Showing papers in "Archiv der Mathematik in 1981"

Finite soluble groups containing an element of prime order whose centralizer is small

Abstract: In [2] we proved that ifG is a finite group containing an involution whose centralizer has order bounded by some numberm, thenG contains a nilpotent subgroup of class at most two and index bounded in terms ofm. One of the steps in the proof of that result was to show that ifG is soluble, then ¦G/F(G) ¦ is bounded by a function ofm, where F (G) is the Fitting subgroup ofG. We now show that, in this part of the argument, the involution can be replaced by an arbitrary element of prime order.

Zweidimensionale Quotientensingularitäten: Gleichungen und Syzygien

Abstract: In [10] wurden minimale Erzeugendensysteme ftir die Algebra C [u, v] ~ der Polynome in zwei Ver/~nderliehen berechnet, die unter der Aktion einer endliehen spiegelungsfreien Gruppe G c GL (2, C) invariant sind. Die erzeugenden Relationen zwisehen diesen Invar ianten k6nnen aufgefaBt werden als die Gleiehungen der zugeh6rigen (algebraisehen oder analytischen) Quotientensing~larit~t (C2/G, 0). Diese Gleiehungen wurden in der eingangs zitierten Arbeit in vielen Fallen bestimmt, wobei jedoeh in den nieht dutch Determinanten besehreibbaren F/~llen kein einfaehes Bildungsgesetz zu erkermen war. Das Ziel der vorliegenden Arbeit ist es, ftir alle Quotientensingularit~ten verh/iltnism/i$ig einfach gebaute Gleichungen anzugeben. Es zeigt sieh, daft sie sieh (mit Ausnahme zweier Serien yon Tetraedersingularit~ten der Einbettungsdimension e _> 6) durch verallgemeinerte Determinantenideale besehreiben lassen: I s t C [u, v]G yon der Einbettungsdimension e, also C[u,v] ~ ----C[zl, . . . , Ze]/a, so l~Bt sieh a minimal erzeugen yon Elementen der Form

Krull dimension, transcendence degree and subalgebras of finitely generated algebras

Abstract: Let A c B be integral domains, and let tr. deg.~ B be the transcendence degree of the quotient field of B over the quotient field of A. The aim of this paper is to study the relation between tr.deg.A B and the increment dim B -- dim A of the Krull dimensions of the rings. It is well-known that dim B -~ dim A ~ tr. deg.A B when A and B are finitely generated algebras over a field. In the general case we prove that if A is noetherian we always have dim B ~ dim A ~- tr. deg.A B, and if moreover B is a finitely generated A-algebra, both numbers differ at most by 1. We caraeterize when we obtain equality in terms of the vanishing of an ideal which we call the dimensional radical of A, defined to be the intersection of the maximal ideals of A whose height equals dim A. This condition enables us to calculate the dimension of a finitely generated A-algebra as in the classical geometric case. Subsequently we deduce some properties of the dimensional radical which show its similarities and differences with the Jaeobson radical. In section 2 we apply the above results to calculate the dimension of a subalgebra B of a finitely generated algebra over a noetherian domain. We obtain in fact almost the same properties as in the finitely generated case, although prime ideals of B can have a bad behavior in the complement of a basic open subset of the spectrum of B. All rings are assumed to be commutative with unity. Dimension means always Krull dimension. If A is a ring, Spec A will be the prime spectrum of A and A I will denote localization at the multiplicative set {/n; n ~ 0}. If P is a prime ideal of A, h(P) denotes the height of P and k(P) the residue field at P. We start from the following basic results:

Characterization of monoids by torsion-free, flat, projective, and free acts

Abstract: A monoidS is susceptible to having properties bearing upon all right acts overS such as: torsion freeness, flatness, projectiveness, freeness. The purpose of this note is to find necessary and sufficient conditions on a monoidS in order that, for example, all flat rightS-acts are free. We do this for all meaningful variants of such conditions and are able, in conjunction with the results of Skornjakov [8], Kilp [5] and Fountain [3], to describe the corresponding monoids, except in the case “all torsion free acts are flat”, where we have only some necessary condition. We mention in passing that homological classification of monoids has been discussed by several authors [3, 4, 5, 8].

