##
**Homological dimensions and abelian model structures on chain complexes.**
*(English)*
Zbl 1375.18071

In 1979, L. Salce introduced the notion of cotorsion pair for abelian groups. The same notion makes sense for module categories or, more generally, for abelian categories. This refers to two classes of objects which are the annihilators of each other with respect to \(\mathrm{Ext}^{1}(-,-)\). In module categories, there always are two trivial examples of cotorsion pairs: \(({\mathbf P}, \mathrm{Mod})\) and \((\mathrm{Mod}, {\mathbf I})\), where \({\mathbf P}\) is the class of projectives, \({\mathbf I}\) is the class of injectives, and Mod denotes the category of all modules. Recall that a module category has enough projectives and enough injectives, i.e., for any module \(M\) there are exact sequences \(0 \to Y_{M} \to P \to M \to 0\), where \(P \in {\mathbf P}\) and (trivially) \(Y_{M} \in \mathrm{Mod}\), and \(0 \to M \to I \to Y^{M} \to 0\), where \(I \in {\mathbf I}\) and (trivially) \(Y^{M} \in \mathrm{Mod}\).

For a nontrivial example, look at finitely generated modules over a commutative Cohen-Macaulay local ring \(R\). Then the classes of maximal Cohen-Macaulay (mCM) modules and modules of finite injective dimension form a cotorsion pair: a module \(M\) is mCM if and only if \(\text{Ext}^{1}(M,I) = 0\) for any module \(I\) of finite injective dimension, and a module \(I\) is of finite injective dimension if and only if \(\text{Ext}^{1}(M,I) = 0\) for any mCM module \(M\). Symbolically, \((mCM, FI)\) is a cotorsion pair. The theory of mCM approximations of Auslander-Buchweitz yields, for any finitely generated \(R\)-module \(M\), an mCM approximation \(0 \to Y_{M} \to X_{M} \to M \to 0\) of \(M\) and a hull of finite injective dimension \(0 \to M \to X^{M} \to Y^{M} \to 0\) of \(M\), where \(X_{M}, X^{M} \in mCM\) and \(Y_{M}, Y^{M} \in FI\). In technical terms, one says that the cotorsion pair \((mCM, FI)\) has enough projectives and enough injectives. In general, when a cotorsion pair has enough projectives and injectives one says that the cotorsion pair is complete.

The question of whether or not a given cotorsion pair is complete could be highly nontrivial. Let \(R\) be a ring, \(\mathbf Fl\) the class of flat (left) \(R\)-modules, and \(\mathbf C\) the right perpendicular of \(\mathbf Fl\), i.e., the class of all \(R\)-modules \(C\) such that the restriction of \(\mathrm{Ext}^{1}(-,C)\) to \(\mathbf Fl\) is zero. The resulting cotorsion pair \((\mathbf Fl, C)\) is complete, but that was established only with the proof of the so-called flat cover conjecture.

A major development in the field occurred in 2002 when Mark Hovey connected cotorsion pairs with abelian model structures on abelian categories. To this end, one looks at a relative homology, i.e., a sub-bifunctor of \(\mathrm{Ext}^{1}(-,-)\), which gives rise to a class \(\mathcal P\) of proper exact sequences. A model structure is said to be compatible with that class if: a) cofibrations are relative monos and b) (trivial) fibrations are precisely the relative epis with a (trivially) fibrant kernels. One the other hand, one looks at cotorsion pairs of the form \((\mathcal A, \mathcal B \cap \mathcal W)\) and \((\mathcal A \cap \mathcal W, \mathcal B)\) for some class of objects \(\mathcal W\). Such pairs are said to be compatible. In rough terms, Hovey’s result can now be stated as follows: there is a 1:1 correspondence between \(\mathcal P\)-compatible model structures and compatible pairs of cotorsion pairs.

