Showing papers on "Field (mathematics) published in 1971"

Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

Abstract: 0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term "perfect" as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection

Families of algebraic curves with fixed degeneracies

Abstract: In this paper we prove that there exist only finitely many nonisomorphic and nonconstant curves of fixed genus, defined over a fixed function field and having bad reductions at a given finite set of points of this field.

The logarithmic limit-set of an algebraic variety

Abstract: Let C be the field of complex numbers and V a subvariety of (C{O})n. To study the "exponential behavior of Vat infinity", we define V(a) as the set of limitpoints on the unit sphere Sn-1 of the set of real n-tuples (u, log I i u.,u log IXn ), where x e V and u, = (1 + (log lx,l)2) -2. More algebraically, in the case of arbitrary base-field k we can look at places "at infinity" on V and use the values of the associated valuations on X1, Xn to construct an analogous set V(b). Thirdly, simply by studying the terms occurring in elements of the ideal I defining V, we define another closely related set, V',). These concepts are introduced to prove a conjecture of A. E. Zalessky on the action of GL(n, Z) on k[X1 1, . . ., Xn 1], then studied further. It is shown among other things that V(b) = V(c) (when defined) V(a. If a certain natural conjecture is true, then equality holds where we wrote "-" and the common set V. Sn1 is a finite union of convex spherical polytopes. 1. A conjecture of Zalessky. Let k be a field, and k[X ]=k[X11,..., Xn l] the ring obtained by adjoining n commuting indeterminates and their inverses to k. This is the group algebra on the free abelian group of rank n, Zn, so GL(n, Z) has a natural action on it. Call a subgroup of Zn nontrivial if it is of infinite order and infinite index in Zn; and call an ideal I( k[X'] nontrivial if it is of infinite dimension (i.e., nonzero) and infinite codimension in k[X+] as k-vector spaces. A. E. Zalessky conjectures in [1, Problem V.9], and we shall here prove: THEOREM 1. Let I be a nontrivial ideal in k[X ], and Hc GL(n, Z) the stabilizer subgroup of I. Then H has a subgroup Ho offinite index, which stabilizes a nontrivial subgroup of Zn (equivalently, which can be put into block-triangular form

The Order of the Antipode of Finite-dimensional Hopf Algebra

Abstract: Examples of finite-dimensional Hopf algebras over a field, whose antipodes have arbitrary even orders ≥4 as mappings, are furnished. The dimension of the Hopf algebra is qn+1, where the antipode has order 2q, q ≥ 2, and n is an arbitrary positive integer. The algebras are not semisimple, and neither they nor their dual algebras are unimodular.

Orbits of linear algebraic groups

Abstract: Let G be a linear algebraic group and let p: G GL(V) be a rational representation of G. When G is linearly reductive, D. Mumford has shown that if a point x C V has 0 in the Zariski-closure cl (Gs x) of its orbit, then there exists a one-parameter subgroup X: Gm G such that x(a) x 0 as a 0 (Theorem (4.1)). (See ? 2 for notation and definitions.) Suppose that G and p :are defined over a field k and that x C Vk. It has been conjectured by Mumford (based on a stronger conjecture of J. Tits see [13, p. 64]) that, when k is perfect, X can be chosen to be defined over k. More generally, one can ask when a linear algebraic k-group G has the following property:

