Resolution of unbounded complexes in Grothendieck categories. (English) Zbl 1033.18007

Author’s abstract: As N. Spaltenstein [Compos. Math. 65, 121–154 (1988; Zbl 0636.18006)] showed, the category of unbounded complexes of sheaves on a topological space has enough \(K\)-injective complexes. We extend this result to the category of unbounded complexes of an arbitrary Grothendieck category. This is important for a construction, by the author, of a triangulated category of equivariant motives.


18E15 Grothendieck categories (MSC2010)
18E30 Derived categories, triangulated categories (MSC2010)
18G35 Chain complexes (category-theoretic aspects), dg categories
14F20 Étale and other Grothendieck topologies and (co)homologies


Zbl 0636.18006
Full Text: DOI


[1] Alonso Tarrı́o, L.; Jeremı́as López, A.; Souto Salorio, M. J., Localization in categories of complexes and unbounded resolutions, Canad. J. Math., 52, 2, 225-247 (2000) · Zbl 0948.18008
[2] Gelfand, S. I.; Manin, Y. I., Methods of Homological Algebra (1996), Springer: Springer Berlin · Zbl 0855.18001
[3] Grothendieck, A., Sur quelques points d’algèbre homologique, Tohoku Math. J., 9, 2, 119-221 (1957) · Zbl 0118.26104
[4] Jech, T., Set Theory (1997), Springer: Springer Berlin · Zbl 0882.03045
[5] Neeman, A., Triangulated categories, Ann. Math. Stud., 148 (2001) · Zbl 0974.18008
[6] C. Serpé, A triangulated category of equivariant motives, 2001, preprint.; C. Serpé, A triangulated category of equivariant motives, 2001, preprint.
[7] Spaltenstein, N., Resolutions of unbounded complexes, Compositio Math., 65, 2, 121-154 (1988) · Zbl 0636.18006
[8] Weibel, C. A., An Introduction to Homological Algebra (1994), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0797.18001
[9] Weibel, C. A., Cyclic homology for schemes, Proc. Amer. Math. Soc., 124 (1996) · Zbl 0855.19002
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.