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I would like to hear the communities' ideas on good Homological Algebra textbooks / references. The standard example is of course Weibel (which I'll leave for someone else to describe).

As usual, the rule is one reference per post. Please include some description which distinguishes it from other texts.

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Charles Weibel's "An Introduction To Homological Algebra" is the gold standard. Very modern, very clear and written by a master. But it may be a bit rough going for beginners. Much more user friendly and still very thorough is the second edition of Joseph Rotman's book of the same name. Like everything by Rotman, it's a wonderful and enlightening read.

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    $\begingroup$ +1 for Rotman. I find that Weibel is the book I turn to for looking things up, but it doesn't hesitate to take a steam-hammer to the walnut $\endgroup$
    – Yemon Choi
    Commented Mar 13, 2010 at 7:46
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    $\begingroup$ +1 to Weibel. I found it the most enlightening source when I started out learning homological algebra myself, and it remains the book that demystified diagram chases for me. $\endgroup$
    – Mikael Vejdemo-Johansson
    Commented Mar 13, 2010 at 14:51
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    $\begingroup$ It is inspiring to complement the Weibel's book which is very reliable and modern, but sometimes a bit dense and dry/terse (theorem, lemma, exercise, proof), with beautiful Weibel's essay on the history of homological algebra, available on his webpage. $\endgroup$
    – Zoran Skoda
    Commented Jun 28, 2011 at 12:09
  • $\begingroup$ Weibel is no doubt a good book, but I think Chapter 1 is too sketchy for beginners (recall that the book is titled "Introduction".) E.g., the proof for the snake diagramme is delegated to actress Jill Clayburgh (the beginning episode of the movie "It's my turn".) I agree this is funny as a joke, but I would have appreciated the authour's effort in listing a detailed account of the lemma (as in "Algebra" by Robert Ash.) Another example: the proposition crucial in proving the existence of chain equivalence is listed as "Exercise 1.5.2." I would like to see an enhancement in the second edition. $\endgroup$
    – eltonjohn
    Commented Jan 24, 2015 at 13:24
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There are two books by Gelfand and Manin, Homological algebra, around 200 pages and Methods of homological algebra, around 350 pages. The first one covers the standard basic topics, and also has chapters on mixed Hodge structures, perverse sheaves, and D-modules. The second one has a different emphasis, with chapters on simplicial sets and homotopical algebra instead of the above-mentioned topics.

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    $\begingroup$ Warning to the reader: Methods of homological algebra is well known for it's many typographical errors. $\endgroup$
    – Anton Geraschenko
    Commented Oct 26, 2009 at 0:54
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    $\begingroup$ Although there are many typos, I find "Methods" excellent for conveying the big picture. $\endgroup$
    – David E Speyer
    Commented Oct 26, 2009 at 2:44
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    $\begingroup$ @Anton: I heard that they recently published an updated edition with many of the typos fixed. $\endgroup$
    – Harry Gindi
    Commented Mar 13, 2010 at 8:00
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    $\begingroup$ As long as you know that there are typos in it, the typos can ultimately be a good things. They keep you on your toes. Most of them a typographical and easily corrected while you read. It is really worth looking at because "Methods" is a great book. There are some real golden nuggets of mathematics hidden in its pages. $\endgroup$
    – Chris Schommer-Pries
    Commented Mar 13, 2010 at 14:00
  • $\begingroup$ "Methods" IS a great book,but I was a little disappointed it delibrately divorced the subject from it's topological roots for most of the book.I like Rotman and particularly Weibel precisely because they DON'T do this-the connections with topology are strongly emphasized. $\endgroup$
    – The Mathemagician
    Commented Apr 12, 2010 at 20:46
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It seems difficult to find good introductions that are freely available online, but a nice set of lecture notes can be be found on Schapira's web page, here.

