Showing papers on "Ring (mathematics) published in 2004"

A New Cohomology Theory of Orbifold

Abstract: Based on the orbifold string theory model in physics, we construct a new cohomology ring for any almost complex orbifold. The key theorem is the associativity of this new ring. Some examples are computed.

The orbifold Chow ring of toric Deligne-Mumford stacks

Abstract: The orbifold Chow ring of a Deligne-Mumford stack, defined by Abramovich, Graber and Vistoli [2], is the algebraic version of the orbifold cohomology ring in troduced by W. Chen and Ruan [7], [8]. By design, this ring incorporates numerical invariants, such as the orbifold Euler characteristic and the orbifold Hodge num bers, of the underlying variety. The product structure is induced by the degree zero part of the quantum product; in particular, it involves Gromov-Witten invariants. Inspired by string theory and results in Batyrev [3] and Yasuda [28], one expects that, in nice situations, the orbifold Chow ring coincides with the Chow ring of a resolution of singularities. Fantechi and G?ttsche [14] and Uribe [25] verify this conjecture when the orbifold is Symn(5) where 5 is a smooth projective surface with Ks = 0 and the resolution is Hilbn(?>). The initial motivation for this project was to compare the orbifold Chow ring of a simplicial toric variety with the Chow ring of a cr?pant resolution. To achieve this goal, we first develop the theory of toric Deligne-Mumford stacks. Modeled on simplicial toric varieties, a toric Deligne-Mumford stack corresponds to a combinatorial object called a stacky fan. As a first approximation, this object is a simplicial fan with a distinguished lattice point on each ray in the fan. More precisely, a stacky fan S is a triple consisting of a finitely generated abelian group N, a simplicial fan E in Q z N with n rays, and a map ?: Zn ?> N where the image of the standard basis in Zn generates the rays in E. A rational simplicial fan E produces a canonical stacky fan S := (N, E, ?) where N is the distinguished lattice and ? is the map defined by the minimal lattice points on the rays. Hence, there is a natural toric Deligne-Mumford stack associated to every simplicial toric variety. A stacky fan ? encodes a group action on a quasi-affine variety and the toric Deligne-Mumford stack #(?) is the quotient. If E corresponds to a smooth toric variety X?E) and S is the canonical stacky fan associated to E, then we simply have #(!?) = X?Z). We show that many of the basic concepts, such as open and closed toric substacks, line bundles, and maps between toric Deligne Mumford stacks, correspond to combinatorial notions. We expect that many more results about toric varieties lift to the realm of stacks and hope that toric Deligne Mumford stacks will serve as a useful testing ground for general theories.

Structured Ring Spectra: Moduli spaces of commutative ring spectra

Abstract: A bstract . Let E be a homotopy commutative ring spectrum, and suppose the ring of cooperations E ∗E is flat over E ∗. We wish to address the following question: given a commutative E ∗-algebra A in E∗E -comodules, is there an E ∞ -ring spectrum X with E∗X ≅ A as comodule algebras? We will formulate this as a moduli problem, and give a way – suggested by work of Dwyer, Kan, and Stover – of dissecting the resulting moduli space as a tower with layers governed by appropriate Andre-Quillen cohomology groups. A special case is A = E∗E itself. The final section applies this to discuss the Lubin-Tate or Morava spectra E n . Some years ago, Alan Robinson developed an obstruction theory based on Hochschild cohomology to decide whether or not a homotopy associative ring spectrum actually has the homotopy type of an A ∞ -ring spectrum. In his original paper on the subject [35] he used this technique to show that the Morava K -theory spectra K(n) can be realized as an A ∞-ring spectrum; subsequently, in [3], Andrew Baker used these techniques to show that a completed version of the Johnson-Wilson spectrum E(n) can also be given such a structure. Then, in the mid-90s, the second author and Haynes Miller showed that the entire theory of universal deformations offinite height formal group laws over fields of non-zero characteristic can be lifted to A ∞-ring spectra in an essentially unique way.

Representation theory of finite reductive groups.

