On supports and associated primes of modules over the enveloping algebras of nilpotent Lie algebras
Boris
Sirola
2131-2170
Abstract: Let $\mathfrak n$ be a nilpotent Lie algebra, over a field of characteristic zero, and $\mathcal U$ its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of $\mathcal U$ related to finitely generated $\mathcal U$-modules $V$, and in particular the set $\operatorname{Ass}V$ of associated primes for such $V$ (note that now $\operatorname{Ass}V$ is equal to the set $\operatorname{Annspec}V$ of annihilator primes for $V$); (2) the problem of nontriviality for the modules $V/\mathcal PV$ when $\mathcal P$ is a (maximal) prime of $\mathcal U$, and in particular when $\mathcal P$ is the augmentation ideal $\mathcal U\mathfrak n$ of $\mathcal U$. We define the support of $V$, as a natural generalization of the same notion from commutative theory, and show that it is the object of primary interest when dealing with (2). We also introduce and study the reduced localization and the reduced support, which enables to better understand the set $\operatorname{Ass}V$. We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when $\mathfrak N$, $\mathfrak N$ as in the theorem, are abelian. We also present some of its interesting consequences. Theorem. Let $\mathfrak Q$ be a finite-dimensional Lie algebra over a field of characteristic zero, and $\mathfrak N$ an ideal of $\mathfrak Q$; denote by $U(\mathfrak N)$ the universal enveloping algebra of $\mathfrak N$. Let $V$ be a $\mathfrak Q$-module which is finitely generated as an $\mathfrak N$-module. Then every annihilator prime of $V$, when $V$ is regarded as a $U(\mathfrak N)$-module, is $\mathfrak Q$-stable for the adjoint action of $\mathfrak Q$ on $U(\mathfrak N)$.
Groups with two extreme character degrees and their normal subgroups
Gustavo
A.
Fernández-Alcober;
Alexander
Moretó
2171-2192
Abstract: We study the finite groups $G$ for which the set $\operatorname{cd}(G)$ of irreducible complex character degrees consists of the two most extreme possible values, that is, $1$ and $\vert G:Z(G)\vert^{1/2}$. We are easily reduced to finite $p$-groups, for which we derive the following group theoretical characterization: they are the $p$-groups such that $\vert G:Z(G)\vert$ is a square and whose only normal subgroups are those containing $G'$ or contained in $Z(G)$. By analogy, we also deal with $p$-groups such that $\vert G:Z(G)\vert=p^{2n+1}$ is not a square, and we prove that $\operatorname{cd}(G) =\{1,p^n\}$ if and only if a similar property holds: for any $N\trianglelefteq G$, either $G'\le N$ or $\vert NZ(G):Z(G)\vert\le p$. The proof of these results requires a detailed analysis of the structure of the $p$-groups with any of the conditions above on normal subgroups, which is interesting for its own sake. It is especially remarkable that these groups have small nilpotency class and that, if the nilpotency class is greater than $2$, then the index of the centre is small, and in some cases we may even bound the order of $G$.
Simple holonomic modules over rings of differential operators with regular coefficients of Krull dimension 2
V.
Bavula;
F.
van Oystaeyen
2193-2214
Abstract: Let $K$ be an algebraically closed field of characteristic zero. Let $\Lambda$ be the ring of ($K$-linear) differential operators with coefficients from a regular commutative affine domain of Krull dimension $2$ which is the tensor product of two regular commutative affine domains of Krull dimension $1$. Simple holonomic $\Lambda$-modules are described. Let a $K$-algebra $D$ be a regular affine commutative domain of Krull dimension $1$ and ${\cal D} (D)$ be the ring of differential operators with coefficients from $D$. We classify (up to irreducible elements of a certain Euclidean domain) simple ${\cal D}(D)$-modules (the field $K$ is not necessarily algebraically closed).
The monopole equations and $J$-holomorphic curves on weakly convex almost Kähler 4-manifolds
Yutaka
Kanda
2215-2243
Abstract: We prove that a weakly convex almost Kähler 4-manifold contains a compact, non-constant $J$-holomorphic curve if the corresponding monopole invariant is not zero and if the corresponding line bundle is non-trivial.
Non-special, non-canal isothermic tori with spherical lines of curvature
Holly
Bernstein
2245-2274
Abstract: This article examines isothermic surfaces smoothly immersed in Möbius space. It finds explicit examples of non-special, non-canal isothermic tori with spherical lines of curvature in two systems by analyzing Darboux transforms of Dupin tori. In addition, it characterizes the property of spherical lines of curvature in terms of differential equations on the Calapso potential of the isothermic immersion, and investigates the effect of classical transformations on this property.
Nonradial Hörmander algebras of several variables and convolution operators
José
Bonet;
Antonio
Galbis;
Siegfried
Momm
2275-2291
Abstract: A characterization of the closed principal ideals in nonradial Hörmander algebras of holomorphic functions of several variables in terms of the behaviour of the generator is obtained. This result is applied to study the range of convolution operators and ultradifferential operators on spaces of quasianalytic functions of Beurling type. Contrary to what is known to happen in the case of non-quasianalytic functions, an ultradistribution on a space of quasianalytic functions is constructed such that the range of the operator does not contain the real analytic functions.
