Showing papers in "Mathematische Zeitschrift in 1980"

Group-graded rings and modules

Abstract: One of the guiding principles of modern mathematics is that the main difficulty in any subject should be to find the right way to look at it. Once this proper viewpoint is found the theorems become obvious and their proofs routine if not trivial. Make the right definitions and the rest will take care of itself! This principle has been of little utility in group theory. However, it certainly applies to the subject under consideration here, which is the study of the representations of a group H related to a normal subgroup N of H. The 'correct ' approach which we shall adopt is to consider the group ring fH of H, over any coefficient ring f, as graded by the factor group G = H/N. Of course, a G-graded ring 91, for any multiplicative group G, is a ring which is a direct sum of additive subgroups 91~, one for each aeG, such that the ring multiplication and the group multiplication are related by:

On a Class of non Linear Schrödinger Equations with non Local Interaction.

Abstract: As regards the first question we prove existence and uniqueness of the solutions of the Cauchy problem with finite initial time; as regards the second one, we prove the existence of solutions of the Cauchy problem with infinite initial time, which implies the existence of the wave operators, and we prove asymptotic completeness for a class of repulsive interactions. The main reason for this investigation is that the Eq. (0.1) can be considered as the classical limit, in the field sense, of the field equation describing a

Constructing Lens Spaces by Surgery on Knots.

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Rings of Fricke characters and automorphism groups of free groups

Abstract: Let F n be the free group on n free generators g~(v= 1, , n) and let ~ , be its group of automorphisms Its quotient group (b* with respect to the inner automorphisms is the group of automorphism classes of F, For n = 2, it is easy to derive a presentation for 45 since ~* is simply the linear group GL(2, 77) of 2 by 2 matrices with integers as entries and determinant __ 1 For n>3 , finding a presentation becomes a difficult problem which was first solved by Nielsen [16] and later by McCool [14] who also proved important theorems about the presentations of subgroups of ~, However, not much is known about the structure of ~, and even less about that of ~b* According to a general theorem of Baumslag [-1], ~b, is residually finite It is already more difficult to prove the same result for ~* if n > 2 ; see Grossman [5] Since there exists a natural mapping of ~, onto GL(n, ~) with a kernel which we shall denote by K,, it seems to be natural to concentrate ones attention on the structure of K, since GL(n, Z) is a well investigated group for all n The quotient group of K, with respect to the inner automorphisms shall be called K* We know [11] that K, (and, therefore, K*) are finitely generated with explicitly known generators We also know that K, is residually torsion free nilpotent This follows immediately from the action of K n on the group ring of F, which is a graded ring [-10] in which the powers of the augmentation ideal provide the grading The action of K, on this ring then provides a faithful representation of K, in terms of upper triangular (infinite) matrices with integers as entries and with terms + 1 in the main diagonal But for K* it is not even known whether it is torsion free or not Nor is it known whether K, (and, therefore, K*) has a finite presentation Also, no finite dimensional matrix representation for ~b,, n > 2 or ~*, n > 3, are known and there is at least some support for the conjecture that none exist (It would be sufficient to show this for ~2; see [13]) However, it has been known for a long time ~4] that ~* acts as a group of automorphisms of a quotient ring Rn of a finitely generated (commutative) ring and it has been shown later [7] that this

Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor

Abstract: S being the Ricci tensor. While every manifold with parallel Ricci tensor has harmonic curvature, i.e., satisfies fiR=O, there are examples ([3], Theorem 5.2) of open Riemannian manifolds with fiR=O and VS+O. In [1] Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this paper is to give examples (see Remark 2) answering this question in the negative. All our examples are conformally flat (Corollary 1). Moreover, we obtain some classification results, restricting our consideration to Riemannian manifolds with fiR = O, VS + 0 and such that the Ricci tensor S has at any point less than three distinct eigenvalues. Starting from a description of their local structure at generic points (Theorem 1), we find all four-dimensional, analytic, complete and simply connected manifolds of this type (Theorem 2). They are all non-compact, but some of them do possess compact quotients. Next we prove (Theorem 3) that all compact four-dimensional analytic Riemannian manifolds with the above properties are covered by S 1 x S 3 with a metric of an explicitly described form. Throughout this paper, by a manifold we mean a connected paracompact manifold of class C ~ or analytic. By abuse of notation, concerning Riemannian manifolds we often write M instead of (M,g) and @ , v ) instead of g(u,v) for tangent vectors u, v.

