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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.

##### MSC:
 13D02 Syzygies, resolutions, complexes and commutative rings
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