Snakes, triangles, 3×3 and what not

And now for some algebra, for a change… One of my teachers (not the pedantic one, in fact a fairly well known topologist) once told us that Homological Algebra should be learnt outside of any class, either alone in one’s room, or with friends (this is in keeping with Lang’s sole exercise in the corresponding chapter of some editions of his Algebra, which asks the reader to just take any book on the topic and prove every statement without looking at the proofs).

If you’ve followed this type of advice, there is a fair chance that you’ve only ever proved the Snake Lemma and its friends for categories of modules over a ring. As it turns out, this is not so restrictive, since some abstract theorems show that any abelian category can be embedded in such a simple one, but it might be argued that this is not very elegant, especially if the theorems in question are taken for granted without proof (or reference).

All this to say that T. Bühler gave a lecture today at our Algebra and Topology seminar, where he explained some of his recent paper (to appear in Expositiones Math.) giving complete detailed proofs of all standard diagrams and diagram chasing lemmas, starting from scratch (or more precisely from axioms for exact categories, which seem quite a bit more general than abelian categories). As he remarked at the end, doing it this way is actually shorter, and it is much more satisfactory.

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I am a professor of mathematics at ETH Zürich since 2008.

4 thoughts on “Snakes, triangles, 3×3 and what not”

  1. According to the Wikipedia page on the 3×3 lemma,
    there is also a sixteen lemma and a twenty-five lemma (but this fact is mentioned jokingly). However, the only reference I could find to the sixteen lemma was a paper on the arXiv that used it; I couldn’t find any other references to the twenty-five lemma at all.

    Do you know if there actually are 4×4 and 5×5 lemmas? How about n x n for larger n? How about on an infinite grid (which would imply all of the others)?

  2. I’m not really the person to ask for non-obvious homological algebra questions… I’ve certainly never gone beyond the 3×3 lemma when I was learning it, and I don’t think I’ve ever used more than the snake lemma in any of my work. Theo Bühler certainly knows much more…

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