Automorphisms of a free associative algebra of rank 2. II

Abstract: Let R be a commutative domain with 1. R(x, y) stands for the free associative algebra of rank 2 over R; R[x, y>] is the polynomial algebra over R in the commuting indeterminates x' and y. We prove that the map Ab: Aut (R(x, y)) Aut (R[x, y ]) induced by the abelianization functor is a monomorphism. As a corollary to this statement and a theorem of Jung [51, Nagata [71 and van der Kulk [81* that describes the automorphisms of FR, y ] (F a field) we are able to conclude that every automorphism of F(x, y) is tame (i.e. a product of elementary automorphisms). R stands for a commutative domain with 1. R(x, y) is the free associative algebra of rank 2 over R on the free generators x and y; R[x, "-1 is the polynomial algebra over R on the commuting indeterminates x and y. We will prove here that the answer to the following conjecture [3, p. 1971 is in the affirmative: If F is a field then the group of automorphisms of F(x, y) is generated by the elementary automorphisms (defined below) of F(x, y) (i.e. every automorphism of F(x, y) is tame). In fact, we are going to prove here that, if R is as above, the map Ab: Aut (R (x, y)) Aut (R[x y71) induced by the abelianization functor is a monomorphism and as a consequence of this statement and a theorem of Jung, Nagata and van der Kulk* that says that every automorphism of F[x, y 1 is tame (for F a field) we will be able to give a complete description of Aut (F(x, y)). The proof is a generalization of the proof of the main theorem of [41; in fact the algorithm we use here to solve a system of equations in R(x, y) is essentially the same we used in the previous paper. We will refer to [41 for additional details in the proofs. I am indebted to G. M. Bergman for making the observation that the tameness result is not true in the generality claimed in our previous paper [41 and announced in the Bulletin of the AMS in November 1971 [Automorphisms of a free associative algebra of rank 2, Bull. Amer. Math. Soc. 77(1971), 992-9941, since the corresponding tameness theorem for the abelian case (i.e. the theorem of Jung, Received by the editors August 5, 1971. AMS 1970 subject classifications. Primary 16AO6, 16A72; Secondary 20F55, 16A02.

Classification of generalized Witt algebras over algebraically closed fields

Abstract: Let D be a field of characteristic p> 0 and m, n1 ..nm be integers ? 1. A Lie algebra W(m : n1, . . . nm) over D is defined. It is shown that if ID is algebraically closed then W(m: nl,..., ntim) is isomorphic to a generalized Witt algebra, that every finite-dimensional generalized Witt algebra over (D is isomorphic to some W(m: ni, . . ., nm), and that W(m : ni . . ., nm) is isomorphic to W(s : r1, . . ., r,) if and only if m=s and ri=na(i) for 1 i m where a is a permutation of {1, . . ., m}. This gives a complete classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The automorphism group of W(m : n1,..., nm) is determined for p > 3. Introduction. Let (d be a field of characteristic p > 0. Kaplansky [5] (generalizing earlier definitions by Witt [1], Zassenhaus [11] and Jacobson [2]) has defined a family of Lie algebras over (D in the following manner: Let I= {i, j, . . . } be a set of indices, 9 be a total additive group of functionals on I with values in (D, and Y be a vector space with basis Ix C. Define a bilinear multiplication in Y by [(i, a), (j, r)] = r(i)(j, g + r) or(j)(i, C + -). It is easily seen that Y is a Lie algebra. Following Ree [7] we will call such algebras generalized Witt algebras. The problem we consider in this paper is the classification of the finite-dimensional generalized Witt algebras over algebraically closed fields. The study of this problem was begun by Ree [7] who showed that generalized Witt algebras over algebraically closed fields are isomorphic to certain algebras of derivations. (We state this result in detail in ?2.) We give a complete solution to this problem by constructing for any field (D of characteristic p > 0 and any integers m, nl, .. n, lm , 1 a Lie algebra W(m : n1, . . ., nm) over (d and proving the following theorem (which was announced in [10]): Received by the editors April 13, 1970. AMS 1969 subject classifications. Primary 1730.

Extensions of derivations

Abstract: We show that for a class of algebras including separable algebras one can extend derivations of the center to derivations of the algebra. The following theorem was proved in the special cases that C is a field by Hochschild (Ho) and C is a semilocal ring by Roy and Sridharan (R, S) (and for any C in (Kn)). It is also a trivial conse- quence of a more general result proved by a short cohomological argument. Theorem 1. Let A be an algebra separable over its center C and M be an A®c A°r'-module. Then any derivation d '. C—>MA extends to a derivation d:A—*M. Since an algebra separable over its center C is C-projective (A, G,

Analytic and algebraic dependence of meromorphic functions

Abstract: Preface.- German letters.- The rank of a holomorphic map.- Product representations.- Meromorphic functions.- Dependence.- Proper, light, holomorphic maps.- The field .- Semi-proper maps.- Quasi-proper maps.- as a finite algebraic extension of .- Quasi-proper maps of codimension k.- Full holomorphic maps.- Globalization.- The schwarz Lemma.- Sections in meromorphic line bundles.- Preparations.- Pseudoconcave maps.- A counter example by Kas.

