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Factorisability, group lattices, and Galois module structure. (English) Zbl 0734.11064
Let \({\mathcal O}\) be a Dedekind domain with field of fractions K and let \(\Gamma\) be a finite abelian group. A. Fröhlich [Ill. J. Math. 32, 407-421 (1988; Zbl 0664.12007)] has introduced the notion of factor equivalence between two \({\mathcal O}\Gamma\)-lattices, which is a weaker relation than that of being in the same genus. (The present paper contains a helpful summary of these definitions and their relationship.) The problem then is to find invariants which describe the genera within a factor equivalence class. The author gives a partial solution in two cases: (i) K is a number field and no prime divisor of \(| \Gamma |\) is ramified in K/\({\mathbb{Q}}\); (ii) K is an absolutely unramified local field.
The main result is as follows. Suppose that X and Y are \({\mathcal O}\Gamma\)- lattices with KX\(\cong KY\) a quotient of the algebra \({\mathbb{Q}}\Gamma\). Then the defect function \(J(X,Y)=0\) if and only if X and Y are both factor equivalent and “\(\circ\)-equivalent”, this last relation also being defined in terms of the module defect. Applying the result when \(Y={\mathfrak A}\) is the associated order of X, a criterion for X to be in the genus of \({\mathfrak A}\) is obtained. The author remarks that the result is not true after any weakening of the hypotheses. The theory is illustrated by a detailed analysis of the possible relationships between the order \({\mathcal O}\Gamma +{\mathfrak b}^{-1}(\sum \gamma)\) and the lattice \({\mathfrak b}{\mathcal O}\Gamma +J_{\Gamma}\), where \({\mathfrak b}\) is an ideal of \({\mathcal O}\) dividing \(| \Gamma |\) and \(J_{\Gamma}\) is the augmentation ideal.

MSC:
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11S23 Integral representations
11R27 Units and factorization
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[1] Burns, D, Factorisability and the arithmetic of wildly ramified Galois extensions, (), 59-66 · Zbl 0734.11065
[2] ()
[3] Fröhlich, A, Invariants for modules over commutative separable orders, Quart. J. math. Oxford, 16, 193-232, (1965) · Zbl 0192.14002
[4] Fröhlich, A, L-values at zero and multiplicative Galois module structure, J. reine angew. math., 397, 42-99, (1989) · Zbl 0693.12012
[5] Fröhlich, A, Module defect and factorisability, Illinois J. math., 32, No. 3, 407-421, (1988) · Zbl 0664.12007
[6] Gillard, R, Remarques sur LES unités cyclotomiques et LES unités elliptiques, J. number theory, 11, 21-48, (1979) · Zbl 0405.12008
[7] Nelson, A, Monomial representations and Galois module structure, ()
[8] Reiner, I, Maximal orders, (1975), Academic Press London · Zbl 0305.16001
[9] Burns, D, Factorisability, group lattices and Galois module structure, () · Zbl 0734.11064
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