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**Homotopy theory of modules and Gorenstein rings.**
*(English)*
Zbl 1023.55009

The main purpose of the paper is to study the stable module categories of a ring from the point of view of modern algebraic homotopy theory. The algebraic contexts are closed model categories [D. G. Quillen, Homotopical algebra, Lect. Notes Math. 43 (Springer, Berlin-N. Y.) (1967; Zbl 0168.20903)], Eckmann-Hilton’s homotopy theory of modules [P. J. Hilton, Homotopy theory and duality (Thomas Nelson and Sons, London) (1967; Zbl 0155.50801)] and homological algebra of rings.

The content summarized:

The stable categories with coproducts and compact objects are studied and consequences of Brown’s representability theorem [E. H. Brown, Abstract homotopy theory, Trans. Am. Math. Soc. 119, 79-85 (1965; Zbl 0129.15301)] in additive categories with coproducts and weak cokernels are given.

An equivalence between functorially finite subcategories \(H\) of \(C\) and closed model structures on \(C\) is given. In this case all objects of \(C\) are bifibrant and the associated homotopy category is equivalent to the stable category of \(C / H\).

Characterization of rings in order to do homotopy theory over the stable categories modulo projectives or injectives is provided. The condition for the ring is to be left coherent and right perfect in case of the projective homotopy, and to be right Morita in case of the injective homotopy.

Rings which admit a projective or injective stable homotopy category are studied. A ring is said to have a projective, resp. injective, stable homotopy category if the stabilization of the stable category modulo projectives, resp. stable category modulo injectives, is compactly generated. The main result here is that any left coherent and right perfect or right Morita right Gorenstein ring has a projective and injective stable homotopy category. Criteria for the stable homotopy category to be phantomless or a Brown category are provided.

The content summarized:

The stable categories with coproducts and compact objects are studied and consequences of Brown’s representability theorem [E. H. Brown, Abstract homotopy theory, Trans. Am. Math. Soc. 119, 79-85 (1965; Zbl 0129.15301)] in additive categories with coproducts and weak cokernels are given.

An equivalence between functorially finite subcategories \(H\) of \(C\) and closed model structures on \(C\) is given. In this case all objects of \(C\) are bifibrant and the associated homotopy category is equivalent to the stable category of \(C / H\).

Characterization of rings in order to do homotopy theory over the stable categories modulo projectives or injectives is provided. The condition for the ring is to be left coherent and right perfect in case of the projective homotopy, and to be right Morita in case of the injective homotopy.

Rings which admit a projective or injective stable homotopy category are studied. A ring is said to have a projective, resp. injective, stable homotopy category if the stabilization of the stable category modulo projectives, resp. stable category modulo injectives, is compactly generated. The main result here is that any left coherent and right perfect or right Morita right Gorenstein ring has a projective and injective stable homotopy category. Criteria for the stable homotopy category to be phantomless or a Brown category are provided.

Reviewer: J.Remedios (La Laguna)

### MSC:

55U35 | Abstract and axiomatic homotopy theory in algebraic topology |

18E30 | Derived categories, triangulated categories (MSC2010) |

18G25 | Relative homological algebra, projective classes (category-theoretic aspects) |