Showing papers on "Ring (mathematics) published in 2001"

Model categories of diagram spectra

Abstract: Working in the category of based spaces, we give the basic theory of diagram spaces and diagram spectra. These are functors for a suitable small topological category . When is symmetric monoidal, there is a smash product that gives the category of -spaces a symmetric monoidal structure. Examples include \begin{enumerate} \item[] prespectra, as defined classically, \item[] symmetric spectra, as defined by Jeff Smith, \item[] orthogonal spectra, a coordinate-free analogue of symmetric spectra with symmetric groups replaced by orthogonal groups in the domain category, \item[] -spaces, as defined by Graeme Segal, \item[] -spaces, an analogue of -spaces with finite sets replaced by finite CW complexes in the domain category. \end{enumerate} We construct and compare model structures on these categories. With the caveat that -spaces are always connective, these categories, and their simplicial analogues, are Quillen equivalent and their associated homotopy categories are equivalent to the classical stable homotopy category. Monoids in these categories are (strict) ring spectra. Often the subcategories of ring spectra, module spectra over a ring spectrum, and commutative ring spectra are also model categories. When this holds, the respective categories of ring and module spectra are Quillen equivalent and thus have equivalent homotopy categories. This allows interchangeable use of these categories in applications.2000 Mathematics Subject Classification: primary 55P42; secondary 18A25, 18E30, 55U35.

Semi-dualizing complexes and their Auslander categories

Abstract: Let R be a commutative Noetherian ring. We study R–modules, and complexes of such, with excellent duality properties. While their common properties are strong enough to admit a rich theory, we count among them such, potentially, diverse objects as dualizing complexes for R on one side, and on the other, the ring itself. In several ways, these two examples constitute the extremes, and their well-understood properties serve as guidelines for our study; however, also the employment, in recent studies of ring homomorphisms, of complexes “lying between” these extremes is incentive.

On the algebraic structure of quasi-cyclic codes .I. Finite fields

Abstract: A new algebraic approach to quasi-cyclic codes is introduced. The key idea is to regard a quasi-cyclic code over a field as a linear code over an auxiliary ring. By the use of the Chinese remainder theorem (CRT), or of the discrete Fourier transform (DFT), that ring can be decomposed into a direct product of fields. That ring decomposition in turn yields a code construction from codes of lower lengths which turns out to be in some cases the celebrated squaring and cubing constructions and in other cases the (u+/spl upsi/|u-/spl upsi/) and Vandermonde constructions. All binary extended quadratic residue codes of length a multiple of three are shown to be attainable by the cubing construction. Quinting and septing constructions are introduced. Other results made possible by the ring decomposition are a characterization of self-dual quasi-cyclic codes, and a trace representation that generalizes that of cyclic codes.

F-regular and F-pure rings vs. log terminal and log canonical singularities

Abstract: In this paper, we investigate the relationship of F-regular (resp. F-pure) rings and log terminal (resp. log canonical) singularities. Also, we extend the notions of F-regularity and F-purity to "F-singularities of pairs." The notions of F-regular and F-pure rings in characteristic $p>0$ are characterized by a splitting of the Frobenius map, and define some classes of rings having "mild" singularities. On the other hand, there are notions of log terminal and log canonical singularities defined via resolutions of singularities in characteristic zero. These are defined also for pairs of a normal variety and a $\Bbb Q$-divisor $\Delta$ on it, and play an important role in birational algebraic geometry. As an analog of these singularities of pairs in characteristic zero, we define the notions of "F-singularities of pairs," namely strong F-regularity, divisorial F-regularity and F-purity for a pair $(A,\Delta)$ of a normal ring $A$ of characteristic $p > 0$ and an effective $\Bbb Q$-divisor $\Delta$ on $Y = Spec A$. The main theorem of this paper asserts that, if $K_Y + \Delta$ is $\Bbb Q$-Cartier, then the above three variants of F-singularitiesof pairs imply KLT, PLT and LC properties, respectively. We also prove some results for F-singularities of pairs which are analoguous to singularities of pairs in characteristic zero.

