Showing papers in "Annals of Mathematics in 1964"

On the Deformation of Rings and Algebras

Abstract: CHAPTER I. The deformation theory for algebras 1. Infinitesimal deformations of an algebra 2. Obstructions 3. Trivial deformations 4. Obstructions to derivations and the squaring operation 5. Obstructions are cocycles 6. Additivity and integrability of the square 7. Restricted deformation theories and their cohomology theories 8. Rigidity of fields in the commutative theory CHAPTER II. The parameter space 1. The set of structure constants as parameter space for the deformation theory 2. Central algebras and an example justifying the choice of parameter space 3. The automorphism group as a parameter space, and examples of obstructions to derivations 4. A fiber space over the parameter space, and the upper semicontinuity theorem 5. An example of a restricted theory and the corresponding modular group CHAPTER III. The deformation theory for graded and filtered rings 1. Graded, filtered, and developable rings 2. The Hochschild theory for developable rings 3. Developable rings as deformations of their associated graded rings 4. Trivial deformations and a criterion for rigidity 5. Restriction to the commutative theory 6. Deformations of power series rings

Remarks on the Cohomology of Groups

Abstract: JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org.

Global Properties of Minimal Surfaces in E 3 and E n

Abstract: : A minimal surface is the surface of least area bounded by a given closed curve. In three dimensional space these surface are realized, for reasonably simple curves, by soap films spanning wire loops. This paper obtains results related to the theorem that the normals to a complete minimal surface, not a plane, are everywhere dense. Examples: l. The same theorem holds for two-dimensional surfaces in Euclidean spaces of dimension three or more. 2. Theorems connecting total curvature, Euler characteristic, and number of boundary components. 3. Theorems about the normals and tangent planes to minimal surfaces, including the capacity of the set of directions omitted. (Author)

Non-Linear Equations of Evolution

Abstract: In the case of equation (2), it is well-known that if we assume that f satisfies a local Lipschitz condition in u, then local solutions of (2) with u(O) = uo will exist. (This is given in detail in [1] and [15]. A straightforward extension of this fact to the case of more general equations of evolution (1) with A(t) = A independent of t is carried through and exploited in [2], [13], and [26].) However, for infinite-dimensional spaces, the continuity hypothesis of Assumption (I) is not enough to assure the existence of local solutions. Our basic result for equation (2) is that Assumptions (I) and (II) above are sufficient for the existence and uniqueness of a strongly C1-function u from R+ to H which satisfies equation (2), and for which u(0) =u0. A similar result

Arithmetic of unitary groups

Abstract: The purpose of this paper is to develop the theory of elementary divisors, to prove the approximation theorem, and to determine the class number for the following two types of algebraic groups: (i) the unitary group of a hermitian form over an algebraic number field; (ii) the unitary group of a hermitian form over a quaternion algebra, having an algebraic number field as center, with respect to the canonical involution. The latter includes the symplectic group as a special case, since we admit the total matric algebra of degree two as a quaternion algebra. Of course, the approximation theorem is obtained only in the case where the hermitian form is indefinite. A problem of the same kind was investigated for the group of regular elements of a simple algebra by Eichler [5], and for the orthogonal group by Eichler [6, 7] and Kneser [9]. In [11], we treated the groups of type (ii) with the same intent and obtained a somewhat weaker result than in the present paper. We now explain our result by giving a summary of each section. Let F be an algebraic number field of finite degree and K a quadratic extension of F. Let V be a vector space over K of dimension n and ?F(x, y) a non-degenerate hermitian form on V with respect to the non-trivial automorphism of K over F. We denote by U( V, F) and G( V, F) respectively the unitary group of iF and the group of similitudes of iF (cf. 1.1 below). Further, we denote by SU(V, (D) the subgroup of U(V, iF) consisting of the elements with determinant 1. If n is even, we can consider the group of direct similitudes H( V, iF) (cf. 2.2). In the first two sections, we give preliminaries for these four groups, and study elementary properties of maximal lattices, whose definition is as follows. Let g and x be the ring of integers in F and in K, respectively. Let L be an x-lattice in V. We denote by Me(L) the ideal in F generated by the elements iF(x, x) for all x e L, and call it the norm of L. We say that L is maximal if L is a maximal one among the x-lattices with the same norm. Let p be a prime ideal of F, and F, the completion of F with respect to P; and let K, = K ?F Fp and V, = V OF Fi. In ?? 3 and 4, we treat the local theory of elementary divisors for the lattices in V,. If p decomposes in F, KP is isomorphic to F, x F,, and therefore things are much easier. Section 3 is concerned with this case. In ? 4, we discuss the case where K, is a field. 369

Finite Groups of Even Order in Which Sylow 2-Groups are Independent

Abstract: In this paper we will study a class of groups of even order which satisfy the following condition: (TI): two different Sylow 2-groups contain only the identity element in common. There are three series of finite non-abelian simple groups known to satisfy the above condition (TI). Let q denote a power of 2 greater than 2. The linear fractional group in 2 variables over the field of q elements, denoted here by L2(q), satisfies the condition (TI). The projective unitary groups U3(q) provide another series of groups satisfying (TI). The third series consists of groups G(q) defined by the author in [7]. The main result of this paper is the converse of the above statement. We will prove the following theorem.

