Inventiones mathematicae

, Volume 148, Issue 2, pp 397–420

A counterexample to a 1961 “theorem” in homological algebra

  • Amnon Neeman

DOI: 10.1007/s002220100197

Cite this article as:
Neeman, A. Invent. math. (2002) 148: 397. doi:10.1007/s002220100197


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.

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Amnon Neeman
    • 1
  1. 1.School of Mathematical Sciences, The Australian National University, Canberra, ACT 0200, Australia (e-mail:

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