Part I of a two-part sequence, describes the bulk microphysics scheme, polarimetric radar forward operator, and numerical model configuration used to simulate supercells in eight idealized, horizontally homogenous environments with different wind profiles. The microphysical structure and evolution of copolar cross-correlation coefficient (ρhv) rings associated with simulated supercells are examined in Part I, whereas Part II examines ZDR columns, ZDR rings, and KDP columns. In both papers, some systematic differences between the signature seen at X and S bands are discussed. The presence of hail is found to affect ρhv much more at X band than at S band (and is found to affect ZDR more at S band than at X band), which corroborates observations. The ρhv half ring is found to be associated with the presence of large, sometimes wet, hail aloft, with an ~20-min time lag between increases in the size of the ρhv ring aloft and the occurrence of a large amount of hail near the ground in some simulations.

An IND-CCA2 Secure Public Key Cryptographic Protocol using Suzuki 2-Group

Objectives: The public key cryptographic protocol is one of the most important fields in computer security. These new public key cryptographic protocols provide high security as compare to past results in the same field. Methods/Statistical Analysis: Public key cryptographic is a protocol of transferring private info and data through open network communication, so only the receiver who has the secret key can read the encrypted messages which might be documents, phone conversations, images or other form of data. To implement privacy simply by encrypting the information intended to remain secret can be achieved by using methods of public key cryptography. Findings: In this study, we propose the new IND-CCA2 secure public key cryptographic protocol using the concept of integral coefficient ring polynomial based on Suzuki 2-group. We demonstrated the security of proposed public key cryptographic protocol in the adaptively chosen cipher text secure (IND-CCA2) in the random oracle model. Application/Improvements: We discussed the new strategy with change over an IND-CPA public key cryptographic protocol into an IND-CCA2 cryptographic protocol.

Singular decompositions of a cap product

In the case of a compact orientable pseudomanifold, a well-known theorem of M. Goresky and R. MacPherson says that the cap product with a fundamental class factorizes through the intersection homology groups. In this work, we show that this classical cap product is compatible with a cap product in intersection (co)homology that we have previously introduced. If the pseudomanifold is also normal, for any commutative ring of coefficients, the existence of a classical Poincaré duality isomorphism is equivalent to the existence of an isomorphism between the intersection homology groups corresponding to the zero and the top perversities.

Optical sensors using chaotic correlation fiber loop ring down

We have proposed a novel optical sensor scheme based on chaotic correlation fiber loop ring down (CCFLRD). In contrast to the well-known FLRD spectroscopy, where pulsed laser is injected to fiber loop and ring down time is measured, the proposed CCFLRD uses a chaotic laser to drive a fiber loop and measures autocorrelation coefficient ring down time of chaotic laser. The fundamental difference enables us to avoid using long fiber loop as required in pulsed FLRD, and thus generates higher sensitivity. A strain sensor has been developed to validate the CCFLRD concept. Theoretical and experiment results demonstrate that the proposed method is able to enhance sensitivity by more than two orders of magnitude comparing to the existing FLRD method. We believe the proposed method could find great potential applications for chemical, medical, and physical sensing.

Gröbner fans of x-homogeneous ideals in R〚t〛[x

We generalise the notion of a Gröbner fan to ideals in R〚t〛[x1,…,xn] for certain classes of coefficient rings R and give a constructive proof that the Gröbner fan is a rational polyhedral fan. For this we introduce the notion of initially reduced standard bases and show how these can be computed in finite time. We deduce algorithms for computing the Gröbner fan, implemented in the computer algebra system Singular. The problem is motivated by the wish to compute tropical varieties over the p-adic numbers.

Steinberg groups as amalgams

For any root system and any commutative ring we give a relatively simple presentation of a group related to its Steinberg group St. This includes the case of infinite root systems used in Kac-Moody theory, for which the Steinberg group was defined by Tits and Morita-Rehmann. In most cases our group equals St, giving a presentation with many advantages over the usual presentation of St. This equality holds for all spherical root systems, all irreducible affine root systems of rank>2, and all 3-spherical root systems. When the coefficient ring satisfies a minor condition, the last condition can be relaxed to 2-sphericity. Our presentation is defined in terms of the Dynkin diagram rather than the full root system. It is concrete, with no implicit coefficients or signs. It makes manifest the exceptional diagram automorphisms in characteristics 2 and 3, and their generalizations to Kac-Moody groups. And it is a Curtis-Tits style presentation: it is the direct limit of the groups coming from 1- and 2-node subdiagrams of the Dynkin diagram. Over non-fields this description as a direct limit is new and surprising. Our main application is that many Steinberg and Kac-Moody groups over finitely-generated rings are finitely presented.

