In this paper R is a commutative noetherian local ring with unit element 1 and M is its maximal ideal. Let K be the residue field R/M and let {t1,t2,…, tn) be a minimal system of generators for M. By a complex R<T1. . ., Tp> we mean an R-algebra* obtained by the adjunction of the variables T1. . ., Tp of degree 1 which kill t1,…, tp. The main purpose of this paper is, among other things, to construct an R-algebra resolution of the field K, so that we can investigate the relationship between the homology algebra H (R < T1,…, Tn>) and the homological invariants of R such as the algebra TorR(K, K) and the Betti numbers Bp = dimk TorR(K, K) of the local ring R. The relationship was initially studied by Serre [5].

Let $R$ be a two-sided Noetherian ring, and let $M$ be a nilpotent $R$-bimodule, which is finitely generated on both sides. We study Gorenstein homological properties of the tensor ring $T_{R}(M)$. Under certain conditions, the ring $R$ is Gorenstein if and only if so is $T_{R}(M)$. We characterize Gorenstein projective $T_{R}(M)$-modules in terms of $R$-modules.

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.

We show that the Kulikov surfaces form a connected component of the moduli space of surfaces of general type with pg = 0 and K2 = 6. We also give a new description for these surfaces, extending ideas of Inoue. Finally, we calculate the bicanonical degree of Kulikov surfaces and prove that they verify the Bloch conjecture.

It is an honour to be invited to contribute a survey article on the work of George Lusztig in celebration of his 60th birthday.

In this article, we prove a strong version of the local Bertini theorem for normality on local rings in mixed characteristic. The main result asserts that a generic hyperplane section of a normal, Cohen–Macaulay, and complete local domain of dimension at least 3 is normal. Applications include the study of characteristic ideals attached to torsion modules over normal domains, which is fundamental in the study of Euler system theory, Iwasawa's main conjectures, and the deformation theory of Galois representations.

We give a classification of Levi-umbilical real hypersurfaces in a complex space form $\widetilde{M}_{n}(c)$, $n\geqslant 3$, whose Levi form is proportional to the induced metric by a nonzero constant. In a complex projective plane $\mathbb{C}\mathbb{P}^{2}$, we give a local construction of such hypersurfaces and moreover, we give new examples of Levi-flat real hypersurfaces in $\mathbb{C}\mathbb{P}^{2}$.

Following eariler work of Kubota and Mennicke, the major work of Bass, Milnor and Serre [1] constructed characters of congruence subgroups of the modular subgroups of SL(n) and Sp(2n) over a totally complex number field, which are related to the power residue symbol. They do not obtain the lowest possible level of these Kubota characters, nor does it appear possible to modify their arguments to extend the characters to the lowest possible level.

The present paper is concerned with the classification and corresponding extension theorem of mappings of the (n+-2)-complex Kn+2 (n>2) into the space Y whose homotopy groups πi(Y) vanish for i < n and i = n+1, and the n-th homotopy group πn(Y) of which has a finite number of generators. Our methods followed here are essentially analogous to those of Steenrod [2].

Let X(t) = X(t,ω), – ∞ < t < ∞, be a stationary stochastic process with

and the continuous covariance function

where F(x) is the spectral distribution function.

Let τ = (τij) be a symmetric complex g = g matrix with the positive definite imaginary part. A theta function of level n means an entire function f(z) in g complex variables z = (zl …, zg) satisfying the difference relations:

Let h: R4 → R3 be a quadratic map defined by

For a natural number t, put

Let X be any Riemann surface. By Koebe’s uniformization theorem we know that the universal covering space of X is conformally equivalent to either Riemann sphere, complex plane, or the unit disc in the complex plane. If X is allowed to vary with parameters we may inquire the parameter dependence of the corresponding family of the universal covering spaces.

Le noyau de Riesz-Frostman dans l’espace euclidien à d(≥3) dimensions est à la puissance fractionnaire du noyau de Newton. On trouve la conformité dans la théorie des espaces de Dirichlet; on construit un noyau à la puissance fractionnaire d’un noyau associé à l’espace de Dirichlet spécial, qui détermine à nouveau un espace de Dirichlet spécial [2]. En notant que le noyau de Dirichlet est positif et symétrique, et satisfait au principe complet du maximum, on s’interroge: est-il possible de construire de bons noyaux à la puissance fractionnaire pour des noyaux de genre plus large?

Very recently, M. Choda, I. Kasahara and R. Nakamoto [3] extend the concept of free action of automorphisms for C*-algebras and prove several theorems which are hither to known for von Neumann algebras. In the present note, we shall concern with freely acting automorphisms on abelian C*-algebras. In § 2, several equivalent conditions for the free action are obtained. In §3, we shall apply them to an automorphism which has a transversal group.

In the previous paper [3] the author has shown a possibility to construct a series of sfields by taking sfields of quotients of split crossed products of a sfield. In this paper the same problem is treated, and, by considering general crossed products, a generalization of the previous result is given: Let K be a sfield and G be the join of a well-ordered ascending chain of groups Gα of outer automorphisms of K such that a) G1 is the identity automorphism group, b) Gα is a group extension of Gα-1 by a torsion-free abelian group for each non-limit ordinal α, and c) for each limit ordinal α. Then an arbitrary crossed product of K with G is an integral domain with a sfield of quotients Q and the commutor ring of K in Q coincides with the centre of K.

In the present paper, we are concerned with the problem to know whether two algebraic Riemann manifolds are isometric or not, where we mean Riemann manifolds of class CΩ (real algebraic smoothness or Nash category) simply by algebraic Riemann manifolds.

For this purpose we introduce notions of minimal differential polynomials and singular base points, both of which are isometry invariants. With the aid of these invariants, we give the following isometry theorem for algebraic Riemann manifolds:

Let R be a (commutative) Noetherian ring, and let {n0, n1 …, nm} {k1 k2, …, km} be two sequences of integers satisfying m > 0, ki ≧ 0 (i = 1, 2, …, m) and ni ≧ ki ki + 1 (i = 1, 2, …, m) with k0 = km + 1 = 0.

During 1934-1936, W. L. Ferrar investigated the relation between summation formulae and Dirichlet series with functional equations, inspired by Voronoi’s works (1904) on summation formula with respect to the numbers of divisors. In [11] A. Weil showed that the automorphic properties of theta series are expressed by generalized Poisson summation formulae with respect to the so-called Weil representation. On the other hand, T. Kubota, in his study of the reciprocity law in a number field, defined a generalized theta series and a generalized Weil type representation of SL(2, C) and obtained similar results to those of A. Weil (1970-1976, [5], [6], [7]). The methods, used by W. L. Ferrar and T. Kubota, to obtain (generalized Poisson) summation formulae depend similarly on functional equations of Dirichlet series (attached to the generalized theta series).

In the theory of functions of several complex variables, the problem about the continuation of meromorphic functions has not been much investigated for a long time in spite of its importance except the deeper result of the continuity theorem due to E. E. Levi [4] and H. Kneser [3], The difficulty of its investigation is based on the following reasons: we can not use the tools of not only Cauchy’s integral formula but also the power series and there are indetermination points for the meromorphic function of many variables different from one variable. Therefore we shall also follow the Levi and Kneser’s method and seek for the aspect of meromorphic completion of a Reinhardt domain in Cn.