A counterexample to a 1961 “theorem” in homological algebra

In 1961, Jan-Erik Roos published a “theorem”, which says that in an AB4^*] abelian category, lim¹ vanishes on Mittag–Leffler sequences. See Propositions 1 and 5 in 4]. This is a “theorem” that many people since have known and used. In this article, we outline a counterexample. We construct some strange abelian categories, which are perhaps of some independent interest.?These abelian categories come up naturally in the study of triangulated categories. A much fuller discussion may be found in 3]. Here we provide a brief, self contained, non–technical account. The idea is to make the counterexample easy to read for all the people who have used the result in their work.?In the appendix, Deligne gives another way to look at the counterexample. Oblatum 22-I-2001 & 15-XII-2001?Published online: 1 February 2002