Cartan-Eilenberg Gorenstein flat complexes. (English) Zbl 1299.13018

Let \(R\) be a ring and let \(\mathcal{F}\) be a class of (left) \(R\)-modules. A complex \(M\) of \(R\)-modules is called a Cartan-Eilenberg \(\mathcal{F}\) complex if every module in each of the complexes \(M\), \(\mathrm{Z}(M)\) (cycles in \(M\)), \(\mathrm{B}(M)\) (boundaries in \(M\)), and \(\mathrm{H}(M)\) (homology of \(M\)) belongs to \(\mathcal{F}\). Cartan-Eilenberg projective, injective, and flat complexes were studied by E. E. Enochs [J. Algebra 342, No. 1, 16–39 (2011; Zbl 1246.18005)]. The paper under review studies Cartan-Eilenberg Gorenstein flat complexes, that is, the case where \(\mathcal{F}\) is the class of Gorenstein flat \(R\)-modules, introduced by Enochs, Jenda, and Torrecillas [E. E. Enochs et al., J. Nanjing Univ., Math. Biq. 10, No. 1, 1–9 (1993; Zbl 0794.16001)]. It is proved that a complex is Cartan-Eilenberg Gorenstein flat if and only if it has a complete Cartan-Eilenberg flat resolution. It is also shown that if \(R\) is (right) coherent, then every complex of \(R\)-modules has a Cartan-Eilenberg Gorenstein flat cover.


13D02 Syzygies, resolutions, complexes and commutative rings
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