Why the Mittag-Leffler condition on a short exact sequence of, say, abelian groups, that ensures that the first derived functor of the inverse limit vanishes, is so named?


The wording of your question suggests that you're familiar with the "classical" Mittag-Leffler theorem from complex analysis, which assures us that meromorphic functions can be constructed with prescribed poles (as long as the specified points don't accumulate in the region).

It turns out - or so I'm told, I must admit to never working through the details - that parts of the proof can be abstracted, and from this point of view a key ingredient (implicit or explicit in the proof, according to taste) is the vanishing of a certain $\lim_1$ group -- as assured by the "abstract" ML-theorem that you mention.

I'm not sure where this was first recorded - I hesitate to say "folklore" since that's just another way of saying "don't really know am and not a historian". One place this is discussed is in Runde's book A taste of topology: see Google Books for the relevant part.

IIRC, Runde says that the use of the "abstract" Mittag-Leffler theorem to prove the "classical" one, and to prove things like the Baire category theorem, can be found in Bourbaki. Perhaps someone better versed in the mathematical literature (or at least better versed in the works of Bourbaki) can confirm or refute this?

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  • $\begingroup$ Gopal Prasad said in class that the abstract M-L condition was discovered by Bourbaki along with a very slick proof. $\endgroup$ – Harry Gindi Feb 9 '10 at 0:53
  • $\begingroup$ Thanks Yemon. I followed your suggestion and took a look at Runde's (Appendix A) and the "abstract" Bourbaki's M-L version for a sequence of complete metric spaces and continuous funcions $f_n:X_n\rightarrow X_{n-1}$ with dense image. It looks that this can be further abstracted to the "algebraic" M-L. I'll do the details later on to see if this is the case. By the way, Runde quotes Bourbaki's "General Topology" volume. I don't have access to that one right now but I'll check it tomorrow. Thanks again. $\endgroup$ – F Zaldivar Feb 9 '10 at 1:26
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    $\begingroup$ This is in Bourbaki's General Topology, Chapter II, section 3.5. The main theorem is attributed to Mittag-Leffler, and is concerned with inverse systems of "complete Hausdorff uniform spaces". The Mittag Leffler condition mentioned there says the functions in the system have dense image. The usual theorem about inverse limits is a corollary, for sets with the 'discrete uniformity'. Classical Mittag-Leffler is given as an example of the main theorem. The spaces there are essentially holomorphic functions on balls centred at 0, continuous on the boundary, with the uniform metric. $\endgroup$ – Zavosh Feb 9 '10 at 2:15
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    $\begingroup$ The condition for groups that the question asks about is in Bourbaki's Algebra. It appears that the name for the condition on groups comes from the name of the condition on complete hausdorff uniform spaces, which indeed comes from the classical case. So it appears that Bourbaki named it the M-L condition because they wrote the book on topology first (yes, I know it is book 3, but it was published before book 2). $\endgroup$ – Harry Gindi Feb 9 '10 at 2:24

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