\(A\)-projective resolutions and an Azumaya theorem for a class of mixed Abelian groups. (English) Zbl 1079.20503

If \(A\) is an Abelian group then any direct summand of a direct power of \(A\) is called \(A\)-projective. If \(\mathcal A\) is a class of groups then an Abelian group \(G\) is said to be \(\mathcal A\)-decomposable if it is of the form \(G\cong\bigoplus_{A\in \mathcal A}P_A\) where each \(P_A\) is \(A\)-projective. Let \(\Gamma\) denote the class of groups \(G\) which are isomorphic to pure subgroups of \(\prod_pG_p\) containing \(\bigoplus_pG_p\). The subclass of \(\Gamma\) consisting of all groups having finite torsionfree rank is denoted by \(\Gamma_\infty\). Now every \(G\in\Gamma_\infty\) contains a finite independent subset \(X\) such that \(F=\langle X\rangle\) is a free subgroup of \(G\) with \(G/F\) torsion. Viewing \(G\) as a pure subgroup of \(\prod_pG_p\) we can write \(X=\{x_i=(x_{ip})\mid i=1,\dots,n\}\). Finally the class \(\mathcal G\) consists of all elements of \(\Gamma_\infty\) for which \(G_p\) is finite for all \(p\) and satisfies \(G_p=\langle x_{1p},\dots,x_{np}\rangle\) for all but finitely many \(p\).
The main purpose of the paper is to prove an Azumaya-style theorem stating that the class of \(\mathcal G\)-decomposable groups is closed under direct summands.


20K21 Mixed groups
20K25 Direct sums, direct products, etc. for abelian groups
20K27 Subgroups of abelian groups
20K40 Homological and categorical methods for abelian groups
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