Finite modules over non-semisimple group rings

Abstract

LetG be an abelian group of ordern and letR be a commutative ring which admits a homomorphism ℤ[ζ n ] →R, where ζ n is a (complex) primitiven-th root of unity. Given a finiteR[G]-moduleM, we derive a formula relating the order ofM to the product of the orders of the various isotypic componentsM x ofM, where ξ ranges over the group ofR-valued characters ofG. ForG cyclic, we give conditions under which the order ofM is exactly equal to the product of the orders of theM x. To derive these conditions, we build on work of Aljadeff and Ginosar and obtain, in particular, a new criterion for cohomological triviality which improves upon the well-known critetion of T. Nakayama. We also give applications to abelian varieties and to ideal class groups of number fields, obtaining in particular some new class number relations. In an Appendix to the paper, we use étale cohomology to obtain some additional class number relations. Our results also have applications to “non-semisimple” Iwasawa theory, but we do not develop these here. In general, the results of this paper could be used to strengthen a variety of known results involving finiteR[G]-modules whose hypotheses include (an equivalent forn of) the following assumption: “the order ofG is invertible inR”.