Abstract. We study the interplay between the notions of n-coherent rings and finitely n-presented modules, and also study the relative homological algebra associated to them. We show that the n-coherency of a ring is equivalent to the thickness of the class of finitely n-presented modules. The relative homological algebra part comes from the study of orthogonal complements to this class of modules with respect to Ext
Abstract.Absolutely clean and level R-modules were introduced in [BGH13] and used to show how Gorenstein homological algebra can be extended to an arbitrary ring R. This led to the notion of Gorenstein AC-injective and Gorenstein AC-projective R-modules. Here we study these concepts in the category of chain complexes of R-modules. We define, characterize and deduce properties of absolutely clean, level, Gorenstein AC-injective, and Gorenstein AC-projective chain complexes. We show that the category Ch(R) of chain complexes has a cofibrantly generated model structure where every object is cofibrant and the fibrant objects are exactly the Gorenstein AC-injective chain complexes.
We study the notions of n-hereditary rings and its connection to the classes of finitely n-presented modules, FPn-injective modules, FPn-flat modules and n-coherent rings. We give characterizations of n-hereditary rings in terms of quotients of injective modules and submodules of flat modules, and a characterization of n-coherent using an injective cogenerator of the category of modules. We show two torsion pairs with respect to the FPn-injective modules and the FPn-flat modules over n-hereditary rings. We also provide an example of a Bézout ring which is 2-hereditary, but not 1-hereditary, such that the torsion pairs over this ring are not trivial.
We investigate conditions for when the t-structure of Happel-Reiten-Smalø associated to a torsion pair is a compactly generated t-structure. The concept of a tCG torsion pair is introduced and for any ring R, we prove that t = (T , F ) is a tCG torsion pair in R-Mod if, and only if, there exists, {T λ } a set of finitely presented R-modules in T , such that F = Ker(Hom R (T λ , ?)). We also show that every tCG torsion pair is of finite type, and show that the reciprocal is not true. Finally, we give a precise description of the tCG torsion pairs over Noetherian rings and von Neumman regular rings.
We prove that, for any n ≥ 2, the classes of FPn-injective modules and of FPn-flat modules are both covering and preenveloping over any ring R. This includes the case of F P∞-injective and F P∞-flat modules (i.e. absolutely clean and, respectively, level modules). Then we consider a generalization of the class of (strongly) Gorenstein flat modules -the (strongly) Gorenstein AC-flat modules (cycles of exact complexes of flat modules that remain exact when tensored with any absolutely clean module). We prove that some of the properties of Gorenstein flat modules extend to the class of Gorenstein AC-flat modules; for example we show that this class is precovering over any ring R. We also show that (as in the case of Gorenstein flat modules) every Gorenstein AC-flat module is a direct summand of a strongly Gorenstein ACflat module. When R is such that the class of Gorenstein AC-flat modules is closed under extensions, the converse is also true. We also prove that if the class of Gorenstein AC-flat modules is closed under extensions, then this class of modules is covering.
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