zbMATH — the first resource for mathematics

Module defect and factorisability. (English) Zbl 0664.12007
From the author’s introduction: “This paper deals with two concepts relating to modules over Abelian group rings. One is factorisability, introduced in Nelson’s thesis [A. Nelson, Ph. D. thesis (London 1979); see also the author, Gaußsche Summen”, Lecture Notes, Univ. Köln 1982 and Cambridge 1983 (Köln 1983; Zbl 0556.12003)], the other is the module defect, first investigated by the author [Quart. J. Math., Oxf. II. Ser. 16, 193-232 (1965; Zbl 0192.140)], in the more general setting of orders in commutative algebras. Both now come into prominence in work on Galois module structure, multiplicative as well as additive...
Factorizability leads to an equivalence relation between lattices over integral group rings, weaker than local isomorphsm, and turning up naturally and significantly when one sets out to compare multiplicative modules of units or additive modules of algebraic integers with certain “standard” modules. The module defect is then the natural channel for information on the structure of our Galois modules. But beyond that the properties of module defects are actually reflected in integral properties of certain L-value-regulator quotients... We are then led to integral formulations of problems and of theorems in the direction of the Stark conjectures...”
For further information the reader may consult the author’s papers: “L- values at zero and multiplicative Galois module structure (also: Galois Gauss sums and additive Galois module structure)”, J. Reine Angew. Math. 397, 42-9 (1989) and “L-functions at \(s=0\) and Galois modules”, Sém. Théor. Nombres, Univ. Bordeaux I 1986/87, Exposé No.32 (1989; Zbl 0642.12015)].
Reviewer: H.Opolka

11R32 Galois theory
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R42 Zeta functions and \(L\)-functions of number fields
11R52 Quaternion and other division algebras: arithmetic, zeta functions