Abstract: Introduction Notations and conventions Part I. Representing Finite BN-Pairs: 1. Cuspidality in finite groups 2. Finite BN-pairs 3. Modular Hecke algebras for finite BN-pairs 4. Modular duality functor and the derived category 5. Local methods for the transversal characteristics 6. Simple modules in the natural characteristic Part II. Deligne-Lusztig Varieties, Rational Series, and Morita Equivalences: 7. Finite reductive groups and Deligne-Lusztig varieties 8. Characters of finite reductive groups 9. Blocks of finite reductive groups and rational series 10. Jordan decomposition as a Morita equivalence, the main reductions 11. Jordan decomposition as a Morita equivalence, sheaves 12. Jordan decomposition as a Morita equivalence, modules Part III. Unipotent Characters and Unipotent Blocks: 13. Levi subgroups and polynomial orders 14. Unipotent characters as a basic set 15. Jordan decomposition of characters 16. On conjugacy classes in type D 17. Standard isomorphisms for unipotent blocks Part IV. Decomposition Numbers and q-Schur Algebras: 18. Some integral Hecke algebras 19. Decomposition numbers and q-Schur algebras, general linear groups 20. Decomposition numbers and q-Schur algebras, linear primes Part V. Unipotent Blocks and Twisted Induction: 21. Local methods. Twisted induction for blocks 22. Unipotent blocks and generalized Harish Chandra theory 23. Local structure and ring structure of unipotent blocks Appendix 1: Derived categories and derived functors Appendix 2: Varieties and schemes Appendix 3: Etale cohomology References Index.

Representations of reductive groups over finite rings

Abstract: We discuss a cohomological construction of representations of a reductive group over the ring of power series over a finite field modulo the r-th power of the maximal ideal. The case r=1 goes back to Deligne and the author. The case where r is greater than 1 was given without proof in the author's work in 1977. Here we supply the proofs and give some further results.

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

Abstract: We 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 use ? ? will be explained at the end of the introduction.)

Polynomial Identity Rings

Abstract: A Combinatorial Aspects in PI-Rings.- Vesselin Drensky.- 1 Basic Properties of PI-algebras.- 2 Quantitative Approach to PI-algebras.- 3 The Amitsur-Levitzki Theorem.- 4 Central Polynomials for Matrices.- 5 Invariant Theory of Matrices.- 6 The Nagata-Higman Theorem.- 7 The Shirshov Theorem for Finitely Generated PI-algebras.- 8 Growth of Codimensions of PI-algebras.- B Polynomial Identity Rings.- Edward Formanek.- 1 Polynomial Identities.- 2 The Amitsur-Levitzki Theorem.- 3 Central Polynomials.- 4 Kaplansky's Theorem.- 5 Theorems of Amitsur and Levitzki on Radicals.- 6 Posner's Theorem.- 7 Every PI-ring Satisfies a Power of the Standard Identity.- 8 Azumaya Algebras.- 9 Artin's Theorem.- 10 Chain Conditions.- 11 Hilbert and Jacobson PI-Rings.- 12 The Ring of Generic Matrices.- 13 The Generic Division Ring of Two 2 x 2 Generic Matrices.- 14 The Center of the Generic Division Ring.- 15 Is the Center of the Generic Division Ring a Rational Function Field?.

Grothendieck ring of pretriangulated categories

Abstract: We consider the abelian group PT generated by quasi-equivalence classes of pretriangulated DG categories with relations coming from semiorthogonal decompositions of corresponding triangulated categories. We introduce an operation of “multiplication” • on the collection of DG categories, which makes this abelian group into a commutative ring. A few applications are considered: representability of “standard” functors between derived categories of coherent sheaves on smooth projective varieties and a construction of an interesting motivic measure.

