In this paper we study Frobenius algebras and quasi-Frobenius rings with particular emphasis on their cohomological dimensions. For definitions of these cohomological dimensions we refer the reader to Cartan-Eilenberg [3] or Eilenberg [4].

It is well-known that the conformal equivalence of a compact simply-connected Riemann surface to the extended plane is readily established once it is shown that given a local uniformizer t(p) which carries a given point p 0 of the surface into 0, there exists a function u harmonic on the surface save at p 0 which admits near p 0 a representation of the form

(α complex 0; h harmonic at p 0). For the monodromy theorem then implies the existence of a meromorphic function on the surface whose real part is u. Such a meromorphic function has a simple pole at p 0 and elsewhere is analytic. It defines a univalent conformal map of the surface onto the extended plane.

The following small remark on the relationship of the (quasi-)regularity and the zero-divisor property may be more or less known but does not seem to the writer to have been explicitly stated in a literature.

Le but de la présente note est de clarifier la notion de variété riemanien-ne ou affine symétrique (et plus généralement, localement reductive) du point de vue de la théorie de connexions dans les espaces fibres. Nous donnerons quelques caractérisations des espaces localement symétriques, ou réductives (Th. 1. 2 et 3). On trouvera la définition de dérivation covariante de tenseurs de courbure et de torsion dans la terminologie de la théorie de connexions dans les espaces fibres. Cette définition peut être généralisée à la dérivation covariante d’un tenseur quelconque. Les résultats concernant la dérivation covariante seront publiés ailleurs. Enfin on retrouvera des résultats de Borel-Lichnerowicz. Même si l’on ne trouvera pas de résultats nouveaux dans cette note, on comprendra mieux les espaces symétriques par la théorie de connexion de Cartan.

Every Riemannian manifold has a unique affine connection without torsion, which is necessarily invariant by any isometrical transformation of the manifold. However, an affine transformation (i.e., transformation leaving invariant the affine connection) is not necessarily an isometrical transformation. (Consider, for example, the ordinary Euclidean space).

The purpose of the present paper is to give the details of the results announced in the C. R. notes [7], [8] and [9].

In I which corresponds to [7], we shall develop a method for the study of affine transformations of a Riemannian manifold and prove Lemma 5 which is fundamental for the results in I and II. This part of our work has been motivated by a result of K. Yano [12] stating that the largest connected group of affine transformations of a compact orientable Riemannian manifold consists of isometries. Our method will have further applications to some other problems concerning or involving the connected group of isometries of a Riemannian manifold (for example, [10]).

The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

Let Λ be a ring with unit. If A is a left Λ-module, the dimension of A (notation: 1.dimΛA) is defined to be the least integer n for which there exists an exact sequence

0 → Xn → … → X0 → A → 0

where the left Λ-modules X 0, …, Xn are projective. If no such sequence exists for any n, then 1. dimAA = ∞. The left global dimension of Λ is

1. gl. dim Λ = sup 1. dimAA

where A ranges over all left Λ-modules, The condition 1. dimA A < n is equivalent with (A, C) = 0 for all left Λ-modules C. The condition 1.gl. dim Λ < n is equivalent with = 0. Similar definitions and theorems hold for right Λ-modules.

This paper is concerned with regular and meromorphic functions in |z| < 1 and their behavior near |z| = 1. Among the results obtained are the following. In section 2 we prove the existence of a non-constant meromorphic function that tends to zero at every point of |z| = 1 along almost all chords of |z| < 1 terminating in that point. Section 3 deals with the impossibility of ex tending this result to regular functions. In section 4 it is shown that a regular function can tend to infinity along every member of a set of spirals approach ing |z| = 1 and exhausting |z| < 1 in a simple manner. Finally, in section 5 we prove that this set of spirals cannot be replaced by an exhaustive set of Jordan arcs terminating in points of |z| = 1; Theorem 3 of this section can be interpreted as a uniqueness theorem for meromorphic functions.

