Le problème de classifier les fibrés vectoriels holomorphes sur une variété complexe M a été résolu récemment dans les deux cas suivants : 1) M est la sphère de Riemann (Grothendieck [4]), 2) M est une courbe elliptique (Atiyah [2]). Mais si la dimension complexe de M est > 1, nous ne connaissons presque rien de ce problème, sauf le cas où la dimension complexe de fibre est égale à 1.

When a Lie group G operates on a differentiable manifold M as a Lie transformation group, the orbit of a point p in M under G, or the G-orbit of p, is by definition the submanifold G(p) = {G(p); g∈G}. The purpose of this paper is to characterize the structure of a non-compact manifold M such that there exists a compact orbit of dimension (n — 1), n — dim M, under a connected Lie transformation group G, which is assumed to be compact or an isometry group of a Riemannian metric on M.

Let F be a commutative ring with unit element 1. A ring R is called an algebraic ring over F by Drazin, in his recent paper [2], when R is an algebra over F and every element x of R satisfies an equation of lower monic form

In [1] we introduced the concept of a non-commutative local ring and studied the structure of such rings. Unfortunately, we were not able to show that the completion of a local ring was a semi-local ring. In this paper we propose to study a class of rings for which the above result is valid. This class of rings is the integral extensions -[4, 5]-of commutative local rings. This class of rings includes the important class of matrix rings over commutative local rings. In part 1 below we study some elementary properties of integral extensions and here we assume merely that the underlying ring is semi-local. In part 2 we discuss some questions of ideal theory for arbitrary local rings as well as for integral extensions. In a later paper we propose to utilize our results to study the deeper properties of these rings including a dimension theory for such rings. We are particularly indebted to the work of Nagata [6, 7, 8, 9] in the preparation of this paper.

Let us consider the following two problems:

Problem A. Let g be a given Lie algebra over the real number field R. Then find all real, irreducible representations of g.

Problem B. Let n be a given positive integer. Then find all irreducible subalgebras of the Lie algebra ôí(w, R) of all real matrices of degree n.

In this note we shall give a modified definition of cohomology groups for algebras. For a class of (infinite rank) algebras (which includes Frobenius algebras, group rings of infinite groups and division algebras of countable rank over fields), these groups can be characterized in a manner similar to the cohomology groups in the * Almost Zero theory’ of B. Eckmann [2]. Actually, in the case of group rings, these coincide with the cohomology groups in the almost zero theory.

The V-system T(V) is defined in §2 by using §1, and its consistency is proved in §3. The definition of T(V) is given in such a way that the consistency proof of T(V) in §3 shows a typical way to prove the consistency of some subsystems of UL. Otherwise we could define T(V) more simply by using truth values. After T(V)-sets are treated in §4, it is proved in §5 as a T(V)-theorem that T(V)-sets are all equal to V.

For an investigation on the foundations of mathematics to get an adequate mutual understanding, it is necessary to describe the generalities of the investigation with the parallel description of its particularities. Although one should obtain an exact and precise knowledge only through the latter, it is almost impossible, without the former, to get the underlying ideas and the fundamental principles, upon which the investigation is based.

The consistency of the natural number theory was proved, as is well known, by G. Gentzen in 1935 for the first time in such generality that the mathematical induction can be consistently used for any arbitrary predicate of natural numbers, which is well-formed in his system so that every quantifier ranges over all natural numbers.

Let f(z) be a holomorphic and unbounded function in | z | < 1, with the property that it remains bounded on some spiral S in | z | < 1 which approaches | z | = 1 asymptotically. The existence of such functions was first established by G. Valiron. Accordingly, we shall refer to such functions as functions of class (V) relative to S. More recently, F. Bagemihl and W. Seidel obtained examples of functions holomorphic and unbounded in | z | < 1 which approach prescribed finite or infinite values as | z | → 1 on any given enumerable set of disjunct spirals which approach | z | =1 asymptotically, as well as on certain sets of such spirals having the power of the continuum.

Let ϕ be a holomorphic mapping of an n-dimensional analytic space E into Cn. If ϕ is non-degenerate at every point of E, we call the pair (E, ϕ) a Riemann domain. The notion of a Riemann domain is a generalization of the notion of a concrete Riemann surface. A Riemann domain (E, ϕ) is said to be unramified if ϕ is a local homeomorphism, and to be ramified if otherwise.

Let k be an algebraic number field of finite degree, and l a rational prime (including 2); k and l being fixed throughout this paper. For any power ln of l, denote by ζn an arbitrarily fixed primitive ln-th root of unity, and put kn = k(ζn). Let r be the maximal rational integer such that ζr∈k i.e. kr = k and kr+1≠k.

The purpose of this paper is to study the relationship between a certain isomorphism of some rings of functions on Riemann surfaces and a quasi-conformal mapping.

It is well known that two compact Hausdorff spaces are topologically equivalent if and only if their rings of continuous functions are isomorphic. We shall establish an analougous result concerning a function ring on a Riemann surface and the quasi-conformal equivalence.

Let k be an algebraically closed field of characteristic p>0. Let K/k be a function field of one variable and L/K be an unramified separable abelian extension of degree pr over K. The galois automorphisms ε1, …, εpr of L/K are naturally extended to automorphisms η(ε1), … , η(εpr) of the jacobian variety JL of L/k. If we take a svstem of p-adic coordinates on JL, we get a representation {Mp(η(εv))} of the galois group G(L/K) of L/K over p-adic integers.