An inverse sequence of groups consists of some groups indexed by the natural numbers and between them group homomorphisms: if , there is a homomorphism and if , , so that we really just need the s to define everything.
An inverse system is said to satisfy the Mittag-Leffler condition if
for any , there is an such that for any ,
This is the ‘classical’ form of the condition. It can also be applied in any category where images make sense.
An inverse sequence is a special type of pro-object and it is well known and quite easy to show that any Mittag-Leffler pro-object like this is isomorphic to one which is essentially epimorphic?.
The condition is one of those used to assure the vanishing of the first derived limit? functor, . This is relevant for the preservation of exactness when applying limiting processes to exact sequences.
Revised on November 17, 2013 11:20:18
by Urs Schreiber