On the functoriality of marked families

The application of methods of computational algebra has recently introduced new tools for the study of Hilbert schemes. The key idea is to define flat families of ideals endowed with a scheme structure whose defining equations can be determined by algorithmic procedures. For this reason, several authors developed new methods, based on the combinatorial properties of Borel-fixed ideals, that allow associating to each ideal $J$ of this type a scheme $\MFScheme {J}$, called a $J$-marked scheme. In this paper, we provide a solid functorial foundation to marked schemes and show that the algorithmic procedures introduced in previous papers do not depend on the ring of coefficients. We prove that, for all strongly stable ideals $J$, the marked schemes $\MFScheme {J}$ can be embedded in a Hilbert scheme as locally closed subschemes, and that they are open under suitable conditions on $J$. Finally, we generalize Lederer's result about Gr\"obner strata of zero-dimensional ideals, proving that Gr\"obner strata of any ideals are locally closed subschemes of Hilbert schemes.

On the cohomology and their torsion of real toric objects

Abstract In this paper, we do the following two things: (i) We present a formula to compute the rational cohomology ring of a real topological toric manifold, and thus that of a small cover or a real toric manifold, which implies the formula of Suciu and Trevisan. Furthermore, the formula also works for an arbitrary coefficient ring G in which 2 is a unit. (ii) We construct infinitely many real toric manifolds and small covers whose integral cohomology rings have a q-torsion for any positive odd integer q.

An Alternative Perspective on Skew Generalized Power Series Rings

This paper continues the ongoing effort to study the structure of the set of nilpotent elements in noncommutative ring constructions. Let R be any ring, \({(S,\leq)}\) a strictly (partially) ordered monoid and also \({\omega:S\rightarrow}\) End(R) a monoid homomorphism. A skew generalized power series ring \({R[[{S,\omega\leq}}]]\) consists of all functions from a monoid S to a coefficient ring R, whose support contains neither infinite descending chains nor infinite antichains, with pointwise addition, and with multiplication given by convolution twisted by an action \({\omega}\) of the monoid S on the ring R. Our studies in this paper is strongly connected to the question of whether or not a skew generalized power series ring \({R[[{S,\omega\leq}}]]\) over a nil coefficient ring R is nil, which is related to the famous question of Amitsur. However, we show that under mild “Armendariz-like” hypothesis on a coefficient ring R, we obtain stronger conditions on the coefficients of elements of a skew generalized power series ring \({R[[{S,\omega\leq}}]]\). We will also explore some annihilator conditions in the skew generalized power series ring setting, unifying and generalizing a number of known Armendariz-like and McCoy-like conditions in the special cases.

Nilradicals of the unique product monoid rings

Armendariz rings are generalization of reduced rings, and therefore, the set of nilpotent elements plays an important role in this class of rings. There are many examples of rings with nonzero nilpotent elements which are Armendariz. Observing structure of the set of all nilpotent elements in the class of Armendariz rings, Antoine introduced the notion of nil-Armendariz rings as a generalization, which are connected to the famous question of Amitsur of whether or not a polynomial ring over a nil coefficient ring is nil. Given an associative ring [Formula: see text] and a monoid [Formula: see text], we introduce and study a class of Armendariz-like rings defined by using the properties of upper and lower nilradicals of the monoid ring [Formula: see text]. The logical relationship between these and other significant classes of Armendariz-like rings are explicated with several examples. These new classes of rings provide the appropriate setting for obtaining results on radicals of the monoid rings of unique product monoids and also can be used to construct new classes of nil-Armendariz rings. We also classify, which of the standard nilpotence properties on polynomial rings pass to monoid rings. As a consequence, we extend and unify several known results.