Let be a nilpotent Lie algebra, over a field of characteristic zero, and its universal enveloping algebra. In this paper we study: (1) the prime ideal structure of related to finitely generated -modules , and in particular the set of associated primes for such (note that now is equal to the set of annihilator primes for ); (2) the problem of nontriviality for the modules when is a (maximal) prime of , and in particular when is the augmentation ideal of . We define the support of , 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 . We prove the following generalization of a stability result given by W. Casselman and M. S. Osborne in the case when , as in the theorem, are abelian. We also present some of its interesting consequences.
Theorem. Let be a finite-dimensional Lie algebra over a field of characteristic zero, and an ideal of ; denote by the universal enveloping algebra of . Let be a -module which is finitely generated as an -module. Then every annihilator prime of , when is regarded as a -module, is -stable for the adjoint action of on .
We study the finite groups for which the set of irreducible complex character degrees consists of the two most extreme possible values, that is, and . We are easily reduced to finite -groups, for which we derive the following group theoretical characterization: they are the -groups such that is a square and whose only normal subgroups are those containing or contained in . By analogy, we also deal with -groups such that is not a square, and we prove that if and only if a similar property holds: for any , either or . The proof of these results requires a detailed analysis of the structure of the -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 , then the index of the centre is small, and in some cases we may even bound the order of .
Let be a finite group and let be a degree 1, -framed map such that and are simply connected, closed, oriented, smooth manifolds of dimension and such that the dimension of the singular set of the -space is at most . In the previous article, assuming is -connected, we defined the -equivariant surgery obstruction in a certain abelian group. There it was shown that if then is -framed cobordant to a homotopy equivalence . In the present article, we prove that the obstruction is a -framed cobordism invariant. Consequently, the -surgery obstruction is uniquely associated to above even if it is not -connected.
Let be an -primary ideal in a Gorenstein local ring (, ) with , and assume that contains a parameter ideal in as a reduction. We say that is a good ideal in if is a Gorenstein ring with . The associated graded ring of is a Gorenstein ring with if and only if . Hence good ideals in our sense are good ones next to the parameter ideals in . A basic theory of good ideals is developed in this paper. We have that is a good ideal in if and only if and . First a criterion for finite-dimensional Gorenstein graded algebras over fields to have nonempty sets of good ideals will be given. Second in the case where we will give a correspondence theorem between the set and the set of certain overrings of . A characterization of good ideals in the case where 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 of good ideals in heavily depends on . The set may be empty if , while is necessarily infinite if and contains a field. To analyze this phenomenon we shall explore monomial good ideals in the polynomial ring in three variables over a field . Examples are given to illustrate the theorems.
This paper proves that a connected matroid in which a largest circuit and a largest cocircuit have and elements, respectively, has at most elements. It is also shown that if is an element of and and are the sizes of a largest circuit containing and a largest cocircuit containing , then . Both these bounds are sharp and the first is proved using the second. The second inequality is an interesting companion to Lehman's width-length inequality which asserts that the former inequality can be reversed for regular matroids when and are replaced by the sizes of a smallest circuit containing and a smallest cocircuit containing . Moreover, it follows from the second inequality that if and are distinct vertices in a -connected loopless graph , then cannot exceed the product of the length of a longest -path and the size of a largest minimal edge-cut separating from .
When does a continuous map have chaotic dynamics in a set ? More specifically, when does it factor over a shift on 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 . If that number is and 1$">, then contains a compact invariant set which factors over a shift on symbols.
For an nonnegative matrix , an isomorphism is obtained between the lattice of initial subsets (of ) for and the lattice of -invariant faces of the nonnegative orthant . Motivated by this isomorphism, we generalize some of the known combinatorial spectral results on a nonnegative matrix that are given in terms of its classes to results for a cone-preserving map on a polyhedral cone, formulated in terms of its invariant faces. In particular, we obtain the following extension of the famous Rothblum index theorem for a nonnegative matrix: If leaves invariant a polyhedral cone , then for each distinguished eigenvalue of for , there is a chain of distinct -invariant join-irreducible faces of , each containing in its relative interior a generalized eigenvector of corresponding to (referred to as semi-distinguished -invariant faces associated with ), where is the maximal order of distinguished generalized eigenvectors of corresponding to , but there is no such chain with more than members. We introduce the important new concepts of semi-distinguished -invariant faces, and of spectral pairs of faces associated with a cone-preserving map, and obtain several properties of a cone-preserving map that mostly involve these two concepts, when the underlying cone is polyhedral, perfect, or strictly convex and/or smooth, or is the cone of all real polynomials of degree not exceeding that are nonnegative on a closed interval. Plentiful illustrative examples are provided. Some open problems are posed at the end.
