Let A be a Noetherian local ring with the maximal ideal m, and let I be an m-primary ideal in A. This paper examines the equality on Hilbert coefficients of I first presented by Elias and Valla, but without assuming that A is a Cohen–Macaulay local ring. That equality is related to the Buchsbaumness of the associated graded ring of I.

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 this paper is to prove the following three theorems which were announced in [2].

The purpose of this note is to study regular actions of simple algebraic groups on projective threefolds as an application of the theory of algebraic threefolds, especially Mori Theory and the theory of Fano threefolds (cf. Mori [11], Iskovskih [7, 8]). The motivation for this study is as follows. In a series of papers, Umemura, in part jointly with Mukai, has classified maximal connected algebraic subgroups of the Cremona group of three variables and also constructed minimal rational threefolds which correspond to such subgroups (cf. Umemura [16-19], Mukai-Ume-mura [12]). In particular, Umemura and Mukai studied in [12] the SL(2, C)-equivariant smooth projectivization of SL(2, C)/G, where G is a binary icosahedral or octahedral subgroup of SL(2, C). The study of equivariant smooth projectivization of SL(2, C)/G for any finite subgroup G has been completed along their lines in Nakano [14]. The main trick of these studies is the investigation of equivariant contraction maps of extremal rays in the context of Mori Theory [11]. In this note, we apply a similar idea to projective threefolds with a regular action of a simple algebraic group and determine which simple algebraic groups can act regularly and nontrivially on projective threefolds and in which fashion. We also need some standard (but difficult) facts from the theory of Fano threefolds. For the precise statement, see Theorem 1 in the main text. For the proof of this theorem, we need a classification of closed subgroups of simple algebraic groups of codimension 1 and 2, which could be derived easily from the classical work of Dynkin [4]. However, we shall give a geometric proof independent of [4] which leads up directly to the proof of Theorem 1. On the whole, we shall establish by geometric methods the scarcity of closed subgroups of small codimension in simple algebraic groups, which is implied in Dynkin [4].

Let be a stochastic process, where T is a finite interval. The ε-entropy is defined as the following quantity:

(1)

Let $\ell $ be a prime number. The Iwasawa theory of multigraphs is the systematic study of growth patterns in the number of spanning trees in abelian $\ell $-towers of multigraphs. In this context, growth patterns are realized by certain analogs of Iwasawa invariants, which depend on the prime $\ell $ and the abelian $\ell $-tower of multigraphs. We formulate and study statistical questions about the behavior of the Iwasawa $\mu $ and $\lambda $ invariants.

We give a formula for the class number of an arbitrary complex mutliplication (CM) algebraic torus over $\mathbb {Q}$ . This is proved based on results of Ono and Shyr. As applications, we give formulas for numbers of polarized CM abelian varieties, of connected components of unitary Shimura varieties and of certain polarized abelian varieties over finite fields. We also give a second proof of our main result.

We study the canonical basis for the negative part U- of the quantum generalized Kac-Moody algebra associated to a symmetric Borcherds-Cartan matrix. The algebras U- associated to two different matrices satisfying certain conditions may coincide (6.3). We show that the canonical bases coincide provided that the algebras U- coincide (Theorem 6.3.5). We also answer partially a question by Lusztig in [L3] (Theorem 7.1.1).

Throughout the paper, a scheme means a noetherian scheme. By a curve C over a scheme S of genus g, we mean a proper and smooth S-scheme with irreducible curves of genus g as geometric fibres. In the previous paper [15], the author showed that the field of moduli for a non-hyperelliptic curve over a field coincides with the one for its canonically polarized jacobian variety, and in [16], he gave a partial result on the coincidence of the fields of rationality for a hyperelliptic curve and for its canonically polarized jacobian variety. In the present paper, we will discuss the isomorphy of the isomorphism schemes of two curves over a scheme and of their canonically polarized jacobian schemes, by using Oort-Steenbrink’s result [12].

Let us consider a surface S of class C3 in Euclidean space E3 and

(1.1) xi = xi(u1, u2) (i, j, k = 1, 2, 3)

be its parametric representation of class C3.

Throughout this note, p denotes a fixed prime number and f denotes a fixed natural number prime to p.

It is easy to see and more or less known that for any natural number n, there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field. In this note, we consider a more precise problem: for any natural number n, decide whether or not there exists an elliptic curve over p whose j-invariant is of degree n over Fp and whose endomorphism ring is isomorphic to an order of an imaginary quadratic field with conductor f.

