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# Homology, Homotopy and Applications

## Volume 10 (2008)

### Number 1

### Cotorsion pairs and degreewise homological model structures

Pages: 283 – 304

DOI: http://dx.doi.org/10.4310/HHA.2008.v10.n1.a12

#### Author

#### Abstract

Let $C$ be an abelian category. We show that under certain hypotheses, a cotorsion pair $(A, B)$ in $C$ may induce two natural homological model structures on $Ch(C)$. One is such that the (trivially) cofibrant objects form the class of (exact) complexes $A$ for which each $A_n ∈ A$. The other is such that the (trivially) fibrant objects form the class of (exact) complexes $B$ for which each $B_n ∈ B$. Special cases of these model structures such as Hovey’s “locally free” model structure and “flasque” model structure have already appeared in the literature. The examples support the belief that any useful homological model structure comes from a single cotorsion pair on the ground category $C$. Furthermore, one of the two types of model structures we consider requires surprisingly few assumptions to exist. For example, Theorem 4.7 implies that every cotorsion pair $(A, B)$ of $R$-modules which is cogenerated by a set gives rise to a cofibrantly generated homological model structure on $Ch(R)$.

#### Keywords

cotorsion pair, chain complexes, model structure

#### 2010 Mathematics Subject Classification

18G55, 55U35