# Cotorsion pairs and the Mittag-Leffler condition

Dolors Herbera1

Departament de Matemàtiques,
Universitat Autònoma de Barcelona,
08193 Bellaterra (Barcelona), Spain
dolors@mat.uab.es

Let be an abelian category with arbitrary direct sums. Let be a sequence of compact objects in , and let be a sequence of morphisms. Then we have the exact sequence

with for any and denoting the canonical morphism.

Let be a subcategory of closed under direct sums. We show that the inverse system

is Mittag-Leffler for any object if and only if the morphism is onto for any object .

This result has some interesting consequences when applied to cotorsion pairs in module categories. For example, it is the key tool in showing that -dimensional tilting modules are of finite type. That is, if is a tilting module over a ring then there exists a set , consisting of finitely presented right -modules of projective dimension at most one, such that

#### Footnotes

... Herbera1
joint work with Silvana Bazzoni

2005-05-23