An Epiperimetric Inequality Related to the Analyticity of Minimal Surfaces

Abstract: The purpose of this paper is to enable us to prove that a minimal surface is analytic at points where its density is one. This will be done elsewhere [1], but the main key to that investigation is the following result which we prove here: Suppose Y is an orientable polyhedral cone, with vertex 0, of dimension m in R., whose boundary lies on the unit sphere with center 0. Then, if Y lies sufficiently near to a diametral plane, we can construct a new surface Y* with the same boundary such that

Good ideals in fields of sets

Abstract: In this paper we shall study some problems which are purely settheoretical in character, but which arose in connection with a problem in the theory of models. The principal results of this paper form an important part of the proofs of the series of results in the theory of models which are announced in [1], [2], [3], and in the appendix of [4]. The present article, however, is entirely free of any notions from the theory of models. It is well-known that, for every infinite cardinal a, there exist nonprincipal prime ideals in the field S(X) of all subsets of a set X of power a, and furthermore there are exactly 221 such prime ideals. The object of this paper is to show that non-principal prime ideals with certain additional properties exist. Let X be a set of power a, and let S( Y) be the set of all non-empty finite subsets of a set Y. If f, g map S.( Y) into the field of subsets of X, we write g > f if, for all a E S.( Y), g(a) -f(a). Let us say that an ideal I in the field S(X) is SJ( Y)-good if, for every monotonic function f on S,(Y) into I, there exists an additive function g > f on S,( Y) into I. If Y is countable, then every ideal I is S.( Y)-good (cf. Corollary 4.2). Our two main results are as follows.

On the Field of Definition for a Field of Automorphic Functions: III

Abstract: The purpose of this paper is to complement the main theorems of the previous papers with the same title, referred to hereafter as [FI] and [FII], and to simplify the proofs. In [FI, II], it was shown that a certain abelian extension k of a certain algebraic number field K' can be determined by a family 1(&2) of abelian varieties belonging to a given type 12 of structures. The structures consist of endomorphism algebra, polarization and points of finite order, of an abelian variety. The case of structures without points of finite order was treated in [FI]. We observed that k is unramified over K', and the law of reciprocity for the extension k/K' can be described in terms of the interrelation between 12 and its "transform". One of the aims of [FII] was to extend this result to the general case involving points of finite order and ramified abelian extensions. However, we determined only the ideal group in K' corresponding to k, but did not give an explicit reciprocity law in such a general case. This will be done in the present paper. We shall also reformulate the results in a more comprehensible manner, and give a simplified proof by means of the characterization of the field k provided by [2, 5.1]. This characterization enables us to dispense with artificial considerations which were necessary in the original proofs. We devote one section to clarify a point concerning the existence of a special member of the family Y(f2), of which the previous proof in [FI] was not with complete clarity. List of symbols. They were introduced in [FI, II], occasionally with a somewhat general meaning.

Existence of Almost Everywhere Convergent Rearrangements for Fourier Series of L 2 Functions

Abstract: It is a fundamental result of Kolmogorov [2] that the Fourier series of a function in L1(-w, w) need not converge almost everywhere. For L, classes with p > 1, no analogous results are available. To the best of our knowledge, to this date, the only result in this direction is again due to Kolmogorov. In a joint paper with D. Menshov [3], Kolmogorov states that there exists a function in L2 whose Fourier series can be so rearranged as to diverge almost everywhere. Kolmogorov's proof of this statement never appeared. Only recently a rather intricate construction of such a function and rearrangement was sketched by Z. Zahorski [8]. More recently Ulyanov [7] extended this construction to general complete orthonormal systems. The results of the present paper are in the opposite direction and lend some support to the conjecture of N. N. Luzin. The latter [4] considered it not unlikely that the Fourier series of every function in L2 should converge almost everywhere. More specifically, we shall show that the following holds:

Asymptotically Neutral Distributions of Electrons and Polynomial Approximation

Abstract: Let D be a bounded simply connected region (connected open set) in the complex plane. We wish to approximate holomorphic functions in D by polynomials; the error must become uniformly small on every fixed closed subset of D. By Runge's theorem such approximation is always possible. However, we impose an additional condition: the zeros of the approximating polynomials must belong to a prescribed set P. Polynomials whose zeros lie in P will be referred to as F-polynomials. It will be assumed for simplicity that F and D are disjoint. P will be called a polynomial approximation set relative to D if and only if every zero free holomorphic function in D can be approximated by F-polynomials. It may be observed that, if F is a polynomial approximation set and J is a simple arc in D, then every continuous function on J can be uniformly approximated by F-polynomials. G. R. MacLane [4] has shown that every rectifiable Jordan curve is a polynomial approximation set relative to its interior. Recently M. D. Thompson [5] gave a new proof of this result, extending it to a different special class of Jordan curves. M.D. Thompson [5] and the author [3] have also considered certain unbounded sets F. In the present paper we determine all bounded sets F that can serve as polynomial approximation sets relative to D. The discussion will be based on the notion of "asymptotically neutral families". A family of finite sequences