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\(\mathrm{FP}_n\)-injective, \(\mathrm{FP}_n\)-flat covers and preenvelopes, and Gorenstein AC-flat covers. (English) Zbl 1427.18007
To understand the scope of this paper, first recall a few definitions. A module is \(FP_{n}\) if it has a projective resolution whose terms of degree up to \(n\) are finitely generated. Thus finitely generated modules are precisely \(FP_{0}\) and finitely presented modules are precisely \(FP_{1}\) (or simply \(FP\)). If a module has a projective resolution all of whose terms are finitely generated one says that the module is \(FP_{\infty}\). By Schanuel’s lemma, this condition is equivalent to saying that the module is \(FP_{n}\) for any \(n \geq 0\). Accordingly, one has \(FP_{n}\)-injectives and \(FP_{n}\)-flats. One also has notions of \(n\)-coherent and \(n\)-hereditary rings. This nomenclature describes the basic objects of study in the paper under review.
The paper consists of two parts. In the first part the authors show, using duality pairs introduced by H. Holm and P. Jørgensen [J. Commut. Algebra 1, No. 4, 621–633 (2009; Zbl 1184.13042)], that over any ring the class of \(FP_{n}\)-flats is both covering and preenveloping for any \(n \geq 2\). A similar result holds for \(FP_{n}\)-injectives. Within the indicated framework, the authors also deal with generalizations of Gorenstein injective and Gorenstein flat modules.
In the second part the authors work with modules of type \(FP_{\infty}\). Injectives are then replaced with absolutely clean (AC) modules and flats are replaced by level modules. Accordingly, Gorenstein injectives (respectively, Gorenstein projectives) are replaced by Gorenstein AC-injectives (respectively, Gorenstein projectives). The authors then introduce Gorenstein AC-flat modules and show that this class inherits some properties of Gorenstein flat modules.

18G25 Relative homological algebra, projective classes (category-theoretic aspects)
18G35 Chain complexes (category-theoretic aspects), dg categories
13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.)
13D07 Homological functors on modules of commutative rings (Tor, Ext, etc.)
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