Baer and Quasi-Baer Modules

Abstract: We introduce the notions of Baer and quasi-Baer properties in a general module theoretic setting. A module M is called (quasi-) Baer if the right annihilator of a (two-sided) left ideal of End(M) is a direct summand of M. We show that a direct summand of a (quasi-) Baer module inherits the property and every finitely generated abelian group is Baer exactly if it is semisimple or torsion-free. Close connections to the (FI-) extending property are investigated and it is shown that a module M is (quasi-) Baer and (FI-) 𝒦-cononsingular if and only if it is (FI-) extending and (FI-) 𝒦-nonsingular. We prove that an arbitrary direct sum of mutually subisomorphic quasi-Baer modules is quasi-Baer and every free (projective) module over a quasi-Baer ring is a quasi-Baer module. Among other results, we also show that the endomorphism ring of a (quasi-) Baer module is a (quasi-) Baer ring, while the converse is not true in general. Applications of results are provided.

Transit Detectability of Ring Systems around Extrasolar Giant Planets

Abstract: We investigate whether rings around extrasolar planets could be detected from those planets' transit light curves. To this end, we develop a basic theoretical framework for calculating and interpreting the light curves of ringed planet transits on the basis of the existing framework used for stellar occultations, a technique that has been effective for discovering and probing ring systems in the solar system. We find that the detectability of large Saturn-like ring systems is largest during ingress and egress and that a reasonable photometric precision of ~(1-3) × 10-4 with 15 minute time resolution should be sufficient to discover such ring systems. For some ring particle sizes, diffraction around individual particles leads to a detectable level of forward-scattering that can be used to measure modal ring particle diameters. An initial census of large ring systems can be carried out using high-precision follow-up observations of detected transits and by the upcoming NASA Kepler mission. The distribution of ring systems as a function of stellar age and as a function of planetary semimajor axis will provide empirical evidence to help constrain how rings form and how long rings last.

Trivial Extensions Defined by Coherent-like Conditions

Abstract: This paper investigates coherent-like conditions and related properties that a trivial extension R ≔ A ∝ E might inherit from the ring A for some classes of modules E. It captures previous results dealing primarily with coherence, and also establishes satisfactory analogues of well-known coherence-like results on pullback constructions. Our results generate new families of examples of rings (with zerodivisors) subject to a given coherent-like condition.

A power structure over the Grothendieck ring of varieties

Abstract: Let R be either the Grothendieck semiring (semigroup with multiplication) of complex quasi-projective varieties, or the Grothendieck ring of these varieties, or the Grothendieck ring localized by the class \L of the complex affine line. We define a power structure over these (semi)rings. This means that, for a power series A(t)=1+∑i=1∞[Ai]ti with the coefficients [Ai] from R and for [M]∈R, there is defined a series (A(t))[M], also with coefficients from R, so that all the usual properties of the exponential function hold. In the particular case A(t)=(1−t)−1, the series (A(t))[M] is the motivic zeta function introduced by M. Kapranov. As an application we express the generating function of the Hilbert scheme of points, 0-dimensional subschemes, on a surface as an exponential of the surface.

Investigation of ring resonators in photonic crystal circuits

Abstract: In this paper, we propose the implementation of waveguide-coupled ring resonators in photonic crystal integrated circuits. Using two-dimensional finite difference time domain (2D FDTD) method, we study the spectral characteristics of a waveguide-coupled ring carved in two-dimensional photonic crystal of square lattice (2D SLPC) and based on the results, we suitably modify the structure geometry to establish its performance as a ring resonator. We further investigate the effects of ring dimension and crystal parameters on the resonance properties of the ring resonator.

Metropolitan area packet-switched WDM networks: A survey on ring systems

Abstract: We provide a comprehensive survey of packet-switched ring metro WDM networks. We first review current standardization and testbed activities, then we provide a categorization of ring WDM networks. We structure our survey according to a classification of the medium access control (MAC) protocols employed in the networks. Throughout the article we pay close attention to the key factors that govern the throughput-delay performance of the networks, such as the source vs. destination stripping of the data packets from the ring and the a priori or a posteriori access strategies. We also consider fairness aspects and QoS support in the networks.

