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Cartan-Eilenberg Gorenstein projective complexes. (English) Zbl 1292.16005
In this paper, the notion of a Cartan-Eilenberg Gorenstein projective complex is introduced. Cartan-Eilenberg Gorenstein projective complex is inspired by similar ideas and problems in [E. E. Enochs, J. Algebra 342, No. 1, 16-39 (2011; Zbl 1246.18005)], where Cartan-Eilenberg Gorenstein injective complex is introduced and studied. The authors describe how the homological theory on Gorenstein projective modules generalizes to a homological theory on Cartan-Eilenberg Gorenstein projective complexes. The author shows that the definitions of the Cartan-Eilenberg Gorenstein injective and projective complexes are the same. The Cartan-Eilenberg Gorenstein balance is also considered: Let \(R\) be a ring of finite left Gorenstein global dimension. Then the functor \(\operatorname{Hom}(-,-)\) on \(C(R\mathrm{-Mod})\times C(R\mathrm{-Mod})\) is right balanced by \(CE(R\mathrm{-Gorproj})\times CE(R\mathrm{-Gorinj})\) where \(CE(R\mathrm{-Gorproj})\) and \(CE(R\mathrm{-Gorinj})\) are the classes of Cartan-Eilenberg Gorenstein projective and injective complexes, respectively.

16E05 Syzygies, resolutions, complexes in associative algebras
16E10 Homological dimension in associative algebras
18G05 Projectives and injectives (category-theoretic aspects)
18G10 Resolutions; derived functors (category-theoretic aspects)
18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G35 Chain complexes (category-theoretic aspects), dg categories
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