A standing conjecture in -cohomology says that every finite -complex is of -determinant class. In this paper, we prove this whenever the fundamental group belongs to a large class of groups containing, e.g., all extensions of residually finite groups with amenable quotients, all residually amenable groups, and free products of these. If, in addition, is -acyclic, we also show that the -determinant is a homotopy invariant -- giving a short and easy proof independent of and encompassing all known cases. Under suitable conditions we give new approximation formulas for -Betti numbers.
Let be a commutative ring and an ideal in which is locally generated by a regular sequence of length . Then, each f. g. projective -module has an -projective resolution of length . In this paper, we compute the homology of the -th Koszul complex associated with the homomorphism for all , if . This computation yields a new proof of the classical Adams-Riemann-Roch formula for regular closed immersions which does not use the deformation to the normal cone any longer. Furthermore, if , we compute the homology of the complex where and denote the functors occurring in the Dold-Kan correspondence.
Kadison has shown that local derivations from a von Neumann algebra into any dual bimodule are derivations. In this paper we extend this result to local derivations from any -algebra into any Banach -bimodule . Most of the work is involved with establishing this result when is a commutative -algebra with one self-adjoint generator. A known result of the author about Jordan derivations then completes the argument. We show that these results do not extend to the algebra of continuously differentiable functions on . We also give an automatic continuity result, that is, we show that local derivations on -algebras are continuous even if not assumed a priori to be so.
The authors get similar results in the case of formal power series satifying growth conditions, of Gevrey type for instance. Moreover, the proofs here give, in the analytic case, a control of the radius of convergence of by the radius of convergence of .
RÉSUMÉ. Soit une application holomorphe de dans définie dans un voisinage de et vérifiant . Si le jacobien de n'est pas identiquement nul au voisinage de , P.M. Eakin et G.A. Harris ont établi le résultat suivant: toute série formelle telle que est analytique est elle-même analytique. Si le jacobien de est identiquement nul, ils montrent que la conclusion précédente est fausse.
Les auteurs obtiennent des résultats analogues pour les séries formelles à croissance contrôlée, du type Gevrey par exemple. De plus, les preuves données ici permettent, dans le cas analytique, un contrôle du rayon de convergence de par celui de .
Let be a closed submanifold of a complete smooth Riemannian manifold and the total space of the unit normal bundle of . For each , let denote the distance from to the cut point of on the geodesic with the velocity vector The continuity of the function on is well known. In this paper we prove that is locally Lipschitz on which is bounded; in particular, if and are compact, then is globally Lipschitz on . Therefore, the canonical interior metric may be introduced on each connected component of the cut locus of and this metric space becomes a locally compact and complete length space.
Fix integers with k>0$"> and . Let be an integral projective curve with and a rank torsion free sheaf on which is a flat limit of a family of locally free sheaves on . Here we prove the existence of a rank subsheaf of such that . We show that for every there is an integral projective curve not Gorenstein, and a rank 2 torsion free sheaf on with no rank 1 subsheaf with . We show the existence of torsion free sheaves on non-Gorenstein projective curves with other pathological properties.
We prove that is sufficient to construct a model in which is measurable and is a closed and unbounded subset of containing only inaccessible cardinals of . Gitik proved that is necessary.
We also calculate the consistency strength of the existence of such a set together with the assumption that is Mahlo, weakly compact, or Ramsey. In addition we consider the possibility of having the set generate the closed unbounded ultrafilter of while remains measurable, and show that Radin forcing, which requires a weak repeat point, cannot be improved on.
Let be the group of automorphisms of a homogeneous tree , and let be a lattice subgroup of . Let be the tensor product of two spherical irreducible unitary representations of . We give an explicit decomposition of the restriction of to . We also describe the spherical component of explicitly, and this decomposition is interpreted as a multiplication formula for associated orthogonal polynomials.
Generalized Eilenberg-Borsuk Theorem. Let be a countable CW complex. If is a separable metrizable space and is an absolute extensor of for some CW complex , then for any map , closed in , there is an extension of over an open set such that .
Theorem. Let be countable CW complexes. If is a separable metrizable space and is an absolute extensor of , then there is a subset of such that and .
Theorem. Suppose are countable, non-trivial, abelian groups and 0$">. For any separable metrizable space of finite dimension 0$">, there is a closed subset of with for .
Theorem. Suppose is a separable metrizable space of finite dimension and is a compactum of finite dimension. Then, for any , , there is a closed subset of such that and .
Theorem. Suppose is a metrizable space of finite dimension and is a compactum of finite dimension. If and are connected CW complexes, then
On the rank of the -class group of . Let be a square-free positive integer and be a prime such that . We set , where or . In this paper, we determine the rank of the -class group of .
RÉSUMÉ. Soit , un corps biquadratique où ou bien un premier et étant un entier positif sans facteurs carrés. Dans ce papier, on détermine le rang du -groupe de classes de .