Showing papers on "Ring (mathematics) published in 1999"

Chern classes of tautological sheaves on Hilbert schemes of points on surfaces

Abstract: We give an algorithmic description of the action of the Chern classes of tautological bundles on the cohomology of Hilbert schemes of points on a smooth surface within the framework of Nakajima's oscillator algebra. This leads to an identification of the cohomology ring of Hilb n (A2) with a ring of explicitly given differential operators on a Fock space. We end with the computation of the top Segre classes of tautological bundles associated to line bundles on Hilb n up to n=7, extending computations of Severi, LeBarz, Tikhomirov and Troshina and give a conjecture for the generating series.

On the Cobordism Class of the Hilbert Scheme of a Surface

Abstract: Let S be a smooth projective surface and S [n] the Hilbert scheme of zerodimensional subschemes of S of length n. We proof that the class of S [n] in the complex cobordism ring depends only on the class of the surface itself. Moreover, we compute the cohomology and holomorphic Euler characteristics of certain tautological sheaves on S [n] and prove results on the general structure of certain integrals over polynomials in Chern classes of tautological sheaves. Let S be a smooth projective surface over the field of complex numbers. For a nonnegative integer n let S [n] denote the Hilbert scheme parameterizing zerodimensional subschemes of length n. By a well-known result of Fogarty [10] the scheme S [n] is smooth and projective of dimension 2n, and is irreducible if S is irreducible. Let = U ⊗ Q be the complex cobordism ring with rational coefficients. Milnor [20] showed that is a polynomial ring freely generated by the cobordism classes [CPi] for i ∈ N. For a smooth and projective complex surface we define

A Conjectural Description of the Tautological Ring of the Moduli Space of Curves

Abstract: We formulate a number of conjectures giving a rather complete description of the tautological ring of M g and we discuss the evidence for these conjectures.

Kostant polynomials and the cohomology ring for $G/B$

Abstract: The Schubert calculus for G/B can be completely determined by a certain matrix related to the Kostant polynomials introduced in section 5 of Bernstein, Gelfand, and Gelfand [Bernstein, I., Gelfand, I. & Gelfand, S. (1973) Russ. Math. Surv. 28, 1–26]. The polynomials are defined by vanishing properties on the orbit of a regular point under the action of the Weyl group. For each element w in the Weyl group the polynomials also have nonzero values on the orbit points corresponding to elements which are larger in the Bruhat order than w. The main theorem given here is an explicit formula for these values. The matrix of orbit values can be used to determine the cup product for the cohomology ring for G/B, using only linear algebra or as described by Lascoux and Schutzenberger [Lascoux, A. & Schutzenberger, M.-P. (1982) C. R. Seances Acad. Sci. Ser. A 294, 447–450]. Complete proofs of all the theorems will appear in a forthcoming paper.

A New Criterion for Normal Form Algorithms

Abstract: In this paper, we present a new approach for computing normal forms in the quotient algebra A of a polynomial ring R by an ideal I. It is based on a criterion, which gives a necessary and sufficient condition for a projection onto a set of polynomials, to be a normal form modulo the ideal I. This criterion does not require any monomial ordering and generalizes the Buchberger criterion of S-polynomials. It leads to a newa lgorithm for constructing the multiplicative structure of a zero-dimensional algebra. Described in terms of intrinsic operations on vector spaces in the ring of polynomials, this algorithm extends naturally to Laurent polynomials.