An interpretation of multiplier ideals via tight closure

Abstract: Hara [Ha3] and Smith [Sm2] independently proved that in a normal Q-Gorenstein ring of characteristic p 0, the test ideal coincides with the multiplier ideal associated to the trivial divisor. We extend this result for a pair (R,∆) of a normal ring R and an effective Q-Weil divisor ∆ on SpecR. As a corollary, we obtain the equivalence of strongly F-regular pairs and klt pairs.

Poincaré series of a rational surface singularity

Abstract: Recently there was found a new method to compute the (generalized) Poincare series of some multi-index filtrations on rings of functions. First the authors had elaborated it for computing the Poincare series of the filtration on the ring OC2,0 of germs of functions of two variables defined by irreducible components of a plane curve singularity. The corresponding formula (announced in [2]) was proved in [3] by another method. The new method uses the notion of the integral with respect to the Euler characteristic over the projectivization of the space of germs of functions. This notion is similar to (and inspired by) the notion of the motivic integration. Here we apply this method for computing the Poincare series of a (natural) multi-index filtration on the ring of germs of functions on a rational surface singularity. We explicitly calculate the coefficients of this series. Let (S, 0) be a germ of an (isolated) rational surface singularity and let π : (X,D) → (S, 0) be its minimal resolution. Here X is a smooth surface, π is a proper analytic map which is an isomorphism outside of D = π−1(0), and the exceptional divisor D is the union of irreducible components Ei (i = 1, . . . , r) transversal to each other, each component Ei is isomorphic to the projective line CP1.

On the splitting of the Bloch-Beilinson filtration

Abstract: For a smooth projective variety X, let CH(X) be the Chow ring (with rational coefficients) of algebraic cycles modulo rational equivalence. The conjectures of Bloch and Beilinson predict the existence of a functorial ring filtration of CH(X). We want to investigate for which varieties this filtration splits, that is, comes from a graduation on CH(X) -- this occurs for K3 surfaces and, conjecturally, for abelian varieties. We observe that, though the Bloch-Beilinson filtration is only conjectural, the fact that it splits has some simple consequences which can be tested in concrete examples. Namely, for a regular variety X, it implies that the sub-Q-algebra of CH(X) spanned by divisor classes injects into the cohomology of X . We give examples of Calabi-Yau threefolds which do not satisfy this property. On the other hand we conjecture that the property does indeed hold for (holomorphic) symplectic manifolds, and we give some (weak) evidence in favour of this conjecture.

Gorenstein derived functors

Abstract: Over any associative ring R it is standard to derive Hom R (-,-) using projective resolutions in the first variable, or injective resolutions in the second variable, and doing this, one obtains Ext n R(-,-) in both cases. We examine the situation where projective and injective modules are replaced by Gorenstein projective and Gorenstein injective ones, respectively. Furthermore, we derive the tensor product - ⊗ R - using Gorenstein flat modules.

Spiraphase-type reflectarrays based on loaded ring slot resonators

Abstract: Reflective periodic arrays based on loaded ring slot resonators are analyzed. A full-wave mathematical model is developed and numerical results are presented. It is proven that the reflection angle for the normally incident circularly polarized wave can be effectively controlled by proper positioning of reactive loads in ring slot resonators. Analysis of the reflection characteristics of the Ka band one-layer reflectarray results in a conclusion that the incident wave can be effectively redirected in the directions determined by elevation angles as high as 65/spl deg/ with conversion coefficient better than -1.5 dB. It is also shown that the usage of the multilayer reflectarray leads to a considerable improvement in the reflection characteristics when it is compared with the one-layer reflectarray. The method of the waveguide simulator has been used to verify the developed mathematical model.