Stability of minimal surfaces and eigenvalues of the laplacian

Abstract: Let \(x:{\text M}\rightarrow \,\tilde{M}^{n}\) be a minimal immersion of a two-dimensional orientable manifold \({\text M}\) into an n-dimensional Riemannian manifold \(\tilde{M}^{n}\)

Algebra structures on minimal resolutions of Gorenstein rings of embedding codimension four

Abstract: Recently Buchsbaum and Eisenbud [3] exploited the algebra structure on a finite free resolution of a Gorenstein ideal of codimension three to obtain a complete determinantal description of such an ideal. As they pointed out, the study of algebra structures on resolutions has for the most part been confined to the (generally infinite) minimal free resolution of the residue field of a local ring or the Koszul resolution of an ideal generated by a regular sequence. They proposed, however, to extend the scope of the study to all minimal free resolutions of cyclic modules. Khinich [1] furnished an example of a grade four ideal I for which the minimal resolution of R/I does not admit the structure of an associative, differential, graded commutat ive algebra (DGC algebra). Khinich's ring R/I is Cohen-Macaulay, but not Gorenstein. We conjecture that minimal finite free resolutions of Gorenstein factor rings R/a admit D G C algebra structures. In this paper we establish the conjecture for R a Gorenstein local ring in which 2 is a unit and a a Gorenstein ideal of grade (or height) four. We begin by clarifying what we mean by an algebra structure on a resolution. Let

Locally Projective-Planar Lattices which Satisfy the Bundle Theorem

Abstract: This investigation originated with a question about inversive planes, which was answered in [19] and [20]. Those results, as well as their analogues for Laguerre and Minkowski planes are special cases of the main result of the present paper; but we are now concerned with a more general class of objects here called "locally projective-planar lattices" in which some properties particular to one or more of the above-mentioned planes are dispensed with. Our main theorem will be introduced in Sect. 2; we begin here with some discussion of the classical examples. An inversive plane is an incidence structure (see Chap. II) ~ = (0, off) (elements of (g are called circles) which satisfies: (0) Circles are nonempty. (1) For each P~(9, ~p is an affine plane (where J e is the "internal" structure whose points are the points of (9 other than P, whose blocks are the circles of c~ which contain P, and whose incidence is that inherited from J ) . The order of J is the (common) order of the affine planes Jp . Inversive planes arise quite naturally in geometry in the following way. Let K be a skewfield and (9 an ovoid in PG(3, K). (I.e., (9 is a set of points satisfying: (1) no three points of (9 are collinear, and (2) if P~(9, then the union of all lines meeting (9 only in P is a plane. PG(3,K) denotes three-dimensional projective space over K.) Then the following incidence structure, ~r is an inversive plane: The points of J((9) are the points of (9. The circles of J((9) are those planes which meet (9 in more than one point. Incidence is inclusion. An inversive plane is said to be egglike 1 if is isomorphic to some J((9). * Supported in part by ONR Contract ~N00014-76-C-0366 1 This term is due to Dembowski and Hughes [131

On equivariant isometric embeddings

Abstract: The representation p can be regarded as a Lie group homomorphism from G into the orthogonal group O(N) which acts on IE N by rotations and reflections; a smooth map X: M ~ I E N is equivariant with respect to p if and only if X(~p) =p(cr) X(p), for all ~r~G, pEM. The main theorem is true in both the C ~ and real analytic categories. We will work in the C ~ category for the time being, and return to the real analytic case in w 4. Moreover, the theorem holds for manifolds with boundary. The main analytic tool used by Nash to prove his isometric embedding theorem is an implicit function theorem based upon the Newton iteration method. The implicit function theorem applies to the equivariant case with virtually no change. In order to apply the implicit function theorem we need to approximate a given G-invariant Riemannian metric on M by a metric induced by an equivariant embedding; we will do this by using the theory of the Laplace operator on compact Riemannian manifolds. According to Gromov and Rokhlin [7], any n-dimensional compact Riemannian manifold can be isometrically embedded in IF, N, where N = (1/2) n(n + 1) + 3 n + 5. No such universal bound is possible in the equivariant case, and in fact, given any positive integer N, it is possible to construct a left invariant