Dualizing Complexes, Morita Equivalence and the Derived Picard Group of a Ring

Abstract: Two rings A and B are said to be derived Morita equivalent if the derived categories Db(Mod A) and Db(Mod B) are equivalent. If A and B are derived Morita equivalent algebras over a field k, then there is a complex of bimodules T such that the functor T[otimes ]LA−[ratio ]Db(Mod A) → Db(Mod B) is an equivalence. The complex T is called a tilting complex.When B = A the isomorphism classes of tilting complexes T form the derived Picard group DPic(A). This group acts naturally on the Grothendieck group Ko(A).It is proved that when the algebra A is either local or commutative, then any derived Morita equivalent algebra B is actually Morita equivalent. This enables one to compute DPic(A) in these cases.Assume that A is noetherian. Dualizing complexes over A are complexes of bimodules which generalize the commutative definition. It is proved that the group DPic(A) classifies the set of isomorphism classes of dualizing complexes. This classification is used to deduce properties of rigid dualizing complexes.Finally finite k-algebras are considered. For the algebra A of upper triangular 2×2 matrices over k, it is proved that t3 = s, where t, s ∈ DPic(A) are the classes of A*:= Homk(A, k) and A[1] respectively. In the appendix, by Elena Kreines, this result is generalized to upper triangular n×n matrices, and it is shown that the relation tn+1 = sn−1 holds.

Geometry of Cross-Stream Mixing in a Double-Gyre Ocean Model

Abstract: New dynamical systems techniques are used to analyze fluid particle paths in an eddy resolving, barotropic ocean model of the Gulf Stream. Specifically, the existence of finite-time invariant manifolds associated with transient, mesoscale events such as ring detachment and merger is proved based on computer-assisted analytic results. These “Lagrangian” invariant manifolds completely organize the dynamics and mark the pathways by which fluid parcels may be exchanged across stream. In this way, the Lagrangian flow geometry of a detaching ring or a ring–jet interaction event, as well as the exact associated particle flux, is obtained. The detaching ring geometry indicates that a significant amount of the fluid entrained by the ring originates in a long thin region on the far side of the jet and that this region extends as far upstream as the western boundary current. In the ring–stream interaction case, particle transport occurs both to and from the ring and is concentrated in thin regions on the ne...

A new proof of D. Popescu’s theorem on smoothing of ring homomorphisms

Abstract: §

The diagonal subring and the Cohen-Macaulay property of a multigraded ring

Abstract: Let T be a multigraded ring defined over a local ring (A, m). This paper deals with the question how the Cohen-Macaulay property of T is related to that of its diagonal subring T∆. In the bigraded case we are able to give necessary and sufficient conditions for the Cohen-Macaulayness of T . If I1, . . . , Ir ⊂ A are ideals of positive height, we can then compare the CohenMacaulay property of the multi-Rees algebra RA(I1, . . . , Ir) with the CohenMacaulay property of the usual Rees algebra RA(I1 · · · Ir). We also obtain a bound for the joint reduction numbers of two m-primary ideals in the case the corresponding multi-Rees algebra is Cohen-Macaulay.

Mixed Bruhat Operators and Yang-Baxter Equations for Weyl Groups

Abstract: We introduce and study a family of operators which act in the group algebra of a Weyl group W and provide a multiparameter solution to the quantum Yang-Baxter equations of the corresponding type. These operators are then used to derive new combinatorial properties of W and to obtain new proofs of known results concerning the Bruhat order of W. The paper is organized as follows. Section 2 is devoted to preliminaries on Coxeter groups and associated Yang-Baxter equations. In Theorem 3.1 of Section 3, we describe our solution of these equations. In Section 4, we consider a certain limiting case of our solution, which leads to the quantum Bruhat operators. These operators play an important role in the explicit description of the (small) quantum cohomology ring of G/B. Section 5 contains the proof of Theorem 3.1. Section 6 is devoted to combinatorial applications of our operators. For an arbitrary element u ∈W,we define a graded partial order onW called the tilted Bruhat order; this partial order has unique minimal element u. (The usual Bruhat order corresponds to the special case where u = e, the identity element.) We then prove that tilted Bruhat orders are lexicographically shellable graded posets whose every interval is Eulerian. This generalizes the well-known results of D.-N. Verma, A. Bjorner, M. Wachs, and M. Dyer.

