nLab
cotorsion pair

Idea

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

Definition

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

Definition

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

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

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

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

Example

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

Definition

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

(1)0XED0. 0 \to X \to E \to D \to 0.

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

(2)0EDX0. 0 \to E \to D \to X \to 0.

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

Definition

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

References

Created on June 7, 2013 at 18:16:08. See the history of this page for a list of all contributions to it.