Introduction to abelian model structures and Gorenstein homological dimensions (Q2803235)

The central ideas of this book start from the notion of cotorsion pair introduced by\textit{L. Salce} [in: Gruppi abeliani e loro relazioni con la teoria dei moduli, Conv. Roma 1977, Symp. Math. 23, 11--32 (1979; Zbl 0426.20044)] for abelian groups but also valid for any abelian categories. This approach is similar to torsion theory but changing the bifunctor Hom by the bifunctor Ext. This notion has been very important in the study of rings and modules using homological methods (in the solution of the flat conjecture by \textit{L. Bican} et al. [Bull. Lond. Math. Soc. 33, No. 4, 385--390 (2001; Zbl 1029.16002)] and in the study of the second finitistic dimension conjecture by \textit{L. Angeleri Hügel} and \textit{O. Mendoza Hernández} [Ill. J. Math. 53, No. 1, 251--263 (2009; Zbl 1205.16005)]. In this book, its relation with model structures, via the \textit{M. Hovey} correspondence [Math. Z. 241, No. 3, 553--592 (2002; Zbl 1016.55010)], is one of the main interest.NEWLINENEWLINEThe book is divided in four parts. The first part deals with the categorical and classical homological concepts, including abelian and Grothendieck categories and extension and torsion functors in this setting.NEWLINENEWLINEThe second part is devoted to the study of the interactions between homological algebra and homotopy theory. After introducing the notion of weak factorizations system in a category, model categories are addressed and the homotopy category of a model category is constructed in two different ways. One of those is using localizations and the alternative one is via cylinder and path objects. Moreover, the definition of monoidal model categories is given. The next chapter studies the cotorsion pairs. The main result of this chapter is the Eklof-Trlifai Theorem that stablishes that any cotorsion pair cogenerated by a set, in a Grothendieck category with enough projectives and, is right complete. The relationship between cotorsion pairs and monoidal structures is given by the Hovey correspondence. This correspondence associates an abelian model structure to a Hovey triple, i.e. two cotorsion pairs with some extra conditions. Using this correpondence, the stable module category of a quasi-Frobenius ring can be constructed as the homotopy category associated to the model category constructed from Hovey triple defined by the projective modules. Analogously, the derived category of the category of left \(R\)-modules can be constructed as the homotopy category of the classic model structure on the category of chain complexes over the ring \(R\).NEWLINENEWLINEHomological dimensions (injective, projective and flat dimension) are analyzed in Part III. Several model structures on chain complexes are constructed using clases of modules of bounded homological dimensions.NEWLINENEWLINEIn the last part of the book, Gorenstein categories are studied. This Grothendieck categories are characterized by the coincidence of the classes of finite projective dimension and finite injective dimension and also the finiteness of big finitistic projective and big finitistic injective dimension. Using Gorenstein projective objects (resp. Gorenstein injective objects) a Hovey triple is constructed, from this a unique model structure is obtained. This model structure is generalized to the case of finite Gorenstein injective dimension. For the case of finite Gorenstein projective dimension the model structure is obtained for Gorenstein rings.NEWLINENEWLINEThis book is a worthy addition to the relative homological algebra literature. Graduate students can use it to introduce themself in the topic and it is also a usefuf reference's book for specialists in the field of homological algebra.