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Large lattices over orders. (English)
Proc. Lond. Math. Soc. (3) 91, No. 1, 105-128 (2005).
The author establishes a very interesting theory in order to characterise when for a classical $R$-order $Λ$ every $R$-projective $Λ$-module $M$ is a coproduct of classical lattices. To be more precise, let $R$ be a Dedekind domain with field of fractions $K$ and let $Λ$ be an $R$-algebra, which is a finitely generated projective $R$-module and suppose that $K\cdotΛ=A$ is a semisimple $K$-algebra. Such an algebra $Λ$ is called an $R$-order. A generalized $Λ$-lattice is an $R$-projective $Λ$-module. A generalized lattice is a lattice if it is finitely generated over $R$. Otherwise the generalized lattice is called large. In a first part the author sets up a quite abstract categorical approach of module categories. In particular a sort of generalization of projective objects is given. In a second section some of the local-global principles known for lattices over orders are shown to be true for generalized lattices over orders. In a third section the author defines a structure, which he calls hypergraph, and which generalizes the notion of a finite graph. In particular there may be more than two vertices associated to an edge. The definition of a hypergraph is still very combinatorial. Then the author associates to a locally lattice-finite order $Λ$ a hypergraph where the edges are the indecomposable lattices over the localisations of the order, and the vertices are the simple modules over localisations of the semisimple algebras. The incidence function is then the dimension of the homomorphism space of a localisation of an indecomposable lattice in a simple module over a localisation of $A$. The dimension is taken over the endomorphism ring of the simple module of a localisation of $A$. This concept yields more sophistications. Then, in a final section, the author shows the main result. This is a complete characterisation of the statement that every generalized lattice is a coproduct of lattices in terms of combinatorial conditions on the hypergraph associated to the order.
Reviewer: Alexander Zimmermann (Amiens)