Caustic ring singularity

Abstract: I investigate the caustics produced by the fall of collisionless dark matter in and out of a galaxy in the limit of negligible velocity dispersion. The outer caustics are spherical shells enveloping the galaxy. The inner caustics are rings. These are located near where the particles with the most angular momentum are at their distance of closest approach to the galactic center. The surface of a caustic ring is a closed tube whose cross section is a D[sub [minus]4] catastrophe. It has three cusps amongst which exists a discrete Z[sub 3] symmetry. A detailed analysis is given in the limit where the flow of particles is axially and reflection symmetric and where the transverse dimensions of the ring are small compared to the ring radius. Five parameters describe the caustic in that limit. The relations between these parameters and the initial velocity distribution of the particles are derived. The structure of the caustic ring is used to predict the shape of the bump produced in a galactic rotation curve by a caustic ring lying in the galactic plane. [copyright] [ital 1999] [ital The American Physical Society]

Cellular algebras: inflations and morita equivalences

Abstract: Cellular algebras have recently been introduced by Graham and Lehrer [5, 6] as a convenient axiomatization of all of the following algebras, each of them containing information on certain classical algebraic or finite groups: group algebras of symmetric groups in any characteristic, Hecke algebras of type A or B (or more generally, Ariki Koike algebras), Brauer algebras, Temperley–Lieb algebras, (q-)Schur algebras, and so on. The problem of determining a parameter set for, or even constructing bases of simple modules, is in this way reduced (but of course not solved in general) to questions of linear algebra.The present paper has two aims. First, we make explicit an inductive construction of cellular algebras which has as input data of linear algebra, and which in fact produces all cellular algebras (but no other ones). This is what we call ‘inflation’. This construction also exhibits close relations between several of the above algebras, as can be seen from the computations in [6]. Among the consequences of the construction is a natural way of generalizing Hochschild cohomology. Another consequence is the construction of certain idempotents which is used in the second part of the paper.The second aim is to study Morita equivalences of cellular algebras. Since the input of many of the constructions of representation theory of finite-dimensional algebras is a basic algebra, it is useful to know whether any finite-dimensional cellular algebra is Morita equivalent to a basic one by a Morita equivalence that preserves the cellular structure. It turns out that the answer is ‘yes’ if the underlying field has characteristic other than 2. However, there are counterexamples in the case of characteristic 2, or more generally for any ring in which 2 is not invertible. This also tells us that the notion of ‘cellular’ cannot be defined only in terms of the module category. However, in any characteristic we find some useful Morita equivalences which are compatible with cellular structures.

Self-dual codes over rings and the Chinese remainder theorem

Abstract: . We give some characterizations of self-dual codes over rings, specifically 1 11e ring \mathbb{Z}_{2k} , where \mathbb{Z}_{2k} denotes the ring \mathbb{Z}/2k\mathbb{Z} of integers modulo 2k , using the (i’,hi11cse Remainder Theorem, investigating Type I and Type II codes The Chinese Re\prime naindt^{\backslash }t. Theorem plays an important role in the study of self-dual codes over \mathbb{Z}_{2k} when 2k is I10\{ a prime power, while the Hensel lift is a powerful tool when 2k is a prime I)ower Ir1 particular, we concentrate on the case k=3 and use construction A to build unimodular and 3-modular lattices.

Products on $MU$-modules

Abstract: In [2, Chapter V], Elmendorf, Kriz, Mandell and May (hereafter referred to as EKMM) use their new technology of modules over highly structured ring spectra to give new constructions of MU -modules such as BP , K(n) and so on, which makes it much easier to analyse product structures on these spectra. Unfortunately, their construction only works in its simplest form for modules over MU [12 ]∗ that are concentrated in degrees divisible by 4; this guarantees that various obstruction groups are trivial. In the present paper we extend the EKMM results to the cases where p = 2 or the homotopy groups are allowed to be nonzero in all even degrees. In this context the obstruction groups are nontrivial. We shall show that there are never any obstructions to associativity. We prove in Section 7 that the obstructions to commutativity are given by a certain power operation; this was inspired by a parallel result of Mironov in Baas-Sullivan theory [6]. In Section 8 we shall use formal group theory to derive various formulae for this power operation. In Section 9 we deduce a number of results about realising 2-local MU∗-modules as MU -modules.

