Cotorsion pairs in abelian categories are a notion generalizing both projective and injective objects.
Let be an abelian category.
A cotorsion pair in is a pair of classes of objects of such that
Let denote the class of injective objects of . Then is a cotorsion pair, where we write for the class of all objects of .
Dually, there is a cotorsion pair where is the class of projective objects of .
The flat cotorsion pair on the category of -modules (for some ring ) is defined by letting be the class of flat -modules and the class of cotorsion modules?, i.e. -modules such that for all flat .
We say that a cotorsion pair has enough injectives if for every object of , there exist objects and such that there is an exact sequence
We say that it has enough projectives if for every object of , there exist objects and such that there is an exact sequence
Note that the category has enough injectives if and only if the cotorsion pair has enough injectives, and has enough projectives if and only if has enough projectives.
A cotorsion pair is called complete if it has enough injectives and enough projectives.
On the generalization of -cotorsion pairs:
Last revised on April 30, 2024 at 09:03:11. See the history of this page for a list of all contributions to it.