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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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References:
 [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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