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On the Irreducible Lattices of Orders

Published online by Cambridge University Press:  20 November 2018

Klaus W. Roggenkamp
Klaus W. Roggenkamp*
Affiliation:
Université de Montréal, Montréal, Québec
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We shall use the following notation:

R = Dedekind domain;

K = quotient field of R;

Rp = ring of p-adic integers in K, p being a prime ideal in R;

A = finite-dimensional separable k-algebra;

G – R-order in A (for the definition cf. (3)).

All modules that occur are assumed to be finitely generated unitary left modules, unless otherwise specified. By a G-lattice we mean a G-module which is torsion-free as R-module. A G-lattice is called irreducible if it does not contain a proper G-submodule of smaller R-rank. If p is a prime ideal in R we shall write Gp = Rp ⊗RG; Mp = RP⊗RM for a G-lattice M, and KM = K ⊗R M. Two G-lattices M and N are said to lie in the same genus (notation M ∨ N) if Mp ≌ Np for every prime ideal p in R.

For any A-module L, let S(L) be the collection of G-lattices M, for which KM ≌ L. Suppose that S(L) splits into rg(L) genera, and into ri(L) classes under G-isomorphism. Maranda (6) has shown: If L is an absolutely irreducible A-module, then

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 21 , 1969 , pp. 970 - 976
Copyright
Copyright © Canadian Mathematical Society 1969

References

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