DERIVED LIMITS IN QUASI-ABELIAN CATEGORIES
Laboratoire Analyse, Géométrie et Applications, UMR 7539, Université Paris 13, Avenue J.-B Clément, 93430 Villetaneuse, prosmans@math.univ-paris13.fr
Abstract
In this paper, we study the derived functors of projective limit functors in quasi-abelian categories. First, we show that if is a quasi-abelian category with exact products, projective limit functors are right derivable and their derived functors are computable using a generalization of a construction of Roos. Next, we study index restriction and extension functors and link them trough the symbolic Hom-functor. If is a functor between small categories and if E is a projective system indexed by , this allows us to give a condition for the derived projective limits of E and E J to be isomorphic. Note that this condition holds, if and are filtering and J is cofinal. Using the preceding results, we establish that the n-th left cohomological functor of the derived projective limit of a projective system indexed by vanishes for n k, if the cofinality of is strictly lower than the k-th infinite cardinal number. Finally, we consider the limits of pro-objects of a quasi-abelian category. From our study, it follows, in particular, that the derived projective limit of a filtering projective system depends only on the associated pro-object.
11991 AMS Mathematics Subject Classification : 18G50, 18A30,46M20