Preprojective algebras, differential operators and a Conze embedding for deformations of Kleinian singularities

Abstract: For any associative algebra A over a field K we define a family of algebras \( \Pi^\lambda(A) \) for \( \lambda \in K \,\otimes_{\Bbb Z}\,{\rm K_0}(A) \). In case A is the path algebra of a quiver, one recovers the deformed preprojective algebra introduced by M. P. Holland and the author. In case A is the coordinate ring of a smooth curve, the family includes the ring of differential operators for A and the coordinate ring of the cotangent bundle for Spec A. In case A is quasi-free and \( \Omega^1 \)A is a finitely generated A-A-bimodule we prove that \( \Pi^\lambda(A) \) is well-behaved under localization. We use this to prove a Conze embedding for deformations of Kleinian singularities.

On semilocal modules and rings

Abstract: It is well-known that a ring Ris semiperfect if and only if RR (orRR ) is a supplemented module. Considering weak supplementsinstead of supplements we show that weakly supplemented modules Mare semilocal (i.e.M/Rad(M) is semisimple) and that R is a semilocal ring if and only if RR (orRR ) is weakly supplemented. In this context the notion of finite hollow dimension (or finite dual Goldie dimension) of modules is of interest and yields a natural interpretation of the Camps-Dicks characterization of semilocal rings. Finitely generated modules are weakly supplemented if and only if they have finite hollow dimension (or are semilocal).

Method for simulating a ring back for a call between parties in different communication networks

Abstract: A ring back signal for a call between a calling party and a called party can be simulated. A ring back message associated with the call is received. The calling party is associated with a first network. The called party is associated with a second network. A prestored ring back signal is selected from a set of prestored ring back signals based on the ring back message and/or a called number for the call. The selected prestored ring back signal is associated with the second network and is different from a second prestored ring back signal associated with the first network. The prestored ring back signal is sent to the calling party.

On ⌖-supplemented Modules

Abstract: Let R be a ring and M a right R-module. M is called ⌖-supplemented if every submodule of M has a supplement that is a direct summand of M, and M is called completely ⌖-supplemented if every direct summand of M is ⌖-supplemented. In this paper various properties of these modules are developed. It is shown that (1) Any finite direct sum of ⌖-supplemented modules is ⌖-supplemented. (2) If M is ⌖-supplemented and (D3) then M is completely ⌖-supplemented.

Equivariant de Rham theory and graphs

Abstract: Goresky, Kottwitz and MacPherson have recently shown that the computation of the equivariant cohomology ring of a G-manifold can be reduced to a computation in graph theory. This opens up the possibility that many of the fundamental theorems in equivariant de Rham theory may, on closer inspection, turn out simply to be theorems about graphs. In this paper we show that for some familiar theorems, this is indeed the case.

On the Cohomology Ring of an Algebra

Abstract: We define several versions of the cohomology ring of an associative algebra. These ring structures unify some well known operations from homological algebra and differential geometry. They have some formal resemblance with the quantum multiplication on Floer cohomology of free loop spaces. We discuss some examples, as well as applications to index theorems, characteristic classes and deformations.

A solution of Deligne's conjecture

Abstract: Deligne asked in 1993 whether the Hochschild cochain complex of an associative ring has a natural action by the singular chains of the little 2-cubes operad. In this paper we give an affirmative answer to this question. We also show that the topological Hochschild cohomology spectrum of an associative ring spectrum has an action of an operad equivalent to the little 2-cubes.