Gröbner bases, H--bases and interpolation
Thomas
Sauer
2293-2308
Abstract: The paper is concerned with a construction for H-bases of polynomial ideals without relying on term orders. The main ingredient is a homogeneous reduction algorithm which orthogonalizes leading terms instead of completely canceling them. This allows for an extension of Buchberger's algorithm to construct these H-bases algorithmically. In addition, the close connection of this approach to minimal degree interpolation, and in particular to the least interpolation scheme due to de Boor and Ron, is pointed out.
Good ideals in Gorenstein local rings
Shiro
Goto;
Sin-Ichiro
Iai;
Kei-ichi
Watanabe
2309-2346
Abstract: Let $I$ be an $\mathfrak{m}$-primary ideal in a Gorenstein local ring ($A$, $\mathfrak{m}$) with $\dim A = d$, and assume that $I$ contains a parameter ideal $Q$ in $A$ as a reduction. We say that $I$ is a good ideal in $A$ if $G = \sum _{n \geq 0} I^{n}/I^{n+1}$ is a Gorenstein ring with $\mathrm{a} (G) = 1 - d$. The associated graded ring $G$ of $I$ is a Gorenstein ring with $\mathrm{a}(G) = -d$ if and only if $I = Q$. Hence good ideals in our sense are good ones next to the parameter ideals $Q$ in $A$. A basic theory of good ideals is developed in this paper. We have that $I$ is a good ideal in $A$ if and only if $I^{2} = QI$ and $I = Q : I$. First a criterion for finite-dimensional Gorenstein graded algebras $A$ over fields $k$ to have nonempty sets $\mathcal{X}_{A}$ of good ideals will be given. Second in the case where $d = 1$ we will give a correspondence theorem between the set $\mathcal{X}_{A}$ and the set $\mathcal{Y}_{A}$ of certain overrings of $A$. A characterization of good ideals in the case where $d = 2$ will be given in terms of the goodness in their powers. Thanks to Kato's Riemann-Roch theorem, we are able to classify the good ideals in two-dimensional Gorenstein rational local rings. As a conclusion we will show that the structure of the set $\mathcal{X}_{A}$ of good ideals in $A$ heavily depends on $d = \dim A$. The set $\mathcal{X}_{A}$ may be empty if $d \leq 2$, while $\mathcal{X}_{A}$ is necessarily infinite if $d \geq 3$ and $A$contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring $k[X_{1},X_{2},X_{3}]$ in three variables over a field $k$. Examples are given to illustrate the theorems.
Relative Embedding Problems
Elena
V.
Black;
John
R.
Swallow
2347-2370
Abstract: We consider Galois embedding problems $G\twoheadrightarrow H\cong \operatorname{Gal}(X/Z)$ such that a Galois embedding problem $G\twoheadrightarrow \operatorname{Gal}(Y/Z)$ is solvable, where $Y/Z$ is a Galois subextension of $X/Z$. For such embedding problems with abelian kernel, we prove a reduction theorem, first in the general case of commutative $k$-algebras, then in the more specialized field case. We demonstrate with examples of dihedral embedding problems that the reduced embedding problem is frequently of smaller order. We then apply these results to the theory of obstructions to central embedding problems, considering a notion of quotients of central embedding problems, and classify the infinite towers of metacyclic $p$-groups to which the reduction theorem applies.
On the computation of stabilized tensor functors and the relative algebraic $K$-theory of dual numbers
Randy
McCarthy
2371-2390
Abstract: We compute the stabilization of functors from exact categories to abelian groups derived from $n$-fold tensor products. Rationally, this gives a new computation for the relative algebraic $K$-theory of dual numbers.
On the telescopic homotopy theory of spaces
A.
K.
Bousfield
2391-2426
Abstract: In telescopic homotopy theory, a space or spectrum $X$ is approximated by a tower of localizations $L^{f}_{n}X$, $n\ge 0$, taking account of $v_{n}$-periodic homotopy groups for progressively higher $n$. For each $n\ge 1$, we construct a telescopic Kuhn functor $\Phi _{n}$ carrying a space to a spectrum with the same $v_{n}$-periodic homotopy groups, and we construct a new functor $\Theta _{n}$ left adjoint to $\Phi _{n}$. Using these functors, we show that the $n$th stable monocular homotopy category (comprising the $n$th fibers of stable telescopic towers) embeds as a retract of the $n$th unstable monocular homotopy category in two ways: one giving infinite loop spaces and the other giving ``infinite $L^{f}_{n}$-suspension spaces.'' We deduce that Ravenel's stable telescope conjectures are equivalent to unstable telescope conjectures. In particular, we show that the failure of Ravenel's $n$th stable telescope conjecture implies the existence of highly connected infinite loop spaces with trivial Johnson-Wilson $E(n)_{*}$-homology but nontrivial $v_{n}$-periodic homotopy groups, showing a fundamental difference between the unstable chromatic and telescopic theories. As a stable chromatic application, we show that each spectrum is $K(n)$-equivalent to a suspension spectrum. As an unstable chromatic application, we determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of infinite loop spaces in terms of $E(n)_{*}$-localizations of spectra under suitable conditions. We also determine the $E(n)_{*}$-localizations and $K(n)_{*}$-localizations of arbitrary Postnikov $H$-spaces.
