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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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