Dans cette note, nous étudions les anneaux d’intégrité primaires et complètement intégralement clos. Krull [1] a posé la question suivante: est-ce qu’un anneau d’intégrité primaire et complètemente intégralement clos est nécessairement un anneau de valuation?

In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

Previously the following question was offered by Zariski [6]:

Is any normal Noetherian local ring analytically irreducible?

In the present note, we will construct a counter-example against the question.

TERMINOLOGY. A ring (integrity domain) means always a commutative ring (integrity domain) with identity. A normal ring is an integrity domain which is integrally closed in its field of quotients. When 0 is an integrity domain, the integral closure of 0 in its field of quotients is called the derived normal ring of 0.

We shall prove in the present note a theorem on units of algebraic number fields, applying one of the strongest formulations, be Hasse [3], of Grunwald’s existence theorem.

M. Brelot [1] has shown that if u(z) is subharmonic in an open set D in the z-plane with boundary C and is bounded from above in a neighborhood of a boundary point z 0, which is contained in a set E ⊂ C of inner harmonic measure zero with respect to D, and such that z 0 is a regular point for Dirichlet problem in D, then

(1) .

Es ist wohlbekannt, dass die Frattini-Gruppe einer endlichen Gruppe G, mit anderen Worten, der Durchschnitt aller maximalen Untergruppen von G, nilpotent ist. Offen ist dagegen bis jetzt die Frage, ob die Faktorgruppe G/F*(G) von G nach dem Erzeugnis F*(G) aller minimalen Untergruppen eine besondere Struktur besitzt, z. B. ob sie auflösbar ist. Mehrere Schwierigkeiten sind der Grund hierfür: Einmal ist G/F*(G) im allgemeinen nicht nilpotent, zum anderen kennen wir die Struktur der einfachen und auflösbaren Gruppen (abgesehen von der Theorie von P. Hall) zu wenig. In der vorliegenden Note sollen teilweise Antworten auf die Frage nach der Auflösbarkeit von G/F*(G) gegeben werden.

In [0], the writer proved some theorems of Hasse type for two orthogonal groups which operate on the same vector space. In this paper, we shall further generalize those results in two directions. One is to consider the propositions of that type for two orthogonal groups which operate respectively on two vector spaces whose dimensions are different from each other, and the other is to deal with some conspicuous subgroups of an orthogonal group simultaneously which play important roles in the structure theory for orthogonal groups. For this reason, the present paper consists of three steps §1, §2 and §3 which give the generalizations in the above sense of the results in the corresponding sections of [0].

The theory of derivations and differents in the algebraic number fields or Dedekind rings has been developped by A. Weil [1], Y. Kawada [2] and M. Moriya [4]. Further Y. Kawada [3] has investigated the derivations in maximal orders of simple algebras over number fields. This note is concerned with the derivations in the simple algebras over fields which are quotient fields of Dedekind rings; the commutative rings in which the fundamental theorem of multiplicative ideal theory holds. The auther gives his hearty thanks to Prof. M. Moriya who gave him valuable remarks.

The main purpose of the present paper is to show that the local homogeneous holonomy group of the conformally flat Riemannian manifold is the full rotation group with some exceptions.

In the present note, applying the theory of harmonic integrals, we shall show some results on cycles on algebraic varieties and give a new birational invariant.

Die klassische Aussagenlogik ist, bekanntlich, in dem Sinne vollständig, dass die Hinzufügung einer klassisch nicht beweisbaren Formel als Axiom dieses Systems stets einen Widerspruch nach sich zieht. Die Vollständigkeit (in diesem Sinne) kann für intuitionistische Aussagenlogik nicht bestehen, weil noch immer die Formel A ∨ Ā (der Satz vom ausgeschlossenen Dritten) hinzugefügt werden kann. Es steht aber die Frage, ob man aus der intuitionistischen Aussagenlogik durch Hinzufügung einer klassisch beweisbaren und intuitionistisch nicht beweisbaren Formel stets die klassischen Aussagenlogik bekommt.