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Cartan-Eilenberg complexes and resolutions. (English) Zbl 1246.18005
The Cartan-Eilenberg injective complexes (CE injective for short) were introduced by Verdier in his thesis. In this paper, the author defines a more general notion, that of a CE($$\mathcal{F}$$) complex, for any class $$\mathcal{F}$$ of left $$R$$-modules (Definition 3.1): $$A$$ is a CE($$\mathcal{F}$$) complex if $$A$$, $$Z(A)$$, $$B(A)$$, and $$H(A)$$ are all complexes of modules from $$\mathcal{F}$$.
The article studies Cartan-Eilenberg injective, projective and flat complexes and shows how they can be used to get Cartan-Eilenberg resolutions. It is proved that every complex has a CE injective envelope (Theorem 4.1). Then it is proved that a complex is Cartan-Eilenberg flat if and only if it is a direct limit of finitely generated Cartan-Eilenberg projective complexes (Theorem 7.2). Proposition 7.3 establishes the existence of the CE flat covers.
The Cartan-Eilenberg balance is also considered (Proposition 6.3): the functor $$\text{Hom}(-,-)$$ on $$C(R\text{-Mod}) \times C(R\text{-Mod})$$ is right balanced by $$CE Proj \times CE Inj$$ where $$CE Proj$$ and $$CE Inj$$ are the classes of Cartan-Eilenberg projective and injective complexes respectively. The right derived functors of $$\text{Hom}(-,-)$$ obtained this way are denoted $$\overline{\text{Ext}}^n(-,-)$$.
The Cartan-Eilenberg Gorenstein injective and projective complexes are also defined. There are two options for defining a CE Gorenstein injective (projective) complex: one is to use Definition 3.1 with $$\mathcal{F}$$ being the class of Gorenstein injective (projective) modules, the other one is to define such a complex in terms of a complete injective (projective) resolution. The author shows (theorem 8.5) that the two definitions are equivalent.
Section 9 of the paper deals with CE cotorsion pairs; the main result here (Theorem 9.4) is that if $$(\mathcal{F}, \mathcal{C})$$ is a hereditary cotorsion pair in $$R\text{-Mod}$$ then $$(CE(\mathcal{F}), CE(\mathcal{C}))$$ is a hereditary cotorsion pair in $$C(R\text{-Mod})$$ relative to $$\overline{\text{Ext}}^1(-,-)$$.

##### MSC:
 18G05 Projectives and injectives (category-theoretic aspects) 18G25 Relative homological algebra, projective classes (category-theoretic aspects) 18G35 Chain complexes (category-theoretic aspects), dg categories
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