Représentations p-adiques et équations différentielles

Abstract: In this paper, we associate to every p-adic representation V a p-adic differential equation D † rig(V), that is to say a module with a connection over the Robba ring. We do this via the theory of Fontaine’s (ϕ,Γ K )-modules.¶This construction enables us to relate the theory of (ϕ,Γ K )-modules to p-adic Hodge theory. We explain how to construct D cris(V) and D st(V) from D † rig(V), which allows us to recognize semi-stable or crystalline representations; the connection is then unipotent or trivial on D † rig(V)[1/t].¶In general, the connection has an infinite number of regular singularities, but V is de Rham if and only if those are apparent singularities. A structure theorem for modules over the Robba ring allows us to get rid of all singularities at once, and to obtain a “classical” differential equation, with a Frobenius structure.¶Using this, we construct a functor from the category of de Rham representations to that of classical p-adic differential equations with Frobenius structure. A recent theorem of Y. Andre gives a complete description of the structure of the latter object. This allows us to prove Fontaine’s p-adic monodromy conjecture: every de Rham representation is potentially semi-stable.¶As an application, we can extend to the case of arbitrary perfect residue fields some results of Hyodo (H 1 g =H 1 st ), of Perrin-Riou (the semi-stability of ordinary representations), of Colmez (absolutely crystalline representations are of finite height), and of Bloch and Kato (if the weights of V are ≥2, then Bloch-Kato’s exponential exp V is an isomorphism).

Extensions of clean rings

Abstract: It is shown that if e is an idempotent in a ring R such that both eRe and (1 − e)R(1 − e) are clean rings, then R is a clean ring. This implies that the matrix ring M n (R) over a clean ring is clean, and it gives a quick proof that every semiperfect is clean. Other extensions of clean rings are studied, including group rings.

Principally quasi-baer rings

Abstract: We say a ring with unity is right principally quasi-Baer (or simply, right pq-Baer) if the right annihilator of a principal right ideal is generated (as a right ideal) by an idempotent This class of rings includes the biregular rings and is closed under direct products and Morita invariance The 2-by-2 formal upper triangular matrix rings of this class are characterized Connections to related classes of rings (eg, right PP, Baer, quasi-Baer, right FPF, right GFC, etc) are investigated Examples to illustrate and delimit the theory are provided

Integration on ${\cal H}_p\times {\cal H}$ and arithmetic applications

Abstract: This article describes a conjectural p-adic analytic construction of global points on (modular) elliptic curves, points which are defined over the ring class fields of real quadratic fields. The resulting conjectures suggest that the classical Heegner point construction, and the theory of complex multiplication on which it is based, should extend to a variety of contexts in which the underlying field is not a CM field.

1-skeleta, Betti numbers, and equivariant cohomology

Abstract: The 1-skeleton of a $G$-manifold $M$ is the set of points $p ∈ M$, where $\dim G_{p}≥ \dim G−1$, and $M$ is a GKM manifold if the dimension of this 1-skeleton is 2. M. Goresky, R. Kottwitz, and R. MacPherson show that for such a manifold this 1-skeleton has the structure of a “labeled” graph, $(Γ,α)$, and that the equivariant cohomology ring of $M$ is isomorphic to the “cohomology ring” of this graph. Hence, if $M$ is symplectic, one can show that this ring is a free module over the symmetric algebra $\mathbb{S}(\mathfrak{g}^*)$, with $b_{2i}(Γ)$ generators in dimension $2 i, b_{2i}(Γ)$ being the “combinatorial” $2i$th Betti number of $Γ$. In this article we show that this “topological” result is, in fact, a combinatorial result about graphs.

An exact sequence for Milnor's K-theory with applications to quadratic forms

Abstract: We construct a four-term exact sequence which provides information on the kernel and cokernel of the multiplication by a pure symbol in Milnor's K-theory mod 2 of fields of characteristic zero. As an application we establish, for fields of characteristics zero, the validity of three conjectures in the theory of quadratic forms - the Milnor conjecture on the structure of the Witt ring, the Khan-Rost-Sujatha conjecture and the J-filtration conjecture. The first version of this paper was written in the spring of 1996.

