nLab cotorsion pair

Idea

Cotorsion pairs in abelian categories are a notion generalizing both projective and injective objects.

Definition

Let $\mathcal{A}$ be an abelian category.

Definition

A cotorsion pair in $\mathcal{A}$ is a pair $(\mathcal{D},\mathcal{E})$ of classes of objects of $\mathcal{A}$ such that

• $D \in \mathcal{D}$ if and only if $Ext^1(D, E) = 0$ for all $E \in \mathcal{E}$
• $E \in \mathcal{E}$ if and only if $Ext^1(D, E) = 0$ for all $D \in \mathcal{D}$.

Example

Let $\mathcal{I}$ denote the class of injective objects of $\mathcal{A}$. Then $(\mathcal{A}, \mathcal{I})$ is a cotorsion pair, where we write $\mathcal{A}$ for the class of all objects of $\mathcal{A}$.

Dually, there is a cotorsion pair $(\mathcal{P}, \mathcal{A})$ where $\mathcal{P}$ is the class of projective objects of $\mathcal{A}$.

Example

The flat cotorsion pair on the category of $R$-modules (for some ring $R$) is defined by letting $\mathcal{D}$ be the class of flat $R$-modules and $\mathcal{E}$ the class of cotorsion modules?, i.e. $R$-modules $E$ such that $Ext^1_R(D,E)=0$ for all flat $D$.

Definition

We say that a cotorsion pair $(\mathcal{D},\mathcal{E})$ has enough injectives if for every object $X$ of $\mathcal{A}$, there exist objects $D \in \mathcal{D}$ and $E \in \mathcal{E}$ such that there is an exact sequence

$$0 \to X \to E \to D \to 0.$$

We say that it has enough projectives if for every object $X$ of $\mathcal{A}$, there exist objects $D \in \mathcal{D}$ and $E \in \mathcal{E}$ such that there is an exact sequence

$$0 \to E \to D \to X \to 0.$$

Note that the category $\mathcal{A}$ has enough injectives if and only if the cotorsion pair $(\mathcal{A}, \mathcal{I})$ has enough injectives, and $\mathcal{A}$ has enough projectives if and only if $(\mathcal{P}, \mathcal{A})$ has enough projectives.

Definition

A cotorsion pair $(\mathcal{D},\mathcal{E})$ is called complete if it has enough injectives and enough projectives.

References

• Mark Hovey, Cotorsion pairs and model categories, Contemp. Math. 436 (2007) 277-296 pdf gBooks doi

On the generalization of $n$-cotorsion pairs:

• Huimin Chang, Panyue Zhou. Mutation of n-cotorsion pairs in triangulated categories (2024). (arXiv:2404.18336).