Equivariant surgery with middle dimensional singular sets. II: Equivariant framed cobordism invariance
Masaharu
Morimoto
2427-2440
Abstract: Let $G$ be a finite group and let $f : X \to Y$ be a degree 1, $G$-framed map such that $X$ and $Y$ are simply connected, closed, oriented, smooth manifolds of dimension $n = 2k \geqq 6$ and such that the dimension of the singular set of the $G$-space $X$ is at most $k$. In the previous article, assuming $f$ is $k$-connected, we defined the $G$-equivariant surgery obstruction $\sigma (f)$ in a certain abelian group. There it was shown that if $\sigma (f) = 0$ then $f$ is $G$-framed cobordant to a homotopy equivalence $f' : X' \to Y$. In the present article, we prove that the obstruction $\sigma (f)$ is a $G$-framed cobordism invariant. Consequently, the $G$-surgery obstruction $\sigma (f)$ is uniquely associated to $f : X \to Y$ above even if it is not $k$-connected.
Model category structures on chain complexes of sheaves
Mark
Hovey
2441-2457
Abstract: The unbounded derived category of a Grothendieck abelian category is the homotopy category of a Quillen model structure on the category of unbounded chain complexes, where the cofibrations are the injections. This folk theorem is apparently due to Joyal, and has been generalized recently by Beke. However, in most cases of interest, such as the category of sheaves on a ringed space or the category of quasi-coherent sheaves on a nice enough scheme, the abelian category in question also has a tensor product. The injective model structure is not well-suited to the tensor product. In this paper, we consider another method for constructing a model structure. We apply it to the category of sheaves on a well-behaved ringed space. The resulting flat model structure is compatible with the tensor product and all homomorphisms of ringed spaces.
Stratified solutions for systems of conservation laws
Andrea
Corli;
Olivier
Gues
2459-2486
Abstract: We study a class of weak solutions to hyperbolic systems of conservation (balance) laws in one space dimension, called stratified solutions. These solutions are bounded and ``regular'' in the direction of a linearly degenerate characteristic field of the system, but not in other directions. In particular, they are not required to have finite total variation. We prove some results of local existence and uniqueness.
Whitney's extension problem for multivariate $C^{1,\omega}$-functions
Yuri
Brudnyi;
Pavel
Shvartsman
2487-2512
Abstract: We prove that the trace of the space $C^{1,\omega}({\mathbb R}^n)$to an arbitrary closed subset $X\subset{\mathbb R}^n$is characterized by the following ``finiteness'' property. A function $f:X\rightarrow{\mathbb R}$belongs to the trace space if and only if the restriction $f\vert _Y$ to an arbitrary subset $Y\subset X$ consisting of at most $3\cdot 2^{n-1}$ can be extended to a function $f_Y\in C^{1,\omega}({\mathbb R}^n)$ such that \begin{displaymath}\sup\{\Vert f_Y\Vert _{C^{1,\omega}}:~Y\subset X, ~\operatorname{card} Y\le 3\cdot 2^{n-1}\}<\infty. \end{displaymath} The constant $3\cdot 2^{n-1}$ is sharp. The proof is based on a Lipschitz selection result which is interesting in its own right.
Topological horseshoes
Judy
Kennedy;
James
A.
Yorke
2513-2530
Abstract: When does a continuous map have chaotic dynamics in a set $Q$? More specifically, when does it factor over a shift on $M$ symbols? This paper is an attempt to clarify some of the issues when there is no hyperbolicity assumed. We find that the key is to define a ``crossing number'' for that set $Q$. If that number is $M$ and $M>1$, then $Q$ contains a compact invariant set which factors over a shift on $M$ symbols.
Complexifications of symmetric spaces and Jordan theory
Wolfgang
Bertram
2531-2556
Abstract: Generalizing Hermitian and pseudo-Hermitian spaces, we define twisted complex symmetric spaces, and we show that they correspond to an algebraic object called Hermitian Jordan triple products. The main topic of this work is to investigate the class of real forms of twisted complex symmetric spaces, called the category of symmetric spaces with twist. We show that this category is equivalent to the category of all real Jordan triple systems, and we can use a work of B.O. Makarevic in order to classify the irreducible spaces. The classification shows that most irreducible symmetric spaces have exactly one twisted complexification. This leads to open problems concerning the relation of Jordan and Lie triple systems.