Permutation Summands Over Z

The Galois module structure of certain arithmetic principal homogeneous spaces

We consider the class of abelian groups possessing partial decomposition bases in L ∞ ω δ L^\delta _{\infty \omega } for δ \delta an ordinal. This class contains the class of Warfield groups which are direct summands of simply presented groups or, alternatively, are abelian groups possessing a nice decomposition basis with simply presented cokernel. We prove a classification theorem using numerical invariants that are deduced from the classical Ulm-Kaplansky and Warfield invariants. This extends earlier work by Barwise-Eklof, Göbel and the authors.

A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G = AT and for any X ≤ A and for any Y ≤ T there exists an element u ∈ ⟨X; Y ⟩ such that XY u ≤ G. The notation H ≤ G means that H is a subgroup of a group G. In this paper, we obtained the estimations of numerical invariants(the derived length, the nilpotent length, the π-length, the nilpotent π-length) of G = AB in terms of invariants of tcc-subgroups A and B.

1. The results. Let l be an odd prime, r ≥ 1, and letbe the quaternion group of order 4lr, as given by generators and relations. Throughout N is a tamely ramified normal number field with Galois group Gal (N/Q) = H (a ‘quaternion field’), and its ring of integers. We are interested in the structure of as a module over the integral group ring ZH. Deriving, first, certain classgroup invariants for locally free ZH-modules, we shall then determine those for the module in terms of the arithmetic invariants of N/Q. When 1 ≡ – 1 (mod 4), this yields again a Galois module interpretation of Artin root numbers quite analogous to that in (2). On the other hand for l ≡ 1 (mode 4), we shall get a weak ‘normal integral basis theorem’. The original impetus for this work came from computations of J. Queyrut, who – in different language – obtained these results in the case l = 3, r = 1 (cf. (7)). The tools, we are using, come from the general theory developed in recent years with such concrete applications in mind, and it is perhaps of interest to see how the various ‘strands’, on root numbers (cf. (3), (4)), on locally free modules (cf. (5)), and on Galois module structure (cf. (6)) are here pulled together. For technical reasons, we shall impose on N the slight further restriction, that l be non-ramified, although our results would remain true without this. Both the statements and the proofs of the theorem depend on ideas contained in (5) and (6). The reader who is prepared to take them for granted should, however, be able to read the present paper independently of those papers.

The structure of a formal module F( $$ \mathfrak{M} $$ ) for a chain of finite extensions M/L/K, where M/L is an unramified p-extension, is studied. It is proved that the first Galois cohomology of a formal module for an unramified extension is trivial for any degree of prime ideal. The presentation of the formal module is constructed in terms of generators and relations. As an application of the main result, the structure of a formal module for generalized Lubin–Tate formal groups is obtained.

Abstract. Let N/Q be a weakly ramified 3-extension of Galois group G. We show that the class of the square root of the inverse different of the extension is of order dividing 3 in the class group of locally free Z[G]-modules. The proof goes through the computation of a sum of cubes of resolvents, using an expression of this sum in terms of symmetric functions.

Let K be a commutative noetherian ring. It is proved that a representation of a finite group on a K-module of finite length or on a K-module of finite exponent has a finite basis for its identities. In particular, this implies an earlier result of Nguyen Hung Shon and the author stating that every representation of a finite group over a field is finitely based. The problem whether every representation of a finite group over a commutative noetherian ring is finitely based still remains open.

Poor modules with no proper poor direct summands

The notion of the orbifold Euler characteristic came from physics at the end of the 1980s. Coincidence (up to sign) of the orbifold Euler characteristics is a necessary condition for crepant resolutions of orbifolds to be mirror symmetric. There were defined higher order versions of the orbifold Euler characteristic and generalized (“motivic”) versions of them. In a previous paper, the authors defined a notion of the Grothendieck ring K 0 fGr ( Var C ) of varieties with actions of finite groups on which the orbifold Euler characteristic and its higher order versions are homomorphisms to the ring of integers. Here, we define the generalized orbifold Euler characteristic and higher order versions of it as ring homomorphisms from K 0 fGr ( Var C ) to the Grothendieck ring K 0 ( Var C ) of complex quasi-projective varieties and give some analogues of the classical Macdonald equations for the generating series of the Euler characteristics of the symmetric products of a space.

Galois module structure of the square root of the inverse different in even degree tame extensions of number fields

A logarithmic approach to classgroups of integral group rings

Overview From class field theory Extension classes Locally free class groups Tate sequences Recognizing $G$-modules Local analogue $\Omega _m$ and the $G$-module structure of $E$ Artin $L$-functions at $s=0$ $q$-indices Parallel properties of$A_\varphi$ and $A_\varphi$ $\mathbb Q$-valued characters Representing the Chinburg class Small $S$ A cyclotomic example Notes Bibliography Subject index.

The so-called class-invariant homomorphism $\psi$ measures the Galois module structure of torsors--under a finite flat group scheme $G$--which lie in the image of a coboundary map associated to an isogeny between (N\'eron models of) abelian varieties with kernel $G$. When the varieties are elliptic curves with semi-stable reduction and the order of $G$ is coprime to 6, is is known that the homomorphism $\psi$ vanishes on torsion points. In this paper, using Weil restrictions of elliptic curves, we give the construction, for any prime number $p>2$, of an abelian variety $A$ of dimension $p$ endowed with an isogeny (with kernel $\mu_p$) whose coboundary map is surjective. In the case when $A$ has rank zero and the $p$-part of the Picard group of the base is non-trivial, we obtain examples where $\psi$ does not vanishes on torsion points.

A note on module structures of source algebras of block ideals of finite groups

A deformation-theoretic version of Maschke's theorem for modular group algebras: The commutative case