Absolutely clean, level, and Gorenstein AC-injective complexes. (English) Zbl 1346.18021

Bravo, Daniel; Gillespie, James

Commun. Algebra 44, No. 5, 2213-2233 (2016).

Let \(R\) be an arbitrary ring. Absolutely clean, level, Gorenstein AC-injective and Gorenstein AC-projective \(R\)-modules were introduced by the authors and M. Hovey in a different paper [“The stable module category of a general ring”, arXiv:1405.5768]. The aim of this article is to extend these concepts to the category of chain complexes of \(R\)-modules. Also, in Sections 2, 3 and 4, important results of that paper are generalised in this context. For example, a chain complex \(X\) is called Gorenstein AC-injective if it is the kernel of some morphism of chain complexes, morphism which appears in an exact complex of injective complexes and this complex of injective complexes has the property that remains exact after applying the covariant external \(\mathrm{Hom}\) functor associated to any absolutely clean chain complex. In Theorem 3.2, Gorenstein AC-injective chain complexes are characterised using Gorenstein AC-injective \(R\)-modules and the internal \(Hom\) functor. Similar and important results are Theorem 3.3 and Theorem 4.13. Most of the used basic concepts and notations are explained in detail in the Introduction and in the beginning of each section.

Reviewer: Constantin-Cosmin Todea (Cluj-Napoca)

MSC:

18G35	Chain complexes (category-theoretic aspects), dg categories
18G25	Relative homological algebra, projective classes (category-theoretic aspects)

Keywords:

chain complexes; Gorenstein homological algebra; model structure

References:

[1]	Bieri, R. (1981). Homological Dimension of Discrete Groups. 2nd ed. Queen Mary College Mathematical Notes, Queen Mary College Department of Pure Mathematics, London. · Zbl 0357.20027
[2]	Enochs E., Math. J. Okayama Univ 39 pp 19– (1997)
[3]	DOI: 10.1515/9783110803662 · doi:10.1515/9783110803662
[4]	García-Rozas J. R., Covers and Envelopes in the Category of Complexes of Modules (1999) · Zbl 0922.16001
[5]	DOI: 10.1090/S0002-9947-04-03416-6 · Zbl 1056.55011 · doi:10.1090/S0002-9947-04-03416-6
[6]	DOI: 10.4310/HHA.2008.v10.n1.a12 · Zbl 1140.18011 · doi:10.4310/HHA.2008.v10.n1.a12
[7]	DOI: 10.4310/HHA.2010.v12.n1.a6 · Zbl 1231.16005 · doi:10.4310/HHA.2010.v12.n1.a6
[8]	DOI: 10.1007/s00209-002-0431-9 · Zbl 1016.55010 · doi:10.1007/s00209-002-0431-9
[9]	DOI: 10.1080/00927872.2011.622326 · Zbl 1273.55009 · doi:10.1080/00927872.2011.622326