Integral structures in algebraic tori

A generalization of the Conner-Floyd map from complex cobordism to complex /C-theory is constructed for formal A-modules when A is the ring of algebraic integers in a number field or its p-adic completion. This map is employed to study the Adams-Novikov spectral sequence for formal A-modules and to confirm a conjecture of D. Ravenel. 0. Introduction. Let BP be the spectrum representing Brown-Peterson cohomology with respect to a prime p and let E be the Adams summand of complex if-theory with respect to this prime. The BP version of the Conner-Floyd map is a map of spectra BP —+ E which induces a natural equivalence BP, X ®Bp. E» ~ E.X. In particular this induces an isomorphism £* ®Bp. BP»BP®bp.£. ~ E.E and so provides a way of computing the Hopf algebra of stable co-operations for E from those for BP. Using this one can obtain a description of E*E similar to that for K*K contained in [AHS]. The study of BP and the computation of BP* BP are based on a study of formal group law, in particular the p-typical formal group law. In [Rl] Ravenel studied a generalization of this situation where the formal group law is replaced by a formal yl-module where A is the ring of integers in an algebraic number field K or its p-adic completion. The purpose of the present paper is to describe the corresponding generalization of the map (BP», BP* BP) —► (E*,E*E) induced by the Conner-Floyd map, and to compute the generalization of E*E. This is of interest because it provides some information about a conjecture (3.10) made in [Rl]. This conjecture concerned the value of a certain Ext group Exty,t(Va, Va) when K is an extension of the field Qp of p-adic numbers. Here (Va, VaT) is the Hopf algebroid corresponding to the A-typical formal A module. This group was conjectured to be, up to small factor, A/J^,^. Here J^iq_1 is the ideal of A generated by the elements of the form aTM — 1 for units a of A congruent to 1 mod(7r) and (it) is the unique prime ideal in A. We will show, using the generalization of the Conner-Floyd map, that A/J*,^ occurs as E\' in the chromatic spectra sequence for formal ^-modules [Rl, Lemma 2.10] and that the small factor in the conjecture is contributed by the nontriviality of the differential d originating from this group. We will analyze this differential and show that it is nonzero for A Received by the editors June 27, 1986. 1980 Mathematics Subject Classification (1985 Revision). Primary 55T25; Secondary 55N22, 14L05. ©1987 American Mathematical Society 0002-9947/87 SI.00 + $.25 per page 319 License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

The purpose of this article to determine and classify the indecomposable RG-lattices, where R is a certain ring of algebraic integers, and G is a cyclic group of prime order.

Given a number field k with ring of algebraic integers o(k), define an equivalence relation by declaring that two ideals a and b lie in the same class if and only if there exist non-zero elements c and d such that ca = db. These classes form an abelian group called the ideal class group of o(k), abbreviated Cl(o(k)). This group, in some sense, measures how different a ring of algebraic integers is from a principal ideal domain.

This paper shows that the group is boundedly elementary generated for and the ring of algebraic integers in a global function field. Contrary to previous statements for number fields and earlier statements for global function fields, the bounds proven in this paper for elementary bounded generation are independent of the underlying global function field and only depend on the integer . Combining our main result with earlier results, we further establish that elementary bounded generation always has bounds independent from the global field in question, only depending on .

The motivation behind the material presented in these notes is the following question:- if N/K is a finite Galois extension of number fields with Galois group Γ, and if \( mathfrak{D}{\text{ and }}\mathfrak{D}N \) are the rings of algebraic integers in K and N respectively, then what can be said about \( mathfrak{D}N \) as a Γ-module? A complete answer to this would be a description of \( mathfrak{D}N \) as a module over the group ring \( mathfrak{D}\Gamma \) , but since in general \( mathfrak{D}N \) need not be free over \( mathfrak{D} \) , it is more fruitful to restrict scalars and view \( mathfrak{D}N \) as a ℤΓ-module.

Let F be a number field, O the ring of algebraic integers in F, and let θ denote the inclusion O ↪ F. The localization exact sequence in algebraic K-theory splits into short exact sequences $$ 0 \to K_n O\xrightarrow{{\theta _\# }}K_n F \to \oplus _m K_{n - 1} \left( {O/m} \right) \to 0 $$ for all positive integers n, where θ # is the homomorphism induced by θ in K-theory and where m runs over the set of maximal ideals of O (see Section 5 of [Q1], Theorem 8 of [Q2] and Theoreme 1 of [S2]); in particular, θ # is always injective. On the other hand, G. Banaszak investigated the subgroup of divisible elements in K n F and explained the important role of these elements in relation with the Lichtenbaum-Quillen conjecture and etale K-theory (see [B1], [B2], [BG], [BZ]). If n is odd, K n F is a finitely generated abelian group and has therefore no non-trivial divisible elements. If n is even, K n F is a large torsion group but all its divisible elements belong to the image of θ # because ⊕ m K n-1 (O/m) is a direct sum of finite cyclic groups and hence contains no non-trivial divisible elements.

