Abstract :
We prove, correct and extend several results of our note [20] (using and recalling our later papers
[21–24]) about the derived functors of projective limit in abelian categories. In Section 3 we prove
(with the help of Ofer Gabber) that if C is an abelian category satisfying the Grothendieck axioms
AB3 and AB4* and having a set of generators then the first derived functor of projective limit
vanishes on so-called Mittag-Leffler sequences in C. The examples given by Deligne [2] and Neeman
[17] show that the condition that the category has a set of generators is necessary. The condition
AB4* is also necessary, and indeed we give for each m 1 an example of a Grothendieck category
Cm and a Mittag-Leffler sequence in Cm for which the derived functors of its projective limit vanish
in all positive degrees except m. The derived functors of infinite products are non-vanishing in
Cm and in Sections 1 and 2 we give a systematic study of these functors in general Grothendieck
categories. As byproducts we get, for example, the examples just mentioned and also the fact
that the study of the vanishing of derived functors of infinite products is related to the study
of calculable sheaves in the sense of Grothendieck [7]. In Section 4 we also recall and use that
we proved in 1965 that the categories studied by Gabber–Ramero in [4] (as generalizations of
examples studied by Faltings [3]) are exactly those abelian categories satisfying the axioms AB4*
and AB6 and having a system of generators. These categories have ‘effacements projectifs’ but
not necessarily sufficiently many projectives [23].