Monomial algebras and polyhedral geometry

Abstract: Publisher Summary This chapter discusses the monomial algebras and its connections to combinatorics, graph theory, and polyhedral geometry. Some important notions from commutative algebra that have played a role in the development of the theory, such as Cohen-Macaulay ring, normal ring, Gorenstein ring, integral closure, Hilbert series, and local cohomology are introduced. The upper bound theorem for the number of faces of a simplicial sphere, a description of the integral closure of an edge subring, a generalized marriage theorem for a certain family of graphs, and a study of systems of binomials in the ideal of an affine toric variety are provided as applications. It illustrates the interplay between several areas of mathematics and the power of combinatorial commutative algebra techniques. There is a connection between monomial rings and monomial subrings due to the fact that the initial ideal of a toric ideal is a monomial ideal. This allows computing several invariants of projective varieties using algebraic systems such as CoCoA and Macaulay2. An important tool to study monomial subrings is Normaliz, which is effective in practice and can be used to find normalizations, Hilbert series, Ehrhart rings, and volumes of lattice polytopes.

Sur l'anneau de cohomologie du schéma de Hilbert de ?

Abstract: We express the product of the cohomology ring of the Hilbert scheme in terms of the center of the algebra of the symmetric group. We give a conjecture for the case of crepant resolutions of symplectic quotient singularities.

Characterization of finite Frobenius rings

Abstract: It is shown that a finite ring R is a Frobenius ring if and only if $_R(R/\hbox {Rad}\, R)\cong \hbox {Soc}\, (_RR)$ . Other combinatorial characterizations of finite Frobenius rings are presented which have applications in the theory of linear codes over finite rings.

Quiver varieties and t-analogs of q-characters of quantum affine algebras

Abstract: Let us consider a specialization of an untwisted quantum affine algebra of type $ADE$ at a nonzero complex number, which may or may not be a root of unity. The Grothendieck ring of its finite dimensional representations has two bases, simple modules and standard modules. We identify entries of the transition matrix with special values of ``computable'' polynomials, similar to Kazhdan-Lusztig polynomials. At the same time we ``compute'' $q$-characters for all simple modules. The result is based on ``computations'' of Betti numbers of graded/cyclic quiver varieties. (The reason why we put `` '' will be explained in the end of the introduction.)

Ring stability of the PROFIBUS token-passing protocol over error-prone links

Abstract: The PROFIBUS is a well-known and widely used fieldbus. On the medium access control layer, it employs a token-passing protocol where all active stations form a logical ring on top of a broadcast medium. This protocol is designed to deliver real-time data transmission services in harsh industrial environments. A necessary prerequisite for timeliness and quality of service is the ring membership stability of the logical ring in the presence of transmission errors, since only ring members are allowed to transmit data. In this paper, the ring membership stability under high error rates and using different error models is analyzed. The choice of the error behavior is in turn inspired by properties of possible future transmission technologies, e.g., wireless LANs. It is shown that the protocol has serious stability problems. To attack these problems, two changes to the protocol and its parameters are proposed, which can be implemented in a purely local manner. We show that they significantly improve ring stability.

Constructions of nontautological classes on moduli spaces of curves

Abstract: We construct explicit examples of algebraic cycles in \bar M_g (for large g congruent to 2 mod 4) and in M_2,20 (no bar) which are not in the tautological ring. In an appendix we give a general method for computing intersections in the tautological ring.

Quantum cohomology rings of Grassmannians and total positivity

Abstract: We give a proof of a result of D. Peterson identifying the quantum cohomology ring of a Grassmannian with the reduced coordinate ring of a certain subvariety of ${\rm GL}_n$. The totally positive part of this subvariety is then constructed, and we give closed formulas for the values of the Schubert basis elements on the totally positive points. We then use the developed methods to give a new proof of a formula of C. Vafa, K. Intriligator, and A. Bertram for the structure constants (Gromov-Witten invariants). Finally, we use the positivity of these Gromov-Witten invariants to prove certain inequalities for Schur polynomials at roots of unity.