F-regularity does not deform

Abstract: We show that the property of F-regularity does not deform, and thereby settle a long- standing open question in the theory of tight closure. Specifically, we construct a three dimensional -graded domain R which is not F-regular (or even F-pure), but has a quotient R tR which is F-regular. Examples are constructed over fields of characteristic p 0, as well as over fields of characteristic zero. 1. Introduction. Throughout this paper, all rings are commutative, Noethe- rian, and have an identity element. The theory of tight closure was developed by Melvin Hochster and Craig Huneke in (HH2) and draws attention to rings which have the property that all their ideals are tightly closed, called weakly F-regular rings. The term F-regular is reserved for rings all of whose localizations are weakly F-regular. A natural question that arose with the development of the the- ory was whether the property of F-regularity deforms, i.e., if (R, m, K) is a local ring such that R tR is F-regular for some nonzerodivisor t m ,m ustR be F- regular? (See the Epilogue of (Ho).) Hochster and Huneke showed that this is indeed true if the ring R is Gorenstein, (HH3), and their work has been followed by various attempts at extending this result, see (AKM), (Si), (Sm3). Our pri- mary goal here is to settle this question by constructing a family of examples to show that F-regularity does not deform. We shall throughout be considering -graded rings, but local examples can be obtained, in all cases, by localizing at the homogeneous maximal ideals. Our main result is: THEOREM 1.1. There exists an -graded ring R of dimension three (finitely generated over a field R 0 = K of characteristic p 2) which is not F-pure, but has an F-regular quotient R tR where t m is a homogeneous nonzerodivisor. Specifically, for positive integers m and n satisfying m m n 2, consider the ring R = K(A, B, C, D, T) I where I is generated by the size two minors of the

A local-to-global principle for deformations of Galois representations

Abstract: Given an absolutely irreducible Galois representation : GE! GLN (k), E a number eld, k a nite eld of characteristic l > 2, and a nite set of places Q of E containing all places above l and1 and all where ramies, there have been dened many functors representing strict equivalence classes of deformations of such a representation, e.g. by Mazur or Wiles in [15] or [26], with various conditions on the behaviour of the deformations at the places in Q and with the condition that the deformations are unramied outside Q. Those functors are known to be representable. For as above, our goal is to present a rather general class of global deformation functors that satisfy local deformation conditions and to investigate for those, under what conditions the global deformation functor is determined by the local deformation functors. We will give precise conditions under which the local functors for all places in Q are sucient to describe the global functor, rst in a coarse form, then in a rened form using auxiliary primes as done by Taylor and Wiles in [24]. This has several consequences. The strongest is that one can derive ring theoretic results for the universal deformation space by Mazur if one uses results of Diamond and Wiles, c.f. [11] and [26], and if one has a good understanding of all local situations. Furthermore it is easier to understand what happens under increasing the ramication

On a Ring of Formal Pseudo-differential Operators

Abstract: We study the notion of non-commumative higher dimensional local fields. A simplest example is the ring P of formal pseudo- differential operators. As an application we extend the KP hierarchy to the space $P^n$.

Integration of solutions of linear functional equations

Abstract: We introduce the notion of the adjoint Ore ring and give a definition of an adjoint polynomial, operator and equation. We apply this for integrating solutions of Ore equations. ∗

Connection set-up and path assignment in wavelength division multiplexed ring networks

Abstract: Techniques for physically implementing fiber ring networks which achieve full mesh connectivity, such networks including a 2-fiber WDM ring network composed of a clockwise ring, and a 4-fiber WDM Self-Healing Ring Network. The number of wavelengths required is derived for both odd and even number of nodes (901-904) on the ring. To physically set-up all required connections in the network, optimal wavelength assignment algorithms are devised so that the wavelength assignment between nodes (901-904) on the ring is systematic and engenders full mesh connectivity while avoiding any possible violation of the color clash constraint. An illustrative algorithm uses a simple matrix approach for calculating the interconnection arrangement.

An infinite analogue of rings with stable rank one

Abstract: Replacing invertibility with quasi-invertibility in Bass' first stable range condition we discover a new class of rings, the QB-rings. These constitute a considerable enlargement of the class of rings with stable rank one (B-rings), and include examples like the ring of endomorphisms of a vector space over a field F, and the ring of all row- and column- finite matrices over F. We show that the category of QB-rings is stable under the formation of corners, ideals and quotients, as well as matrices and direct limits. We also give necessary and sufficient conditions for an extension of QB-rings to be again a QB-ring, and show that extensions of B-rings often lead to QB-rings. Specializing to the category of exchange rings we characterize the subset of exchange QB-rings as those in which every von Neumann regular element extends to a maximal regular element, i.e. a quasi-invertible element. Finally we show that the C*- algebras that are QB-rings are exactly the extremally rich C*-algebras previously studied by L.G. Brown and the second author.