A positive proof of the Littlewood-Richardson rule using the octahedron recurrence

Abstract: We define the hive ring , which has a basis indexed by dominant weights for $GL_n({\Bbb C})$, and structure constants given by counting hives [Knutson-Tao, "The honeycomb model of $GL_n$ tensor products"] (or equivalently honeycombs, or BZ patterns [Berenstein-Zelevinsky, "Involutions on Gel$'$fand-Tsetlin schemes$\dots$ "]). We use the octahedron rule from [Robbins-Rumsey, "Determinants$\dots$"] to prove bijectively that this "ring" is indeed associative. This, and the Pieri rule, give a self-contained proof that the hive ring is isomorphic as a ring-with-basis to the representation ring of $GL_n({\Bbb C})$. In the honeycomb interpretation, the octahedron rule becomes "scattering" of the honeycombs. This recovers some of the "crosses and wrenches" diagrams from Speyer's very recent preprint ["Perfect matchings$\dots$"], whose results we use to give a closed form for the associativity bijection.

The sigma orientation is an H ∞ map

Abstract: In an earlier paper, the authors constructed a natural map, called the sigma orientation, from the Thom spectrum MU� 6� to any elliptic spectrum. MU� 6� is an H∞ ring spectrum, and in this paper we show that if (E, C, t) is the elliptic spectrum associated to the universal deformation of a supersingular elliptic curve over a perfect field of characteristic p > 0, then the sigma orientation is a map of H∞ ring spectra.

Finite quasi-frobenius modules and linear codes

Abstract: The theory of linear codes over finite fields has been extended by A. Nechaev to codes over quasi-Frobenius modules over commutative rings, and by J. Wood to codes over (not necessarily commutative) finite Frobenius rings. In the present paper, we subsume these results by studying linear codes over quasi-Frobenius and Frobenius modules over any finite ring. Using the character module of the ring as alphabet, we show that fundamental results like MacWilliams' theorems on weight enumerators and code isometry can be obtained in this general setting.

Units of ring spectra and their traces in algebraic K-theory

Abstract: Let GL1(R) be the units of a commutative ring spectrum R. In this paper we identify the composition

Partial classification of modules for Lie-algebra of diffeomorphisms of d-dimensional torus

Abstract: We consider the Lie-algebra of the group of diffeomorphisms of a d-dimensional torus which is also known to be the algebra of derivations on a Laurent polynomial ring A in d commuting variables denoted by Der A. The universal central extension of Der A for d=1 is the so-called Virasoro algebra. The connection between Virasoro algebra and physics is well known. See, for example, the book on Conformal Field Theory by Di Francesco, Mathieu, and Senechal. In this paper we classify (A, Der A) modules which are irreducible and have finite dimensional weight spaces. Earlier Larsson constructed a large class of modules, the so-called tensor fields, based on gld modules which are also A modules. We prove that they exhaust all (A, Der A) irreducible modules.

Cyclotomic Completions of Polynomial Rings

Abstract: The main object of study in this paper is the completion Z[q]^N=\varprojlim_n Z[q]/((1-q)(1-q^2)...(1-q^n)) of the polynomial ring Z[q], which arises from the study of a new invariant of integral homology 3-spheres with values in Z[q]^N announced by the author, which unifies all the sl_2 Witten-Reshetikhin-Turaev invariants at various roots of unity. We show that any element of Z[q]^N is uniquely determined by its power series expansion in q-\zeta for each root \zeta of unity. We also show that any element of Z[q]^N is uniquely determined by its values at the roots of unity. These results may be interpreted that Z[q]^N behaves like a ring of ``holomorphic functions defined on the set of the roots of unity''. We will also study the generalizations of Z[q]^N, which are completions of the polynomial ring R[q] over a commutative ring R with unit with respect to the linear topologies defined by the principal ideals generated by products of powers of cyclotomic polynomials.

Indecomposable parabolic bundles

Abstract: We study the possible dimension vectors of indecomposable parabolic bundles on the projective line, and use our answer to solve the problem of characterizing those collections of conjugacy classes of n×n matrices for which one can find matrices in their closures whose product is equal to the identity matrix. Both answers depend on the root system of a Kac-Moody Lie algebra. Our proofs use Ringel’s theory of tubular algebras, work of Mihai on the existence of logarithmic connections, the Riemann-Hilbert correspondence and an algebraic version, due to Dettweiler and Reiter, of Katz’s middle convolution operation.