Let k be a field and A a noetherian k-algebra. In this note, we shall study the universal finite module of differentials of A over k, which is denoted by Dk(A). When the characteristic of k is zero, detailed results have been obtained by Scheja and Storch [8]. So we shall treat the positive characteristic case. In § 1, we shall study differential modules of a local ring over subfields. We obtain a criterion of regularity (Theorem (1.14)). In § 2, we shall study the formal fibres and regular locus of A with Dk(A). Our main result is Theorem (2.1) which shows that, if Dk(A) exists, then A is a universally catenary G-ring under a certain assumption. In the local case, this is a generalization of Matsumura’s theorem ([5] Theorem 15), where regularity of A is assumed.

In [16] we studied the following problems which had been asked by H. Matsumura (cf. [11]):

• (I) What is the difference between smoothness and J-smoothness? In particular, concerning the characterization of smoothness,

• (II) When is a ring A[X1 …, Xn]/a smooth over A?

In this paper, according to these problems, we will study J-smoothness further when rings are noetherian as in [16].

We study the multiple Eisenstein series introduced by Gangl, Kaneko and Zagier. We give a proof of (restricted) finite double shuffle relations for multiple Eisenstein series by revealing an explicit connection between the Fourier expansion of multiple Eisenstein series and the Goncharov co-product on Hopf algebras of iterated integrals.

For positive integers h, k and m, the higher-order Dedekind sums are defined by

Let X be an irreducible complex analytic space with j: U ↪ X an immersion of a smooth Zariski-open subset, and let 𝕍 be a variation of Hodge structure of weight n over U. Assume that X is compact Kähler. Then, provided that the local monodromy operators at infinity are quasi-unipotent, IHk (X, 𝕍) is known to carry a pure Hodge structure of weight k+n, while Hk (U, 𝕍) carries a mixed Hodge structure of weight at least k +n. In this note it is shown that the image of the natural map IHk (X, 𝕍) → Hk (U, 𝕍) is the lowest-weight part of this mixed Hodge structure. In the algebraic case this easily follows from the formalism of mixed sheaves, but the analytic case is rather complicated, in particular when the complement X — U is not a hypersurface.

To any Nash germ invariant under right composition with a linear action of a finite group, we associate its equivariant zeta functions, inspired from motivic zeta functions, using the equivariant virtual Poincaré series as a motivic measure. We show Denef–Loeser formulas for the equivariant zeta functions and prove that they are invariants for equivariant blow-Nash equivalence via equivariant blow-Nash isomorphisms. Equivariant blow-Nash equivalence between invariant Nash germs is defined as a generalization involving equivariant data of the blow-Nash equivalence.

In this paper we use a general technique, due to Larfeldt-Lech [9], to show that there exist local noetherian rings which are “bad” with respect to some properties which are obviously verified in the algebro-geometric case. In particular we show that there exist local Gorenstein rings which are not homomorphic images of regular (local) rings and that there exist local rings which do have canonical module but do not have canonical algebra.

The examples on p. 44 of metabelian Lie algebras L1 and L2 are isomorphic over certain base fields contrary to the claim in the paper. More specifically, the error lies in the incorrect statement that the rank of ad x for any x in L1 is either zero or is at least 3. This statement does hold over Q, but if τ is a scalar satisfying τ2 − τ − 1 = 0 then ad (x1 + τx2 + τ2x3 + x4 + τ2x5 has rank 2.

In 1969, Mumford [8] proved that, for a complete non-singular algebraic surface F over the complex number field C, the dimension of the Chow group of zero-cycles on F is infinite if the geometric genus of F is positive. To this end, he defined a regular 2-form ηf on a non-singular variety S for a regular 2-form η on F and for a morphism f: S → SnF, where SnF is the 72-th symmetric product of F, and he showed that ηf vanishes if all 0-cycles f(s), s ∈ S, are rationally equivalent. Roitman [9] later generalized this to a higher dimensional smooth projective variety V.

Consider a nonsingular vector field X on a closed manifold Mn. As a matter of fact, X always admits a transverse codimension one plane field, which however may fail to be integrable. In fact it is well known that there are many examples of vector fields which do not admit transverse foliations.