Transitivitätsfragen bei linearen Liegruppen

Abstract: Einleitung. Sei K der KSrper der reellen oder komplexen Zah]en oder der (reellen) Quaternionen. Es werden hier diejenigen Lie-Untergruppen der Gruppen GL (n, K) gefunden, welche in einem bestimmten Sinn ,,grol]\" sind, n~mlieh gewisse Transitivit~tseigenschaften haben. Genauer gesagt, geht es darum, f'fir gewisse Mannigfaltigkeiten M, auf denen eine abgeschlossene Untergruppe Go yon GL(n, K) in natiirlicher Weise (transitiv) operiert, alle auf M transitiven Lie-Untergruppen G yon Go zu bestimmen. Als M werden bier die Gral3mann'schen Manni~oSaltigkeiten yon K n sowie projektive Quadriken yon Bilinearformen auf K n zugelassen. Ausgangspunkt war ein Resultat yon Tits [9, IV.C], worin die auf dem projektiven Raum yon K n transitiven, abgeschlossenen Untergruppen yon GL (n, K) bestimmt werden (mit Hilfe der Darstellungstheorie halbeinfacher Liealgebren). Mit Mitteln der algebraischen Topologie gelang es Oniw f'fir jede transitive Wirkung einer kompakten, lokal-einfachen Liegruppe alle transitiven Lie-Untergruppen festzustellen [5, 4]; sowie fiir gewisse kompakte Ma~nnigfaltigkeiten alle transitiven Liegruppen-Wirkungen darauf zu bestimmen [7]. Aus diesen allgemeineren Resultaten ergibt sieh die L5sung obigen Problems f'tir den Fall, dal3 M eine Gra~mann'sehe Mannigfaltigkeit yon K n ist oder die projektive Quadrik einer komplexen quadratischen Form. Ferner hat Wolf diejenigen abgeschlossenen Untergruppen der Invarianzgruppen yon (niehtdegenerierten) reellen quadratisehen Formen festgestellt, welche transitiv wirkea auf den Zusammenhangskomponenten der Menge aller isotropen Vektoren und der Menge aller Vektoren yore ,,L~ngenquadrat\" 1 [10, 3]. Hier wird nun im wesentlichen obiges Problem gelSst ffir den Fall, dal3 M die projektive Quadrik einer (nichtdegenerierten) reellen quadratisehen Form ist. Zu Begirm wird kurz dargestellt, wie der Fall der Gragmann-l~Iannigfaltigkeiten und komplexen Quadriken aus den Resultaten yon Oniw folgt. Ich bedanke reich bei Herrn Prof. K. Strambach, unter dessen Anleitung die dieser Note zugrundeliegende Diplomarbeit entstand.

The imbedding of bounding manifolds in Euclidean space

Abstract: Theorem 1. I / n ~_ 5, then any n-dimensional mani/old which is the boundary o/a~ orientable mani/old will be also the boundary o / a simple connected manifold which imbeds in R 2n-i. In [4] M. Hirsch proved the Theorem above for many values of n, depending upon the dyadic expansion of n. By "manifold" we mean a compact C ~~ manifold. Let y be a cohomology class of the classifying space BSO, with any group of coefficients and let W be an oriented manifold, (with boundary or not). Then y(W) will denote the corresponding normal characteristic class of W. By Wi we denote the/-dimensional Stiefel-Whitney class of the universal oriented bundle. I f i equals 1 or an even number, then Wi is an element of H i (BSO; Z2). On the other hand ff i is an odd number greater than 1, then Wi is an element of H i (BSO; Z). Let wi be the rood 2 reduction of Wi. The main result of this paper, which is exactly what is needed to complete Hirseh's proof, is the following:

Totally umbilical submanifolds of Kaehler manifolds

Abstract: By BAI~G-YEN CHEN 1. Introduction. Let N be a submanifold of a Riemannian manifold M with g as its first fundamental form. Let V and V be the covariant differentiations on iV and M, respectively. The second fundamental form h of the immersion is given by (1.1) h(X, r) = vxr - vxr where X and Y are vector fields tangent to N. It is well-known that h is a normal- bundle-valued symmetric 2-form on N. If the first and second fundamental forms are proportional, there is a normal vector field H, called the mean-curvature vector, on N such that (1.2) h(X, r) -~ g(X, Y)H. In this case, N is called a totally umbilical submani/old of M. The snbmanifold h r is called a totally geodesic submani/old ff h vanishes identically. Totally umbilical submanifolds (including totally geodesic submanifolds) are the simplest submani- folds of Riemannian manifolds. In this paper we shall study totally umbilical sub- manifolds in Kaehler manifolds. In particular we shall prove the following Main Theorem. Let M be a Hermitian symmetric space and N a totally umbilical hypersur/ace o/M. (A) I] N is totally geodesic, then M is the Riemannian product o/ a Hermitian sym- metric space M1 and a two-dimensional Hermitian symmetric space Ms such that !Y is locally the Riemannian product o/M1 and a geodesic o/Ms. (B) I] N is not totally geodesic, then one o/the ]ollowing three statements holds: (a) M is a complex projective line C,P 1 (c), a complex line C 1, or a disk D1 (--c) in G 1 and N any curve in M. Here c and --c denote the sectional curvature o/C,P 1 and D 1, reslaectively. (b) N is a totally umbilical hypersur/ace o/M and M is the Riemannian product CP 1 (c) X D 1 (-- c). (c) M is the complex number m-space C m and N is an open portion o/an ordinary hypersphere o/G m. I/statement (b) holds, then, locally, N is the con/ormal image o/a hypersphere S a or a hyperplane Ea o/C ~ under a con/ormal mapping/tom C ~ into M. 6*

Characterization of Buekenhout-Metz unitals

Abstract: Lemmas 5 and 6 show thatU is the cone with double pointx e P and basis 0, where 0 is a 3-dimensional ovoid tangent toH at a pointy e P. Following the definition of Buekenhout-Metz unitals, this proves our theorem.