Symmetric and Exterior Powers, Linear Source Modules and Representations of Schur Superalgebras

Abstract: We study modules for the general linear group (over an infinite field of arbitrary characteristic) which are direct summands of tensor products of exterior powers and symmetric powers of the natural module. These modules, which we call listing modules, include the tilting modules and the injective modules for Schur algebras. The modules are studied via their relationship to linear source modules for symmetric groups on the one hand, and simple modules for Schur superalgebras on the other. Listing modules are parametrized by certain pairs of partitions. They are used to describe, by generators and relations, the Grothendieck ring of polynomial functors generated by the symmetric and exterior powers. We also (continuing work of J. Grabmeier) describe the vertices and sources of linear source modules for symmetric groups.

Influence of topological constraints on the statics and dynamics of ring polymers.

Abstract: We report a computer simulation study of the influence of topological constraints on the statics and dynamics of single ring polymers and ring polymers in the melt. We show that single rings have identical static and dynamic scaling behavior regardless of the presence of topological constraints. For rings in the melt we find that the scaling behavior is significantly influenced by the presence of topological constraints.

Tangential Structures on Toric Manifolds, and Connected Sums of Polytopes

Abstract: We extend work of Davis and Januszkiewicz by considering omnioriented toric manifolds, whose canonical codimension-2 submanifolds are independently oriented. We show that each omniorientation induces a canonical stably complex structure, which is respected by the torus action and so defines an element of an equivariant cobordism ring. As an application, we compute the complex bordism groups and cobordism ring of an arbitrary omnioriented toric manifold. We consider a family of examples Bi,j, which are toric manifolds over products of simplices, and verify that their natural stably complex structure is induced by an omniorientation. Studying connected sums of products of the Bi,j allows us to deduce that every complex cobordism class of dimension >2 contains a toric manifold, necessarily connected, and so provides a positive answer to the toric analogue of Hirzebruch's famous question for algebraic varieties. In previous work, we dealt only with disjoint unions, and ignored the relationship between the stably complex structure and the action of the torus. In passing, we introduce a notion of connected sum # for simple n-dimensional polytopes; when Pn is a product of simplices, we describe Pn#Qn by applying an appropriate sequence of pruning operators, or hyperplane cuts, to Qn.

p-Adic Fourier Theory

Abstract: In this paper we generalize work of Amice and Lazard from the early sixties. Amice determined the dual of the space of locally Qp-analytic functions on Zp and showed that it is isomorphic to the ring of rigid functions on the open unit disk over Cp. Lazard showed that this ring has a divisor theory and that the classes of closed, flnitely generated, and principal ideals in this ring coincide. We study the space of locally L-analytic functions on the ring of integers in L, where L is a flnite extension of Qp. We show that the dual of this space is a ring isomorphic to the ring of rigid functions on a certain rigid variety X. We show that the variety X is isomorphic to the open unit disk over Cp, but not over any discretely valued extension fleld of L; it is a "twisted form" of the open unit disk. In the ring of functions on X, the classes of closed, flnitely generated, and invertible ideals coincide, but unless L=Qp not all flnitely generated ideals are principal. The paper uses Lubin-Tate theory and results on p-adic Hodge theory. We give several applications, including one to the construction of p- adic L-functions for supersingular elliptic curves.

Modules over Iwasawa algebras

Abstract: Let $p$ be a prime number, and $G$ a compact $p$-adic Lie group. We recall that the Iwasawa algebra $\Lambda(G)$ is defined to be the completed group ring of $G$ over the ring of $p$-adic integers. Interesting examples of finitely generated modules over $\Lambda(G),$ in which $G$ is the image of Galois in the automorphism group of a $p$-adic Galois representation, abound in arithmetic geometry. The study of such $\Lambda(G)$-modules arising from arithmetic geometry can be thought of as a natural generalization of Iwasawa theory. One of the cornerstones of classical Iwasawa theory is the fact that, when $G$ is the additive group of $p$-adic integers, a good structure theory for finitely generated $\Lambda(G)$-modules is known, up to pseudo-isomorphism. The aim of the present paper is to extend as much as possible of this commutative structure theory to the non-commuta tive case.