Characters and decomposition of a representation in a number field

Information sets in quadratic-residue codes

Using arithmetic intersection theory, a theory of heights for projective varieties over rings of algebraic integers is developed. These heights are generalizations of those considered by Weil, Schmidt, Nesterenko, Philippon, and Faltings. Several of their properties are proved, including lower bounds and an arithmetic Bézout theorem for the height of the intersection of two projective varieties. New estimates for the size of (generalized) resultants are derived. Among the analytic tools used in the paper are “Green forms” for analytic subvarieties, and the existence of positive Green forms is discussed.

In this paper, for any given natural numbersn anda, we can construct explicitly positive definite indecomposable integral Hermitian forms of rankn over\(\mathbb{Q}(\sqrt { - 3} )\) with discriminanta, with the following ten exceptions:n=2,a=1, 2, 4, 10;n=3,a=1, 2, 5;n=4,a=1;n=5,a=1; andn=7,a=1. In the exceptional cases there are no forms with the desired properties. The method given here can be applied to solving the problem of constructing indecomposable positive definite HermitianRm-lattices of any given rankn and discriminanta, whereRm is the ring of algebraic integers in an imaginary quadratic field\(\mathbb{Q}(\sqrt { - m} )\) with class number unity.

The integral structure of a simple Lie algebra L of Chevalley type over a field F of fractions of an integral domain D is studied. Sandwich relations for sufficiently large orders are obtained, including a new general sandwich relation for orders of L in case D is an integrally closed Noetherian domain. Generalizations of the principal results of Hyman (1966) in the case when D is a ring of algebraic integers are obtained, using techniques developed by the author and Stewart (1973) which are applied to certain orders in L that arise in a natural fashion from the Chevalley basis.Subject classification (Amer. Math. Soc. (MOS) 1970): primary 17 B 20; secondary 17 B 10, 17 B 45, 20 G 15, 20 G 05.

Let Cl(OK[G]) denote the locally free class group, that is the group of stable isomorphism classes of locally free OK[G]-modules, where OK is the ring of algebraic integers in the number field K and G is a finite group. We show how to compute the Swan subgroup, T(OK[G]), of Cl(OK[G]) when K = Q(ζp), ζp a primitive p-th root of unity, G = C2, where p is an odd (rational) prime so that hp+ = 1 and 2 is inert in K/Q. We show that, under these hypotheses, this calculation reduces to computing a quotient ring of a polynomial ˚ we do the computations obtaining for several primes p a nontrivial divisor of Cl(Z[ζp]C2). These calculations give an alternative proof that the fields Q(ζp) for p=11, 13, 19, 29, 37, 53, 59, and 61 are not Hilbert-Speiser.

A means to construct dense, full-diversity STBCs from maximal orders in central simple algebras is introduced for the first time. As an example we construct an efficient ST lattice code with non-vanishing determinant for 4 transmit antenna MISO application. Also a general algorithm for testing the maximality of a given order is presented. By using a maximal order instead of just the ring of algebraic integers, the size of the code increases without losses in the minimum determinant. The usage of a proper ideal of a maximal order further improves the code, as the minimum determinant increases. Simulations in a quasi-static Rayleigh fading channel show that our lattice outperforms the DAST-lattice due to the properties described above.

The problem of locating and classifying the normal subgroups of GLn(R), the general linear group of degree n over a commutative ring R with an identity element, has received considerable attention. The solution when R is a field is well known (of. Dieudonné(5), Artin(1)): unless n is equal to two and R has two or three elements, normal subgroups of GLn(R) either lie in the centre of GLn(R) or contain the special linear group SLn(R). However, if R is not a field, then for each ideal I of R the natural map R → R/I induces a homomorphismand, if 0 < I < R, the kernel of θ1 is a non-central normal subgroup of GLn(R) which does not contain SLn(R). The most that may be expected is that each normal subgroup determines an ideal I of R, in such a way that all normal subgroups determining the same ideal I lie between suitably defined greatest and smallest normal subgroups of GLn(R) corresponding to I. For example, write ZI for the inverse image of the centre of GLn(R/I) under the homomorphism θI, and write KI for the intersection of SLn(R) with the kernel of θI. Then KI ≽ ZI, and the results of Klingenberg(7) and Mennicke(9) show that if n ≥ 3, and if R is either a local ring or the ring of rational integers, then any normal subgroup H of GLn(R) satisfiesfor some uniquely determined ideal I. There are many similar theorems. That the above result breaks down for arbitrary rings, even for large n, follows easily from the negative solution to the congruence subgroup problem for certain rings of algebraic integers (see Bass, Milnor and Serre(3)).

Direct constructions of Diophantine representations of linear recurrent sequences are discussed. These constructions generalize already known results for second-order recurrences. Some connections of this problem with the theory of units in rings of algebraic integers are shown. It is proved that the required representations exist only for second-, third-, and fourth-order sequences. In the two last-mentioned cases certain additional restrictions on their coefficients must be imposed. Bibliography:14 titles.