A Cohomological Property of Regular p-Groups

Abstract: Peter Schmid Mathematisches Institut der Universit~it, Auf der Morgenstelle 10, D-7400-Tiibingen1, Federal Republic of Germany To Helmut Wielandt, on his seventieth birthday, 19 December, 1980 Let A be a Q-module where A and Q are finite p-groups. By a theorem of Gaschtitz and Uchida A is cohomologically trivial provided the (Tare) cohomology H"(Q, A)=0 for just one integer n (cf. [-1], p. 110). This has been used in order to produce noninner p-automorphisms for p-groups. In fact, when N is a normal subgroup of a finite p-group G and Q = G/N, it often happens that the Q-mod- ule A =Z(N), the centre of N, has nontrivial cohomology. The object of this paper is to record the following result. Theorem. Let G be a regular p-group and N a nontrivial normal subgroup of G. If Q = G/N is not cyclic, H"(Q, Z(N)) =t= 0 for all n. If A is a cyclic p-group, p an odd prime, and Q a p-subgroup of Aut(A), the semidirect product G=Q. A is regular but Hn(Q,A)=O. Thus the hypothesis in the theorem cannot be omitted. ' One might ask when extensions of regular p-groups are aga!n regular. Some necessary condition may be taken from Proposition 2 below.' Note, however, that even direct products of regular p-groups need not be regular; see Wielandt's example in [-2], p. 323. 1. Cohomological Triviality and Fixed Points We begin with some basic observations concerning cohomologically trivial mod- ules for p-groups. If A is a (right) Q-module, A o denotes the submodule of fixed points under Q and [A,Q] the commutator submodule. The trace map a ~-+ a ~, x of A is written z = zQ, its image A t and its kernel A,. In dealing with

The Curvature of Most Convex Surfaces Vanishes Almost Everywhere

Abstract: At least since 1904 we know about the existence of singular monotone functions, i.e. continuous monotone functions of one real variable with vanishing derivative a.e. In that year Lebesgue [7] and Minkowski [8] gave their famous examples. In 1910 Faber [4], in 1916 Sierpinski [10] and then many others greatly enriched the variety of known examples. By integrating anyone of these functions we get a differentiable convex function with vanishing second derivative a.e. Thus we obtain smooth convex curves with vanishing curvature a.e. A very beautiful and simple example is due to de R h a m [3J: Take a convex polygon, consider on each side the two points dividing it into three equal parts and take the convex polygon with all these division points as vertices. Repeat the procedure. As a limit we get a smooth, strictly convex curve with a.e. vanishing curvature! The purpose of this paper is to show that, in the sense of Baire categories, most convex curves have the above property. More generally, most convex surfaces in IR" have a.e. a vanishing sectional curvature in every tangent direction. We start with some definitions. By a convex surface we shall always understand a closed convex surface in Busemann's sense (see [11, p. 3). A convex curve is a one-dimensional convex surface. A point x of a convex surface S is called smooth if S is differentiable at x. S is smooth if each of its points is smooth. Any half-line in a supporting plane (this is a hyperplane, see [1], p. 4) of a convex surface S, originating at a point x ~ S will be called supporting direction at x. A supporting direction at a smooth point is called tangent direction. The union of a circular with a square 2-cell, such that the radius and centre of the circle coincide with the side-length and a vertex of the square, is called corner-disk. The two points lying on both the circle and the square are called touching points of the corner-disk. A 2-cell having the union of half a circle with a segment as boundary is called semidisk. The points of the boundary of a semidisk which are not smooth are called corners of the semidisk. The radius of a corner-disk or a semidisk is the radius of the circle appearing in their definitions.