Les fonctions périodiques d’une seule variable ont été étudiées depuis longtemps. On sait que toute fonction méromorphe de n variables 2n fois périodiques peut s’écrire comme quotient de deux fonctions thêta. Ceci est a l’origine de l’essor des fonctions thêta. Au contraire, l’étude des fonctions périodiques de n variables r fois périodiques (r < 2n) a pris du retard. Il nous semble que P. Cousin ([4] et [5]) a étudié ces fonctions pour la première fois.

In this paper, we shall study a generalization of the Maillet determinant. hetp be an odd prime, and G = (Z/pZ)*. We shall identify any integer and its image in G if there is no fear of confusion. For any integer a, let R(a) denote an integer satisfying

Let D be a bounded pseudoconvex domain in ℂn and ζ ∈ D. By KD and BD we denote the Bergman kernel and metric of D, respectively. Given a ball B = B(ζ, R), we study the behavior of the ratio KD/KD∩B(w) when w ∈ D ∩ B tends towards ζ. It is well-known, that it remains bounded from above and below by a positive constant. We show, that the ratio tends to 1, as w tends to ζ, under an additional assumption on the pluricomplex Green function D(·, w) of D with pole at w, namely that the diameter of the sublevel sets Aw :={z ∈ D | D(z, w) < −1} tends to zero, as w → ζ. A similar result is obtained also for the Bergman metric. In this case we also show that the extremal function associated to the Bergman kernel has the concentration of mass property introduced in [DiOh1], where the question was discussed how to recognize a weight function from the associated Bergman space. The hypothesis concerning the set Aw is satisfied for example, if the domain is regular in the sense of Diederich-Fornæss, ([DiFo2]).

There are some errors in Theorems 3.3 and 4.2 in [2]. In this note we would like to correct them.

1) In Theorem 3.3 (and [IV]), the condition (1) must be replaced by the following one;

(1) П is (i) a cyclic group, (ii) a dihedral group of order 2m, m odd, (iii) a direct product of a cyclic group of order qf, q an odd prime, f ≧ 1, and a dihedral group of order 2m, m odd, where each prime divisor of m is a primitive qf-1(q — 1)-th root of unity modulo qf, or (iv) a generalized quaternion group of order 4m, m odd, where each prime divisor of m is congruent to 3 modulo 4.

The deflection, under a point load, of a thin elastic plate clamped at the edges is the biharmonic Green’s function β with the boundary data β = ∂β/∂n = 0. If the boundary of the region is reasonably smooth, the construction of β offers no difficulty. In contrast, nothing is known about the existence of β in the general case. The purpose of our study is to give a sufficient condition for the existence of β on a given Riemannian manifold of arbitrary dimension and to construct β. Our results will have, apart from their physical meaning in elasticity, consequences in the biharmonic classification theory of Riemannian manifolds.

We use higher ideles and duality theorems to develop a universal approach to higher dimensional class field theory.

Consider a one-dimensional differential algebraic function field K over an algebraically closed ordinary differential field k of characteristic 0. We shall prove the following theorem:

Suppose that the group of all automorphisms of K over k is infinite. Then, K is either a differential elliptic function field over k or K = k(ν) with ν′ = ξ or ν′ = ην, where ξ, η ϵ k.

Let Mn be an n-dimensional Riemannian manifold minimally immersed in the unit sphere Sn+p (1) of dimension n + p. When Mn is compact, Chern, do Carmo and Kobayashi [1] proved that if the square ‖h‖2 of length of the second fundamental form h in Mn is not more than , then either Mn is totally

geodesic, or Mn is the Veronese surface in S4 (1) or Mn is the Clifford torus .

In this paper, we generalize the results due to Chern, do Carmo and Kobayashi [1] to complete Riemannian manifolds.

The tangent bundle of a differentiable manifold is an important invariant of a differentiable structure. It is determined neither by the topological structure nor by the homotopy type of a manifold. But in some cases tangent bundles depend only on the homotopy types of manifolds.

Let (M, g) be a Riemannian manifold of dimension n≥ 3 and ĝanother metric on M which is pointwise conformai to g. It can be written where u is a positive smooth function on M. Then the curvature of g is computable in terms of that of g and the derivatives of u up to second order. In particular, if S and S denote the scalar curvature of g and g respectively, they are related by the equation

where ▽u denotes the Laplacian of u, defined with respect to the metric g.