On infinite rank integral representations of groups and orders of finite lattice type.
Classically, an algebra \(\Lambda\) over a Dedekind domain \(R\) is said to be an \(R\)-order if \(\Lambda\) belongs to the category \(R\)-proj of finitely generated projective \(R\)-modules. A \(\Lambda\)-module in \(R\)-proj is called a \(\Lambda\)-lattice. In this paper, the authors extend the scope of integral representation theory, replacing \(R\)-proj by the category \(R\)-Proj of all projective \(R\)-modules. Accordingly, for an \(R\)-order \(\Lambda\), they define a generalised \(\Lambda\)-lattice to be an \(R\)-projective \(\Lambda\)-module. The paper was motivated by the problem to classify involutions on a free Abelian group, which arose independently in the context of Lie algebras (2000) and \(C^*\)-algebras (2002). The authors prove, more generally, that every generalised \(\mathbb{Z} C_p\)-lattice, where \(C_p\) denotes the cyclic group of prime order \(p\), decomposes into \(\mathbb{Z} C_p\)-lattices. By an old result of Diederichsen and Reiner, there are just three types of indecomposable \(\mathbb{Z} C_p\)-lattices, namely, the trivial \(\mathbb{Z} C_p\)-lattice \(\mathbb{Z}\), the cyclotomic \(\mathbb{Z} C_p\)-lattice \(\mathbb{Z}[\zeta_p]\), where \(\zeta_p\) denotes a primitive \(p\)-th root of unity, and the regular representation \(\mathbb{Z} C_p\). In case \(\Lambda\) is representation-finite, i.e. the number of isomorphism classes of indecomposable \(\Lambda\)-lattices is finite, the authors prove that every generalised \(\Lambda\)-lattice is a direct summand of a coproduct \(\coprod E_i\) with \(\Lambda\)-lattices \(E_i\). As a consequence, using a theorem of Kaplansky, it follows that every generalised \(\Lambda\)-lattice decomposes into countably generated \(\Lambda\)-lattices. If, in addition, \(R\) is a complete discrete valuation domain, this implies that every generalised \(\Lambda\)-lattice decomposes into \(\Lambda\)-lattices. (The latter result can also be obtained directly from a theorem of Crawley, Jónsson, and Warfield.) For a hereditary \(R\)-order \(\Lambda\) satisfying the Jordan-Zassenhaus theorem, it follows that the generalised \(\Lambda\)-lattices coincide with the projective \(\Lambda\)-modules. (For further results on this new topic, see our forthcoming paper ``Large lattices over orders'', to appear in Proc. Lond. Math. Soc., III. Ser.)
scientific article; zbMATH DE number 2130537