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    $\begingroup$ Another nice set of lecture notes is the one by Moerdijk, available at staff.science.uu.nl/~lukac101/homalg2007.pdf. $\endgroup$
    – skupers
    Commented May 7, 2010 at 10:14
  • $\begingroup$ Wow,didn't know about the Moerdijk notes-they are quite nice indeed,skupers. $\endgroup$
    – The Mathemagician
    Commented May 7, 2010 at 16:40
  • $\begingroup$ @skupers Your link is broken. $\endgroup$
    – LeBlanc
    Commented Sep 23, 2012 at 3:34
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    $\begingroup$ Here's a fresh link to Moerdijk's notes $\endgroup$
    – Amartya
    Commented Jun 13, 2022 at 3:29
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I agree the best reference is Weibel, and GM's Methods is really good, but for starting out I'd recommend Mac Lane's Homology (which is just about homological algebra). This is much more readable for someone coming from an undergraduate degree.

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Appendix 3 of Eisenbud's "Commutative Algebra" is the best short treatment I know. I find it fantastic. It clearly and concisely covers a surprising number of topics in homological algebra.

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  • $\begingroup$ I was about to suggest the same. Quite surprising for a simple appendix :) $\endgroup$
    – Antoine
    Commented Mar 14, 2018 at 22:17
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There's a basic book by Northcott; it does everything only for the category of modules over a ring and does not go far, but it has essentially no prerequisites.

It was written soon after Cartan and Eilenberg, which probably explains the old-fashioned style.

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    $\begingroup$ I guess, thanks to Freyd-Mitchell (en.wikipedia.org/wiki/Mitchell%27s_embedding_theorem), this should be essentially sufficient. It's always a shame (to me) when people take this ``applied'' approach, though. $\endgroup$
    – alekzander
    Commented Oct 26, 2009 at 18:04
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    $\begingroup$ I must spread the word that character limits are of no consequence any longer. $\endgroup$
    – Steven Gubkin
    Commented May 7, 2010 at 13:14
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    $\begingroup$ Behold! ${ }$ $\endgroup$
    – Steven Gubkin
    Commented May 7, 2010 at 13:14
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    $\begingroup$ (Behold has only 7 characters) $\endgroup$
    – Steven Gubkin
    Commented May 7, 2010 at 13:15
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    $\begingroup$ Older books are not without value, including Cartan-Eilenberg, but it's hard to recommend them currently when books by Weibel, Rotman, and Gelfand-Manin are available. Probably the 1971 Springer text A Course in Homological Algebra by Hilton-Stammbach is a better choice among the early books than Northcott. But for later books the choice depends a lot on your preferred style and whether you want to study derived categories, Freyd-Mitchell, etc. Also whether your motivation for the subject comes from topology, algebra, representation theory, ... $\endgroup$
    – Jim Humphreys
    Commented May 7, 2010 at 17:10
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I liked Rotmans book a lot.

http://www.amazon.com/Introduction-Homological-Algebra-Universitext/dp/0387245278

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Basic Homological Algebra by Scott Osbourne is a nice beginners text. It is very thorough and detailed yet well motivated and conversational with a particularly engaging style.

Although old fashioned and outdated in many respects; I would have to say that Cartan-Eilenberg is still of great value as a reference.

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Part IV of Lang's 'Algebra', especially Chapter XX, covers almost everything you want to learn about homological algebra in a first course.

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A less well-known book is Vermani: An elementary approach to homological algebra. This was the first book I ever read on homological algebra, and I loved it. I would recommend it to anyone who has not seen much of the subject, as a starting point before going on to more advanced texts. Here is a Google Books preview.

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There is also an interesting lectures on homological algebra of I.Moerdijk, which his notes are on the following link http://www.staff.science.uu.nl/~lukac101/homalg2007.pdf

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I have used Weibel in the past as my reference in a graduate course, but I think the less confident students can have trouble getting into it. I've always enjoyed the way it is organized, somehow. The books by Rotman and Scott Osborne (Basic Homological Algebra) seem friendlier for students, but I like to have spectral sequences early on, not just in the last chapter. (Who likes to balance Tor by hand?)

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