Arnold, D. M.; Vinsonhaler, C. I. Duality and invariants for Butler groups. (English) Zbl 0752.20026 Pac. J. Math. 148, No. 1, 1-10 (1991). 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) Cited in 12 Documents MSC: 20K15 Torsion-free groups, finite rank 20K40 Homological and categorical methods for abelian groups 20K99 Abelian groups Keywords:Butler groups; finite distributive lattice; types; typeset; numerical quasi-isomorphism invariants; strongly indecomposable \(T\)-groups; category equivalence Citations:Zbl 0673.20033 PDFBibTeX XMLCite \textit{D. M. Arnold} and \textit{C. I. Vinsonhaler}, Pac. J. Math. 148, No. 1, 1--10 (1991; Zbl 0752.20026) Full Text: DOI