##
**Homological and homotopical aspects of torsion theories.**
*(English)*
Zbl 1124.18005

Mem. Am. Math. Soc. 883, 207 p. (2007).

The treatise is an extended version of a former preprint on “Homological aspects of torsion theories”, now with more connections to closed model categories which are also used to reinspect (co)torsion-theoretic structures from this different point of view. Similar results of Mark Hovey are revisited.

The authors define the concept of pretriangulated category to cover preabelian (i.e., additive with kernels and cokernels) and triangulated categories, as well as homotopy categories of additive closed model categories. The idea is to replace the suspension, respectively loop functor by an adjoint pair which becomes zero in the preabelian case. In this framework, many common features of structures like torsion theories in abelian and triangulated categories can be developed in general. For example, if \(\omega\) is a functorially finite subcategory of an abelian category \({\mathcal{C}}\), the stable category \({\mathcal{C}}/[\omega]\) can be regarded as a homotopy category of a closed model structure on \({\mathcal{C}}\) which arises from left and right approximations by objects in \(\omega\). Therefore, the stable category \({\mathcal{C}}/[\omega]\) is pretriangulated. In particular, \(\omega\) can be chosen to be \(\text{add}(T)\) for a tilting module \(T\).

Tilting theory with its various extensions and ramifications has by now influenced and fermented a wide area of mathematics with its peculiar categorical mechanism. In a sense, the authors’ comprehensive treatment can be viewed as a long walk through that wide field, regarding those different facets of tilting. Along this walk, some important achievements have been simplified, e.g., Rickard’s equivalence criterion for derived equivalence in terms of a tilting complex. Of course, the fundamental but still unsolved homological conjectures on artinian algebras are also touched in several respects. Similar to Hovey’s results, the authors classify cotorsion pairs \(({\mathcal{F}},{\mathcal{C}})\) with \({\mathcal F}\cap{\mathcal C}\) functorially finite in terms of closed model structures, which leads to characterizations of tilting modules over an artinian algebra.

Generalizing module categories over Gorenstein or Cohen-Macaulay rings, the authors define an abelian category \({\mathcal{C}}\) with enough projectives and injectives to be Gorenstein if the projective dimension of the injectives and the injective dimension of the projectives are bounded. They define \({\mathcal{C}}\) to be Cohen-Macaulay if there is an adjoint pair of endofunctors which induces an equivalence between the objects of finite projective, respectively injective dimension. Relationships to finitistic dimensions are studied. In case they are finite, it is shown that for a Cohen-Macaulay category \({\mathcal{C}}\) with a dualizing adjoint pair \((F,G)\), the trivial extension \({\mathcal{C}}\ltimes F\) is Gorenstein.

For an abelian category \({\mathcal{C}}\), the universal homology extension functors induced by a hereditary torsion theory in the stable category \({\mathcal{C}}/[\omega]\) with \(\omega\) functorially finite, are investigated. It is shown that these functors generalize the Tate-Vogel homology functors. In particular, the special case is studied where the hereditary torsion theory comes from the stable category of Cohen-Macaulay objecs, provided that \({\mathcal{C}}\) admits an adjoint pair of Nakayama functors. Such abelian categories \({\mathcal{C}}\) are therefore called Nakayama categories. The standard example is the category of finitely generated or all modules over an artinian algebra.

The authors define the concept of pretriangulated category to cover preabelian (i.e., additive with kernels and cokernels) and triangulated categories, as well as homotopy categories of additive closed model categories. The idea is to replace the suspension, respectively loop functor by an adjoint pair which becomes zero in the preabelian case. In this framework, many common features of structures like torsion theories in abelian and triangulated categories can be developed in general. For example, if \(\omega\) is a functorially finite subcategory of an abelian category \({\mathcal{C}}\), the stable category \({\mathcal{C}}/[\omega]\) can be regarded as a homotopy category of a closed model structure on \({\mathcal{C}}\) which arises from left and right approximations by objects in \(\omega\). Therefore, the stable category \({\mathcal{C}}/[\omega]\) is pretriangulated. In particular, \(\omega\) can be chosen to be \(\text{add}(T)\) for a tilting module \(T\).

Tilting theory with its various extensions and ramifications has by now influenced and fermented a wide area of mathematics with its peculiar categorical mechanism. In a sense, the authors’ comprehensive treatment can be viewed as a long walk through that wide field, regarding those different facets of tilting. Along this walk, some important achievements have been simplified, e.g., Rickard’s equivalence criterion for derived equivalence in terms of a tilting complex. Of course, the fundamental but still unsolved homological conjectures on artinian algebras are also touched in several respects. Similar to Hovey’s results, the authors classify cotorsion pairs \(({\mathcal{F}},{\mathcal{C}})\) with \({\mathcal F}\cap{\mathcal C}\) functorially finite in terms of closed model structures, which leads to characterizations of tilting modules over an artinian algebra.

Generalizing module categories over Gorenstein or Cohen-Macaulay rings, the authors define an abelian category \({\mathcal{C}}\) with enough projectives and injectives to be Gorenstein if the projective dimension of the injectives and the injective dimension of the projectives are bounded. They define \({\mathcal{C}}\) to be Cohen-Macaulay if there is an adjoint pair of endofunctors which induces an equivalence between the objects of finite projective, respectively injective dimension. Relationships to finitistic dimensions are studied. In case they are finite, it is shown that for a Cohen-Macaulay category \({\mathcal{C}}\) with a dualizing adjoint pair \((F,G)\), the trivial extension \({\mathcal{C}}\ltimes F\) is Gorenstein.

For an abelian category \({\mathcal{C}}\), the universal homology extension functors induced by a hereditary torsion theory in the stable category \({\mathcal{C}}/[\omega]\) with \(\omega\) functorially finite, are investigated. It is shown that these functors generalize the Tate-Vogel homology functors. In particular, the special case is studied where the hereditary torsion theory comes from the stable category of Cohen-Macaulay objecs, provided that \({\mathcal{C}}\) admits an adjoint pair of Nakayama functors. Such abelian categories \({\mathcal{C}}\) are therefore called Nakayama categories. The standard example is the category of finitely generated or all modules over an artinian algebra.

Reviewer: Wolfgang Rump (Stuttgart)

### MSC:

18E40 | Torsion theories, radicals |

18-02 | Research exposition (monographs, survey articles) pertaining to category theory |

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

18E35 | Localization of categories, calculus of fractions |

18G55 | Nonabelian homotopical algebra (MSC2010) |

18G60 | Other (co)homology theories (MSC2010) |

16G10 | Representations of associative Artinian rings |

18G30 | Simplicial sets; simplicial objects in a category (MSC2010) |

18E10 | Abelian categories, Grothendieck categories |

20C05 | Group rings of finite groups and their modules (group-theoretic aspects) |

20J05 | Homological methods in group theory |

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