Ideal cotorsion theories in triangulated categories (Q2214136)

From the introduction of the article: ``Approximations of objects by some better understood ones are important tools in the study of various categories. For example they are used to construct resolutions and to do homological algebra: in module theory the existence of injective envelopes, projective precovers and flat covers are often used for definig derived functors [...]. The central role in approximation theory for the case of module, or more general abelian or exact, categories is played by the notion of \textit{cotorsion pair} [...]. In the context of triangulated cetegories, the cotorsion pairs are replaced by \(t\)-structures.'' The article deals with triangulated categories and their \textit{almost exact structures} (see its \S 2.1). It makes also a wide use of the notion of \textit{phantom ideal} introduced in [\textit{I. Herzog}, Adv. Math. 215, No. 1, 220--249 (2007; Zbl 1128.16005)] and studies the property, for a phantom idal, to be \textit{precovering} (meaning that each object admits a precover relative to it), or \textit{preenveloping} (the dual notion) -- see section 3. The heart of the paper is Section 5, which studies together \textit{ideal cotorsion pairs} and \textit{relative phantom ideals}. As the authors indicate in their introduction: ``As an application of the theory developed here we prove in the last section of the paper a generalization to projective classes of a result by \textit{I. Herzog} [Invent. Math. 139, No. 1, 99--133 (2000; Zbl 0937.18013)] for smashing subcategories of compatly generated triangulated categories (see Proposition 6.2.5).''