Two remarks about hereditary orders

Abstract: In the first remark it is shown that, over a Dedekind ring, hereditary orders in a separable algebra are precisely the "maximal" orders under a relation stronger than inclusion (Theorem 1). At the same time simple proofs for known structure theorems of hereditary orders are obtained. In the second remark a complete classification is given of lattices over a hereditary order, provided the underlying Dedekind ring is contained in an algebraic number field and the lattices satisfy the Eichler condition (Theorem 2). Let o be a Dedekind ring with quotient field k, A/k a separable finite-dimensional algebra over k and R an o-order in A (i.e. a finitely generated o-algebra in A, containing the identity and such that kR =A). An order R is hereditary, if every left ideal is a projective R-module. It is a classical result-apart from terminology-that maximal orders are hereditary, but the converse of this is false: there are nonmaximal hereditary orders. Our first remark is, that if inclusion is replaced by a stronger relation, hereditary orders are characterized by the property of being locally maximal everywhere under this relation. To avoid confusion, we will use the term extremal orders instead. This characterization of hereditary orders can be used to give very simple proofs of some known properties of hereditary orders, which were obtained by Harada [4] and Brumer [2]. Since Brumer [2] is not available in print, we include proofs of the main results given there. In the complete local case, the structure of Rp-lattices is well known (Brumer [2 ]). The basic fact is that indecomposable Rp-lattices are in fact lattices over a maximal order containing Rp. This does not hold globally and only partial results are known in that case. Using results of an earlier paper (Jacobinski [5]), we give a complete classification of lattices over a hereditary o-order, provided the quotient field of 0 is an algebraic number field. The local theory yields a classification of genera of R-lattices. Our result is that the lattices in a restricted genus are isomorphic. This means that two R-lattices M and N are Received by the editors April 24, 1970. AMS 1970 subject classifications. Primary 16A18, 16A14; Secondary 20C0, 16A50.

Inseparable splitting theory

Abstract: If L is a purely inseparable field extension of K, we show that, for large enough extensions E of K, the E algebra L &OK, E splits to become a truncated polynomial algebra. In fact, there is a unique smallest extension E of K which splits L/K and we call this the splitting field S(L/K) of LIK. Now L --S(L/K) and the extension S(L/K) of K is also purely inseparable. This allows us to repeat the splitting field construction and obtain inductively a tower of fields. We show that the tower stabilizes in a finite number of steps and we study questions such as how soon must the tower stabilize. We also characterize in many ways the case when L is its own splitting field. Finally, we classify all K algebras A which split in a similar way to purely inseparable field extensions. Introduction. In Chapter 1, we introduce a splitting theory for purely inseparable field extensions. The idea is that if L/K is purely inseparable then for suitable base extensions E of K the algebra L /)K E over E has a very simple form, namely, it is a special case of what we call a simply truncated polynomial algebra. This fits into a pattern long familiar in the theory of separable extensions or the theory of central simple algebras, namely, that after suitable base extensions the initial object reduces to a standard form. The splitting theory starts from the structure equations of a purely inseparable extension first discovered by Pickert [3]. Let L/K be purely inseparable and let xl ... ., xr be what we call a normal generating sequence for L/K. Set Ki = K[xl, . . ., xi] and qi =[Ki:Ki Then the structure theorem says that xgi EK [Xqi ... ., Xqi1 From this we obtain structure equations of the form

On a theorem of Malcev

Abstract: For any pair of distinct elements a, b in a finitely generated abelian semigroup S, we indicate what are the homomorphisms 4 of S onto a finite semigroup such that 4(a) 34 +(b). This improves a previous result of Malcev which states that the considered semigroups are residually finite. A semigroup S is residually finite if for any pair of distinct elements a, bES, a and b can be separated by a congruence of finite index (i.e. there exists a congruence p on S such that S/p is finite and such that the classes of a and b are distinct). In 1958 A. I. Malcev proved the following theorem: Any finitely generated abelian semigroup is residually finite [6]. Malcev's elegant proof is based on two other results: (1) Every finitely generated commutative algebra over a field of characteristic 0 is isomorphic to an algebra of n Xn matrices over a field of characteristic 0 [5 ]. (2) Every finitely generated group or semigroup of n X n matrices over a field of characteristic 0 is residually finite [4 ]. The theorem of (1) is not easily extendable to noncommutative algebras. Furthermore in the proof of (2) Malcev used the Hilbert basis theorem which allows one to establish the result without an explicit construction of a congruence separating two elements. For possible extensions of Malcev's theorem to nonabelian semigroups, it is of interest to have an explicit description of congruences of finite index separating two given elements. A solution to this problem is provided by our Theorem 3. For connections of residual finiteness with the problem of embedding a semigroup in a compact semigroup the reader is referred to [1]. 1. Notation and conventions. To avoid notation complications we assume that the semigroups discussed have an identity. If S has no Received by the editors February 26, 1970. AMS 1969 subject classifications. Primary 2093, 2093; Secondary 2205, 0288.

