A. Weil a généralisé la notion de variété algébrique d’un espace affine (ou projectif ) en introduisant la notion de variété abstraite (nous dirons simplement “variété” au lieu de “variété abstraite,” appelant variétés affines les variétés algébriques d’un espace affine). La définition donnée par A. Weil des variétés algébriques s’inspire manifestement de la notion de variété differentiate, conçue comme réunion d’un nombre fini de morceaux dont chacun peut être considéré comme une partie ouverte d’un espace numérique, les variétés des espaces affines se substituant aux ensembles ouverts des espaces euclidiens.

Es bezeichne Sn die n-dimensionale Sphäre, Eine solche ist der Rand einer (n + 1)-dimensionalen euklidischen Vollkugel vom Radius 1.

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Recently Prof. Chevalley in Nagoya suggested to the author the following problem: Let k be a field, K 5 = k(x 1, x 2, x 3, x 4, x 5) be a purely transcendental extension field (of transcendental degree 5) of k, s5 be the cyclic permutation of x: S 5 X 1 = x 2 s 5 x 2 = x 3 s 5 x 3 = x 4 s 5 x 4 = x 5 s 5 x 5 = x 1, and let L 5 be the field of invariants of s5 in K 5. Is L 5 then purely transcendental over k or not? When the characteristic p of k is not equal to 5, it is answered in the following positively. When the characteristic p of k is equal to 5, it is answered also positively by Mr. Kuniyoshi’s result in [2].

In the present note we wish to deal with the same problem as the preceding paper [1] for the case of modular fields.

Let k be a ñeiá of characteristic and K = k(x1 …, xp ) a purely-transcendental extension of k, Let S be the automorphism of K which is induced by the cyclic permutation (x1 ., .., xp ) and L the fixed subfield of S. Then L is a purely transcendental extension field over k.

This short note gives the generalized sum theorem for Lebesgue dimension of paracompact Hausdorff spaces. Our theorem, though it is a generalization of Mr. Morita’s sum theorem for fully normal spaces [3, Theorem 3. 2] which is essentially based on his generalized sum theorem for normal spaces [3, Theorem 3.1], is obtained by very brief arguments, using only the usual sum theorem for normal spaces.

The purpose of the present note is to reform Mr. K. Nomizu’s result on the group of all affine transformations of an affinely connected manifold. We shall prove the following.

Throughout this paper, a space means a T1 -space. A space is called fully normal if every open covering of it has a Δ-refinement , that is, an open covering for which the stars (x, ) form a covering which refines . A space is called paracompact if every open covering of it has a locally finite (= neighborhood finite) open covering which refines . It is well known that paracompactness is identical with full normality in a Hausdorff space ([3], [7]).