Cotorsion pairs, model category structures, and representation theory
The paper under review deals with Quillen model structures [\textit{D.G. Quillen}, Homotopical algebra, Lect. Notes Math. 43, Springer, Berlin-N.Y. (1967; Zbl 0168.20903)]. The author makes a general study of Quillen model category structures on abelian categories, showing that these structures are closely related to cotorsion pairs. The main results are the following. For a bicomplete abelian category with a model structure, in a predefined way, the full subcategories of cofibrant, fibrant and trivial objects produce functorially complete cotorsion pairs under certain conditions. Conversely, given classes satisfying these conditions, there is a unique model structure such that these classes are the cofibrant, fibrant and trivial objects. A functorially complete cotorsion pair is described in a Grothendieck category with enough projectives. Amodel structure defined by cotorsion pairs is a monoidal model category (compatible with a given tensor product on the abelian category), if certain conditions are satisfied. This theory is applied to (possibly non-commutative) Gorenstein rings to obtain two model structures on the category of \(R\)-modules compatible with all exact sequences, where the class of trivial objects is the class of modules of finite projective dimension. In the projective model structure, every module is fibrant, and the cofibrant objects are the Gorenstein projective modules. In the injective model structure every module is cofibrant and the fibrant objects are the Gorenstein injective modules. The homotopy categories in these two model structures are equivalent, and they are called the stable module category. The stable module category is shown to be a compactly generated triangulated category. When the ring is \(R=K[G]\), where \(K\) is a principal ideal domain and \(G\) is finite, then the stable module category is a symmetric monoidal as well. Finally, some of the known examples of model structures are shown to fit into the framework of the first result commented above.
scientific article; zbMATH DE number 1889153