Two books discussing more recent results are Weibel, An Introduction to Homological Algebra, 1994, and Gelfand- Manin, Methods of Homological Algebra, 2003.

In their Foreword, Gelfand and Manin divide the history of Homological Algebra into three periods: the rst period ended in the early 1960s, culminating in applications of Ho- logical Algebra to regular local rings.

The second period, greatly in uenced by the work of A.

Grothendieck and J. -P. Serre, continued through the 1980s; it involves abelian categories and sheaf cohomology.

The third period, - volving derived categories and triangulated categories, is still ongoing.

Both of these newer books discuss all three periods (see also Kashiwara-Schapira, Categories and Sheaves).

The original version of this book discussed the rst period only; this new edition remains at the same introductory level, but it now introduces thesecond period as well.

This change makes sense pe- gogically, for there has been a change in the mathematics population since 1979; today, virtually all mathematics graduate students have learned so- thing about functors and categories, and so I can now take the categorical viewpoint more seriously.

When I was a graduate student, Homological Algebra was an unpopular subject.

The general attitude was that it was a grotesque formalism, boring to learn, and not very useful once one had learned it.