Monte Carlo simulation studies of the size and shape of ring polymers

Abstract: The present work gives a comparison between the properties of athermal ring polymers and those of linear chains. Based on bond fluctuation (BF) and pivot algorithm (PIV) for the construction of molecules, a new algorithm was developed, which proved most efficient due to large acceptance fractions and small (integral) auto-correlation times of global properties, in addition having the advantage of a large set of different bond vectors. While the topological state of ring polymers remains unchanged by exclusive use of BF, knotted structures (which were identified with the help of Alexander polynomials) can by formed and removed by the use of PIV. In accordance with previous work, it turned out that the probability of unknotted rings (in principle) exponentially decreases with an increasing number of segments, however, so slowly that the appearance of knotted structures (ca. 0.1 % for N)512) is a rare event in the range of chain-lenths evaluated (N=132-8192). The chain-length dependence of global quantities of ring polymers are described by the use of scaling relations with proper short chain corrections, in analogy to linear chains. The instantaneous shape of ring polymers is more symmetric than that of linear chains. Local quantities, i.e., mean squared bond lengths and mean bond angles are the same for both systems, at least in the limit of an infinite number of segments

On the Chow motive of some moduli spaces

Abstract: We study the motive of moduli spaces of stable vector bundles over a smooth projective curve. We prove this motive lies in the category generated by the motive of the curve and we compute its class in the Grothendieck ring of the category of motives. As applications we compute the Poincare-Hodge polynomials and the number of points over a finite field and we study some conjectures on algebraic cycles on these moduli spaces.

Some structural properties of the topological ring of colombeau's generalized numbers

Abstract: Let denote the commutative ring with identity of Colombeau's generalized numbers. (K will denote either R or C). This ring can be endowed with an ultra-metric in such a way that is a topological ring. There are many interesting questions about in the frameworks of Commutative Algebra and General Topology as well as of the superposition of these two subjects. This paper is meant to represent an initial step in this direction. In a few simple cases the study is extended to the ring of Colombeau's generalized functions on an open subset of R n .

On minimal graded free resolutions

Abstract: Minimal graded free resolutions are an important and central topic in algebra. They are a useful tool for studying modules over finitely generated graded K- algebras. Such a resolution determines the Hilbert series, the Castelnuovo-Mumford regularity and other invariants of the module. This thesis is concerned with the structure of minimal graded free resolutions. We relate our results to several recent trends in commutative algebra. The first of these trends deals with relations between properties of the Stanley- Reisner ring associated to a simplicial complex and the Stanley-Reisner ring of its Alexander dual. Another development is the investigation of the linear part of a minimal graded free resolution as defined by Eisenbud and Schreyer. Several authors were interested in the problem to give lower bounds for the Betti numbers of a module. In particular, Eisenbud-Koh, Green, Herzog and Reiner- Welker studied the graded Betti numbers which determine the linear strand of a minimal graded free resolution. Bigraded algebras occur naturally in many research areas of commutative algebra. A typical example of a bigraded algebra is the Rees ring of a graded ideal. Herzog and Trung used this bigraded structure of the Rees ring to study the Castelnuovo- Mumford regularity of powers of graded ideals in a polynomial ring. Conca, Herzog, Trung and Valla dealt with diagonal subalgebras of bigraded algebras. Aramova, Crona and De Negri studied homological properties of bigraded K-algebras.

Higher algebraic K-theory for actions of diagonalizable groups

Abstract: We study the K-theory of actions of diagonalizable group schemes on noetherian regular separated algebraic spaces: our main result shows how to reconstruct the K-theory ring of such an action from the K-theory rings of the loci where the stabilizers have constant dimension. We apply this to the calculation of the equivariant K-theory of toric varieties, and give conditions under which the Merkurjev spectral sequence degenerates, so that the equivariant K-theory ring determines the ordinary K-theory ring. We also prove a very refined localization theorem for actions of this type.

On derivations of lattices

Abstract: This de nition clearly coincides with the usual (algebraic) notion of derivation when [A; +, ·] is a ring However, it can be formally stated for every algebraic structure endowed with two binary operations In this paper, we will consider the special case in which [A; +, ·] is a lattice, so that + and · are, respectively, the join and the meet operations These ideas have been introduced and developed by Szasz in a series of papers (here we recall [S1, S2]), in which he established the main properties of derivations of lattices Also Kolibiar [K] gave his contribution, for example in the study of the case of the chain of natural numbers However, it seems that these investigations only scratched the surface of the subject