In the paper under review, the author uses the above approach to construct new abelian model structures on the category of complexes over a ring. He extends a number of previously established results, where model structures on complexes require the use of projectives, by allowing the requisite modules to be of bounded projective dimension. The finiteness of projective dimension can be imposed either degree-wise, or on a complex as a whole (by viewing the complex as a module over a ringoid). Similar results are established when projective dimension is replaced with flat dimension.

For a nontrivial example, look at finitely generated modules over a commutative Cohen-Macaulay local ring \(R\). Then the classes of maximal Cohen-Macaulay (mCM) modules and modules of finite injective dimension form a cotorsion pair: a module \(M\) is mCM if and only if \(\text{Ext}^{1}(M,I) = 0\) for any module \(I\) of finite injective dimension, and a module \(I\) is of finite injective dimension if and only if \(\text{Ext}^{1}(M,I) = 0\) for any mCM module \(M\). Symbolically, \((mCM, FI)\) is a cotorsion pair. The theory of mCM approximations of Auslander-Buchweitz yields, for any finitely generated \(R\)-module \(M\), an mCM approximation \(0 \to Y_{M} \to X_{M} \to M \to 0\) of \(M\) and a hull of finite injective dimension \(0 \to M \to X^{M} \to Y^{M} \to 0\) of \(M\), where \(X_{M}, X^{M} \in mCM\) and \(Y_{M}, Y^{M} \in FI\). In technical terms, one says that the cotorsion pair \((mCM, FI)\) has enough projectives and enough injectives. In general, when a cotorsion pair has enough projectives and injectives one says that the cotorsion pair is complete.

The question of whether or not a given cotorsion pair is complete could be highly nontrivial. Let \(R\) be a ring, \(\mathbf Fl\) the class of flat (left) \(R\)-modules, and \(\mathbf C\) the right perpendicular of \(\mathbf Fl\), i.e., the class of all \(R\)-modules \(C\) such that the restriction of \(\mathrm{Ext}^{1}(-,C)\) to \(\mathbf Fl\) is zero. The resulting cotorsion pair \((\mathbf Fl, C)\) is complete, but that was established only with the proof of the so-called flat cover conjecture.

A major development in the field occurred in 2002 when Mark Hovey connected cotorsion pairs with abelian model structures on abelian categories. To this end, one looks at a relative homology, i.e., a sub-bifunctor of \(\mathrm{Ext}^{1}(-,-)\), which gives rise to a class \(\mathcal P\) of proper exact sequences. A model structure is said to be compatible with that class if: a) cofibrations are relative monos and b) (trivial) fibrations are precisely the relative epis with a (trivially) fibrant kernels. One the other hand, one looks at cotorsion pairs of the form \((\mathcal A, \mathcal B \cap \mathcal W)\) and \((\mathcal A \cap \mathcal W, \mathcal B)\) for some class of objects \(\mathcal W\). Such pairs are said to be compatible. In rough terms, Hovey’s result can now be stated as follows: there is a 1:1 correspondence between \(\mathcal P\)-compatible model structures and compatible pairs of cotorsion pairs.

In the paper under review, the author uses the above approach to construct new abelian model structures on the category of complexes over a ring. He extends a number of previously established results, where model structures on complexes require the use of projectives, by allowing the requisite modules to be of bounded projective dimension. The finiteness of projective dimension can be imposed either degree-wise, or on a complex as a whole (by viewing the complex as a module over a ringoid). Similar results are established when projective dimension is replaced with flat dimension.

Reviewer: Alex Martsinkovsky (Boston)

### MSC:

18G20 | Homological dimension (category-theoretic aspects) |

18G35 | Chain complexes (category-theoretic aspects), dg categories |

### Keywords:

homological dimensions; chain complexes; Abelian model structures; cotorsion pairs; Hovey’s correspondence### Citations:

Zbl 1268.18011
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\textit{M. A. Pérez}, Rocky Mt. J. Math. 46, No. 3, 951--1010 (2016; Zbl 1375.18071)

### References:

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