group theory

# Contents

## Idea

An inverse sequence (sequential diagram) $G_\bullet$ of groups is said to satisfy the Mittag-Leffler condition if the images of groups from far down the sequence do not get smaller.

This is a condition used to assure the vanishing of the first derived functor of the limit-functor, $\underset{\longleftarrow}{lim}^1 G_\bullet$. See at lim^1 and Milnor sequences.

This is relevant for the preservation of exactness when applying limiting processes to exact sequences. _

## Definition

An inverse sequence of groups consists of some groups $G_n$ indexed by the natural numbers and between them group homomorphisms: if $m \gt n$, there is a homomorphism $p^m_n : G_m \to G_n$ and if $l\gt m \gt n$, $p^m_n p^l_m= p^l_n$, so that we really just need the $p^{n+1}_n$s to define everything.

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

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

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

## 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?.

Revised on February 15, 2016 13:55:54 by Urs Schreiber (31.55.9.40)