Rationality of certain algebraic tori

Abstract: It is shown that the necessary condition for pure transcendence of a field of invariants given by Swan is also sufficient. We prove the rationality of factors of the type , where the are tori of a certain special form, as well as the extension of the statement to all connected linear groups with cyclic algebraic splitting field.

Application of rational approximation in calculation of the second derivative of the gravity field

Abstract: A rational approximation method has been introduced to increase the convergence of the Taylor series, which is the fundamental basis for calculation of the second derivative of the gravity field or any field satisfying Laplace’s equation. Six weight coefficient sets, three each using rational and Taylor series approximations for the same number of rings, have been developed. The frequency responses of the proposed weight coefficient sets establish the superiority of rational approximation over Taylor series approximation. The amplitude responses of all derived coefficient sets, along with those of some of the existing coefficient sets, are also presented.

On the notion of algebraic closedness for noncommutative groups and fields

Abstract: ?1. The notion of algebraic closedness plays an important part in the theory of commutative fields. The corresponding notion in the theory of ordered fields is (not only intuitively but in a sense which can be made precise in a metamathematical framework, compare [4]) that of a real closed ordered field. Several suggestions have been made (see [2] and [8]) for the formulation of corresponding concepts in the theory of groups and in the theory of skew fields (division rings, noncommutative fields). Here we present a concept of this kind, which preserves the principal metamathematical properties of algebraically closed commutative fields and which applies to a wide class of first order theories K, including the theories of commutative and of skew fields and the theories of commutative and of general groups. The precise choice of the vocabularies for these theories is irre, levant. For example, in the case of field theory it may consist of a binary relation, x = y, and of the two ternary relations x + y = z and xy = z, or of the relation x = y and of the two operations x + y and xy. In fact, the definition and some of the principal properties of our concept are based on the sole assumption that the theory K under consideration is inductive, i.e. that the union (inductive limit) of a monotonic set {M,} of models of K is again a model of K. The framework presented here is the outcome of some ongoing research on the forcing method in model theory whose results will be, or have been, published more fully elsewhere [1], [5], [6], [7]. The reader should have no difficulty in adapting the proofs given in those references to the somewhat different situation considered here. However, the formulation of the concepts and results of the present paper does not involve the forcing notion and only a knowledge of the elements of the predicate calculus is required in order to understand it. The title of our paper includes the term closedness rather than closure. Thus, our analysis will not touch upon the question of the existence of a "closed" extension which is in some sense minimal as is the algebraic closure of a commutative field.

$G_{a}$-action of the affine plane

Abstract: Recently, R. Rentschler [4] proved that if k is a field of characteristic 0, any action of the additive group G a on the affine plane is equivalent to an action of the form t (x, y) = (x, y + tf(x) , where t∈G a (k), (x, y)∈k 2 and f∈k[X] .

