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Ideal cotorsion theories in triangulated categories. (English) Zbl 1461.18007

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 cotorsion pair [...]. In the context of triangulated cetegories, the cotorsion pairs are replaced by \(t\)-structures.”
The article deals with triangulated categories and their almost exact structures (see its §2.1). It makes also a wide use of the notion of phantom ideal introduced in [I. Herzog, Adv. Math. 215, No. 1, 220–249 (2007; Zbl 1128.16005)] and studies the property, for a phantom idal, to be precovering (meaning that each object admits a precover relative to it), or preenveloping (the dual notion) – see section 3. The heart of the paper is Section 5, which studies together ideal cotorsion pairs and 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 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).”

MSC:

18E40 Torsion theories, radicals
18G10 Resolutions; derived functors (category-theoretic aspects)
18G80 Derived categories, triangulated categories
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