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 \(\overline{\text{Ext}}^1(-,-)\).


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


[1] Bican, L.; El Bashir, R.; Enochs, E., All modules have flat covers, Bull. Lond. Math. Soc., 33, 385-390 (2001) · Zbl 1029.16002
[2] Cartan, H.; Eilenberg, S., Homological Algebra (1956), Princeton University Press · Zbl 0075.24305
[3] Dold, A., Zur Homotopietheorie der Ketten Komplexe, Math. Ann., 140, 278-298 (1960) · Zbl 0093.36903
[4] Enochs, E.; Jenda, O., Relative Homological Algebra, de Gruyter Exp. Math., vol. 30 (2000), Walter de Gruyter: Walter de Gruyter New York · Zbl 0952.13001
[5] Enochs, E.; Jenda, O.; Xu, J., Orthogonality in the category of complexes, Math. J. Okayama Univ., 38, 25-46 (1997) · Zbl 0940.18006
[6] Enochs, E.; Jenda, O.; Torrecillas, B., Gorenstein flat modules, Nanjing Daxue Shuxue Bannian Kan, 10, 1-9 (1993) · Zbl 0794.16001
[7] Gillespie, J., Cotorsion pairs and degreewise homological model structures, Homology, Homotopy Appl., 10, 1, 283-304 (2008) · Zbl 1140.18011
[8] Gillespie, J., Kaplansky classes and derived categories, Math. Z., 257, 4, 811-843 (2007) · Zbl 1134.55016
[9] Göbel, R.; Trlifaj, J., Approximations and Endomorphism Algebras of Modules (2006), Walter de Gruyter: Walter de Gruyter Berlin · Zbl 1121.16002
[10] Hovey, M., Cotorsion theories, model category structures and representation theory, Math. Z., 241, 553-592 (2002) · Zbl 1016.55010
[11] Lazard, D., Autour de la platitude, Bull. Soc. Math. France, 97, 31-128 (1969) · Zbl 0174.33301
[12] Rump, W., Elementary varieties and existence of flat covers, J. Algebra, 322, 2131-2149 (2009) · Zbl 1189.18001
[13] Rump, W., Flat covers in abelian and non-abelian categories, Adv. Math., 225, 1589-1615 (2010) · Zbl 1200.18006
[14] Salce, L., Cotorsion theories for abelian groups, (Sympos. Math., vol. XXIII (1979), Academic Press), 11-32 · Zbl 0426.20044
[15] Verdier, J.-L., Des catégories dérivées des catégories abéliennes, Astérisque, 239 (1997)
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