# Famous exercise from Lang's Algebra

There's a famous story about an exercise from Lang's Algebra that says something along the lines of "pick up a homological algebra book and prove all of the theorems yourself". I cannot find it in the third revised edition, and I'm wondering if it's still in the third revised edition, if it's only in the older editions, or if it's an urban legend.

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I'm sure your prof went back to his office laughing thinking,"That'll keep them busy while I do my real work-research.Imagine the nerve of them,asking me to TEACH......." Got news for you:SPANIER couldn't prove everything in Spanier. –  Andrew L Apr 12 '10 at 20:37
@Andrew L: Please refrain from making disrespectful comments like that, especially on my posts. –  Harry Gindi Apr 12 '10 at 20:56
@Harry Gindi I can't believe this.Harry,what about this was disrespectful?!? I was giving my opinion! I don't understand people in here sometimes,seriously! –  Andrew L Apr 13 '10 at 2:33
First of all, your comment is disrespectful to Steve, and second of all, it is disrespectful to Spanier as well. Also, it was clearly meant as a joke. You seemto have some kind of axe to grind with the mathematical community. I don't care if you do or not, but I'd appreciate you doing it somewhere else. MO is not your web-log, and my posts definitely aren't. –  Harry Gindi Apr 13 '10 at 12:50
@Harry:First of all,I can't see how my comment was disrespectful to Steve-it was a general comment on a situation he was describing from his student days. Second-the Spanier comment was a critism of his textbook,which is compressed to within a ludicrously small epsilon of readability.My comment was intended as a sardonic expression of that opinion. Thirdly-I canno understand the ridiculous level of hyper-sensitivity in here,even towards innocuous commentary.Do I REALLY have to keep defending myself in here when no harm is clearly intended?!? –  Andrew L Apr 13 '10 at 17:54

It's real, but only in the first and second editions. (I don't have any electronic proof, but I've seen it in my copy of the second edition and someone else's copy of the first edition.) It's the only exercise in the chapter.

The full quote in the second edition is:

Take any book on homological algebra, and prove all the theorems without looking at the proofs given in that book.

Homological algebra was invented by Eilenberg-MacLane. General category theory (i. e. the theory of arrow-theoretic results) is generally known as abstract nonsense (the terminology is due to Steenrod).

If I'm not mistaken, the quote is the same in the first edition. First edition: page 105; Second edition: page 175; so you can look in your library to see if I messed up the quote!

And I do have an electronic copy of the third edition, which I've searched to confirm it is not there. The historical remarks were expanded, written less dismissively and put at the intro to Part Four.

Update: I've scanned the evidence. First, Second.

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If you have the means, the time, and the will, is there any chance you could scan that page? –  Harry Gindi Jan 6 '10 at 10:07
Done!  –  Jonas Meyer Jan 6 '10 at 16:07
Thank you for your help, sir! –  Harry Gindi Jan 6 '10 at 17:42
Joseph Rotman was the first to tell me about this exercise in an email correspondence.I told him this is why everyone hates Lang except the do-it-yourself crew. I seriously doubt anyone but the most brillant of students ever succeeded in proving everything fom scratch in any homological algebra text worth looking at. –  Andrew L Apr 12 '10 at 20:40
@Minhyong: you and Joe Rotman (in his homological algebra book) are rather generous to Lang. I am very glad Lang had a change of mind and actually wrote something responsible on homological algebra in the third edition; the original exercise strikes me as an obnoxious joke, a kind of then-fashionable sneer at categorical subjects. –  Todd Trimble Dec 13 '10 at 16:50
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In the Russian 1968 translation of Lang's Algebra (which is done from the Addison-Wesley 1965 edition), this exercise is there on page 126, exactly as quoted in Jonas' answer. It is complemented by two footnotes by the editor of the tranlation.

The footnotes say: "1) We suggest to skip these exercises on the first reading." and "2) It should be pointed out that the term `abstract nonsense' in this book has a positive character and is used below in the serious sense."

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That is interesting! –  Jonas Meyer Jan 6 '10 at 16:06
I've also seen the footnotes in the Polish translation I used. –  Michał Kukieła Jan 30 '10 at 16:37

It is not an urban legend at all. It's certainly in the second edition, which is the one I studied from when I was a grad student. However it seems to be missing from the Springer 2002 edition. Certainly a loss, in my opinion.

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Probably because in almost 30 years since the 1st edition,no student succeeded in doing it,Jose.I'm sure students were able to get big chunks of the task done,but I doubt the whole thing. –  Andrew L Apr 12 '10 at 20:42