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

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