nLab
Mittag-Leffler condition

Contents

Idea

An inverse sequence (sequential diagram) G 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, lim 1G \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 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.

An inverse system G={G n,p n m}G = \{G_n,p^m_n\} is said to satisfy the Mittag-Leffler property (or 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.

Any Mittag-Leffler pro-object is known to be essentially epimorphic in the sense that it is isomorphic to a pro-object whose connecting morphisms are epis, that is to a strict pro-object.

Literature

Related nnLab entries include movable pro-object.

Mittag-Leffler property of pro-objects in the category of pointed sets and in the category of (nonabelian) groups is studied in Ch.II, Sec. 6.2 of

Revised on September 16, 2017 11:37:22 by Zoran Škoda (212.91.105.243)