The purpose of this paper is to show that the corona theorem is valid on any unbounded finitely sheeted covering surface if and only if the corona theorem is valid on its base surface.

Let Rn+1 = Rn X R denote the (n + l)-dimensional Euclidian space (n ≧ 1). For X ε Rn+1 we write X = (x, t) with x ε Rn and t ε R. For an α with 0 < α < 1, we write

where Δ is the Laplacian on Rn and (– Δ)α is the α-fractional power of - Δ on R*

As a generalization of the conformal structure of type (2, 2), we study Grassmannian structures of type (n, m) for n, m ≥ 2. We develop their twistor theory by considering the complete integrability of the associated null distributions. The integrability corresponds to global solutions of the geometric structures.

A Grassmannian structure of type (n, m) on a manifold M is, by definition, an isomorphism from the tangent bundle TM of M to the tensor product V ⊗ W of two vector bundles V and W with rank n and m over M respectively. Because of the tensor product structure, we have two null plane bundles with fibres Pm-1(ℝ) and Pn-1(ℝ) over M. The tautological distribution is defined on each two bundles by a connection. We relate the integrability condition to the half flatness of the Grassmannian structures. Tanaka’s normal Cartan connections are fully used and the Spencer cohomology groups of graded Lie algebras play a fundamental role.

Besides the integrability conditions corrsponding to the twistor theory, the lifting theorems and the reduction theorems are derived. We also study twistor diagrams under Weyl connections.

We present results on the Watanabe–Yoshida conjecture for the Hilbert–Kunz multiplicity of a local ring of positive characteristic. By improving on a “volume estimate” giving a lower bound for Hilbert–Kunz multiplicity, we obtain the conjecture when the ring has either Hilbert–Samuel multiplicity less than or equal to 5 or dimension less than or equal to 6. For nonregular rings with fixed dimension, a new lower bound for the Hilbert–Kunz multiplicity is obtained.

Let k be an algebraic number field and K be a finite extension of k. Recently, T. Ono defined positive rational numbers E(K/k) and E′(K/k) for K/k. In [7], he investigated some relations between E(K/k) and other cohomological invariants for K/k. He obtained a formula when K is a normal extension of k. In our paper [3], we obtained a similar formula for E′(K/k) in the case of normal extensions K/k. Both proofs essentially use Ono’s results on the Tamagawa number of algebraic tori, on which the formulae themselves do not depend. Hence, in [8], T. Ono posed a problem to give direct proofs of these formulae.

QF-3 algebras R are classified according to their second commutator algebras R′ with respect to the minimal faithful module, which satisfy dom.dim. R′ ≧ 2. The class C(S) of all QF-3 algebras whose second commutator is S, contains besides S only algebras R with dom.dim. R = 1. C(S) contains a unique (up to isomorphism) minimal algebra which can be represented as a subalgebra S0 of S describable in terms of the structure of S, and C(S) consists just of the algebras S0 ⊂ R ⊂ S (up to isomorphism).

Some years ago, Kuramochi gave in his paper [5] a very interesting theorem, which can be stated as follows.

THEOREM OF KURAMOCHI. Let R be a hyperbolic Riemann surface of the class Of OHR(OHD,resp.). Then, for any compact subset K of R such that R—K is connected, R—K as an open Riemann surface belongs to the class 0AB(OAD resp.).

In the first half of this paper, all the limits of irreducible characters of Gn = 𝔖n(T) as n → ∞ are calculated. The set of all continuous limit functions on G = 𝔖 ∞(T) is exactly equal to the set of all characters of G determined in [HH6]. We give a necessary and sufficient condition for a series of irreducible characters of Gn to have a continuous limit and also such a condition to have a discontinuous limit. In the second half, we study the limits of characters of certain induced representations of Gn which are usually reducible. The limits turn out to be characters of G, and we analyse which of irreducible components are responsible to these limits.

Several years ago, Huneke and Leuschke proved a theorem solving a conjecture of Schreyer. It asserts that an excellent Cohen-Macaulay local ring of countable Cohen-Macaulay type which is complete or has uncountable residue field has at most a one-dimensional singular locus. In this paper, it is verified that the assumption of the excellent property can be removed, and the theorem is considered over an arbitrary local ring. The main purpose of this paper is to prove that the existence of a certain prime ideal and a certain totally reflexive module implies the existence of an uncountably infinite number of isomorphism classes of indecomposable totally reflexive modules.