nLab
Mittag-Leffler condition

Contents

Idea

An inverse sequence of groups satisfies the Mittag-Leffler condition if the images of groups from far down the sequence do not get smaller.

An inverse sequence of groups consists of some groups G nG_n indexed by the natural numbers and between them group homomorphisms: if m>nm \gt n, there is a homomorphism p n m:G mG np^m_n : G_m \to G_n and if l>m>nl\gt m \gt n, p n mp m l=p n lp^m_n p^l_m= p^l_n, so that we really just need the p n n+1p^{n+1}_ns to define everything.

Definition

An inverse system G={G n,p n m}G = \{G_n,p^m_n\} is said to satisfy the Mittag-Leffler condition if

for any nn, there is an n >nn^\prime \gt n such that for any n >n n^{\prime\prime} \gt n^\prime,

p n n (G n )=p n n (G n ).p^{n^{\prime\prime}}_n(G_{n^{\prime\prime}}) = p^{n^\prime}_n(G_{n^\prime}).

Discussion

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, lim 1Glim^1G. 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 (89.204.135.147)