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Duality and invariants for Butler groups. (English) Zbl 0752.20026

The authors classify a special class of Butler groups up to quasi- isomorphism by numerical invariants. The class of groups is defined as follows. Let \(A_ 1,A_ 2,\dots,A_ n\) be non-zero subgroups of the additive group of rational numbers. The group \(G[A_ 1,A_ 2,\dots,A_ n]\) is the cokernel of the diagonal embedding \[ \bigcap^ n_{i=1}A_ i\to A_ 1\oplus A_ 2\oplus\dots\oplus A_ n: a\mapsto (a,a,\dots,a). \] Fix a finite distributive lattice \(T\) of types. A \(T\)- group is a torsion-free group whose typeset is a subset of \(T\). The main result of the paper is the following. A complete set of numerical quasi- isomorphism invariants for strongly indecomposable \(T\)-groups of the form \(G=G[A_ 1,A_ 2,\dots,A_ n]\) is given by \(\{r_ G(M): M\subset T\}\), where \(r_ G(M)=\text{rank}(\sum\{G(\tau): \tau\in M\})\).
The proof reduces the classification to a previous classification result by the same authors in the following way. Let \(A_ 1,A_ 2,\dots,A_ n\) be a rational groups as before and let \(G(A_ 1,A_ 2,\dots,A_ n)\) denote the kernel of the map \(A_ 1\oplus A_ 2\oplus \dots\oplus A_ n\to \mathbb{Q}: (a_ 1a_ 2,\dots,a_ n)\mapsto\sum^ n_{i=1}a_ i\). It was shown [in Proc. Am. Math. Soc. 105, No. 2, 293-300 (1989; Zbl 0673.20033)] that the strongly indecomposable \(T\)-groups of the form \(G(A_ 1,A_ 2,\dots,A_ n)\) are classified up to quasi-isomorphism by the invariants \(\{r_ G[M]: M\subset T\}\) where \(r_ G[M]=\text{rank}(\bigcap\{G[\sigma]: \sigma\in M\})\). The connection between the groups \(G(A_ 1,A_ 2,\dots,A_ n)\) and \(G[A_ 1,A_ 2,\dots,A_ n]\) is made by means of a category equivalence which is established by utilizing suitable representations of groups. Let \(T\) and \(T'\) be finite distributive lattices of types and suppose that there is a lattice anti-isomorphism \(\alpha: T\to T'\). Let \(B_ T\) (\(B_{T'}\)) be the category of all Butler groups that are \(T\)-groups (\(T'\)-groups) with \(\mathbb{Q}\otimes \text{Hom}(G,H)\) as morphisms. Then there is a contravariant exact category equivalence \(D=D(\alpha): B_ T\to B_{T'}\) defined by \(D(G)=H\), where \(\mathbb{Q} H=\text{Hom}_ \mathbb{Q}(\mathbb{Q} G,\mathbb{Q})\) and \(\mathbb{Q} H[\alpha(\tau)]=\mathbb{Q} G(\tau)^{\perp}\) for each \(\tau\in T\). The equivalence has (among others) the property that \(D(G(A_ 1,A_ 2,\dots,A_ n))\) is quasi-isomorphic to \(G[D(A_ 1),D(A_ 2),\dots,D(A_ n)]\) for each \(n\)-tuple (\(A_ 1,A_ 2,\dots,A_ n\)) of subgroups of \(\mathbb{Q}\) with types in \(T\).
Reviewer: A.Mader (Honolulu)

MSC:

20K15 Torsion-free groups, finite rank
20K40 Homological and categorical methods for abelian groups
20K99 Abelian groups

Citations:

Zbl 0673.20033
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