Embedding free algebras in skew fields

Abstract: This paper constructs a minimal element in the partial order on the set of skew fields generated by a free algebra, and shows that the partial order contains a certain sub partial order. Examples of embedding free algebras in skew fields of heights one and two are also given. Let R be an integral domain (noncommutative) with identity. We will assume that the integral domain R is always embedded in some skew field (this need not be the case in general [2], [3], [6], [7]). There are many skew fields which contain R but in the commutative case there is only one field which is generated by R, namely the field of fractions. In the noncommutative case there may be several distinct skew fields generated by R [4, pp. 277]. For skew fields D1 and D2 generated by R, we say D1 > D2 if there exists a place from D1 to D2 which extends the natural isomorphism between the embeddings of R. This paper shows that this is a partial order on the set (P (we identify isomorphic embeddings) of skew fields generated by R and in the case where R is the free algebra on two generators we show that (P contains the subposet ? where C(x) is the unique maximal element [1, Theorem 27 ] of (P and Ki are distinct elements of (P with K2 minimal in (P. We also examine the height of an integral domain and give examples of embeddings of different heights of a free algebra in two skew fields. I. The partial order and the chain of domains Qi(K, ?(R)). DEFINITION. Let D be a skew field and 0 an isomorphism of R into D. R is fully embedded in D if the smallest sub skew field of D containing + (R) is D itself. We will denote a full embedding of (D, 4 (R)) by (D, q). DEFINITION. If D1 and D2 are two division rings we say 0 is a place from D1 to D2 if 0 is a homomorphism from a local subring S of D, onto D2 [local means the set of nonunits is an ideal ]. DEFINITION. For full embeddings (D, a) and (K, y) we say (D, a) > (K, Py) if there is a place q from D to K such that qV'(K) Doa(R) and f1 a (R) =,y-c'. Received by the editors October 9, 1970. AMS 1969 subject classifications. Primary 1646, 1615.

On the block of defect zero

Abstract: Let G be a finite group and let p be a fixed prime number. If D is any p-subgroup of G, then the problem whether there exists a p-block with D as its defect group is reduced to whether NG(D)/D possesses a p-block of defect 0. Some necessary or sufficient conditions for a finite group to possess a p-block of defect 0 have been known (Brauer-Fowler [1], Green [3], Ito [4] [5]). In this paper we shall show that the existences of such blocks depend on the multiplicative structures of the p-elements of G. Namely, let p be a prime divisor of p in an algebraic number field which is a splitting one for G, o the ring of p-integers and k = o/p, the residue class field.

Global dimension of triangular orders

Abstract: The triangular orders of finite global dimension in nXn matrices over the quotient field of a DVR are found and a bound is given for their dimensions.

Modular arithmetic and finite field theory: A tutorial

Abstract: The paradigm of algorithm analysis has achieved major pre-eminence in the field of symbolic and algebraic manipulation in the last few years. A major factor in its success has been the use of modular arithmetic. Application of this technique has proved effective in reducing computing times for algorithms covering a wide variety of symbolic mathematical problems. This paper is intended to review the basic theory underlying modular arithmetic. In addition, attention will be paid to certain practical problems which arise in the construction of a modular arithmetic system. A second area of importance in symbol manipulation is the theory of finite fields. A recent algorithm for polynomial factorization over a finite field has led to faster algorithms for factorization over the field of rationals. Moreover, the work in modular arithmetic often consists of manipulating elements in a finite field. Hence, this paper will outline some of the major theorems for finite fields, hoping to provide a basis from which an easier grasp of these new algorithms can be made.

Application of an Algebraic Technique to the Solution of Laplace’s Equation in Three Dimensions

Abstract: Solutions to Laplace’s equation in three dimensions are obtained by the introduction of a suitable commutative and associative algebra over the field of the complex numbers. In one formulation these solutions turn out to be the coefficients of powers of $\sigma $ in the expression of a function of $w = z + ( i\rho /2 )( {\sigma e^{i\phi } + \sigma ^{ - 1} e^{ - i\phi } } )$ as a formal series. A variety of relations from the subject of spherical harmonics are shown to follow from the consideration of simple functions of w. While there is a condition on the gradients of the coefficients of powers of $\sigma $ that is analogous to the orthogonality of the gradients of the real and imaginary parts of an analytic function of $x + iy$, since here the number of coefficients may be infinite, the relation is much more complicated than for the two-dimensional case.

The category of representations of a completely 0-simple semigroup

Abstract: A. H. Clifford [1], [2] has shown that all finite dimensional representations of a completely 0-simple semigroup S over a field Φ.phi; can be obtained as extensions of those of its maximal subgroups and has given a method for constructing all such representations. This representation theory depends strongly on the fact the representations under consideration are finite dimensional and is not adequate to deal with the infinite dimensional case or with representations over arbitrary rings. In order to determine the structure of the (contracted) algebra Φ( S ) of S modulo its radical, one has to consider representations which are not finite dimensional or over fields; c.f. ‘6’. Hence Clifford's theory does not suffice for this purpose.