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\(A\)-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.

MSC:

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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