September 14th, 2017


Since its inception around 1960, there has been very little literature on the theory of derived categories. In some respect, the only detailed account for many years was the original book [RD], written by Hartshorne following notes by Grothendieck. Several accounts appeared later as parts of the books [We], [GeMa], [KaSc1], [KaSc2], [Huy], and maybe a few others – but none of these accounts provided enough detailed content to make it possible for a mathematician to learn how to work with derived categories, beyond a rather superficial level. The theory thus remained mysterious.

A personal belief of mine is that mathematics should not be mysterious. Some mathematics is very easy to explain. However, a few branches of mathematics are truly hard; among them are algebraic geometry and derived categories. My moral goal in this book is to demonstrate that the theory of derived categories is difficult, but not mysterious.

The [EGA] series by Grothendieck and Dieudonné, and then the book [Har] by Hartshorne, have shown us that algebraic geometry is difficult but not mysterious. The definitions and the statements are precise, and the proofs are available (to be read or to be taken on trust, as the reader prefers). I hope that the present book will do the same for derived categories. (Although I doubt I can match the excellent writing talent of the aforementioned authors!)

К предыдущему

As compared to Amnon's view expressed in his statement below, my view is that the main problem of the present-day homological algebra is that it has become too democratized, too accessible, looking from the superficial point of view of all too many of its students.

I agree that all too many algebraists (commutative algebraists, representation theorists etc.) are still afraid of homological algebra deep in their hearts, having not mastered some of its long well-known, elementary constructions and arguments. I certainly hope that Amnon's book, as well as other books, will help these people feel more at ease with the subject.

What worries me, though, is a different phenomenon most commonly observed in young people doing what they call "mathematical physics", or "homological field theory", or suchlike. It looks as if these people sincerely believed that all the homological algebra anyone needs to know can be presented in a single year-long graduate course, or at most in two such courses.

The mindset is that homological algebra is a rather small toolbox, a closed list of universally applicable constructions, definitions, theorems, and points of view perfectly capable of resolving all the homological algebra-related problems that may appear in one's research. No new tools or new concepts ever need to be developed, learned, or added to the box; rather, all one ever needs to do with one's homological algebra problems is to apply the given tools in the right order.

Many mathematicians may be totally unconcerned about whatever winds are blowing in the mathematical physics community. Perhaps it is just a peculiarity of my personal background, in which mathematical physicists -- meaning mathematicians by training who turned to quantum field theory-related mathematics later in their research lives -- played a prominent role, that forces me to pay attention.

Be it as it may, my chief concern in the later yeas has been not with how people learn to use derived categories in their research, but with how people unlearn the notion that all they need to know about the derived categories is contained in their preferred textbook. Thus, while Amnon's efforts may be directed towards making homological algebra more accessible to its students, my efforts tend to be directed towards making it less accessible to the ignoramuses.

The point is that the people who are not afraid of the mysterious, the people who are prepared to learn and study nontrivial, counterintuitive things, should have a competitive advantage over those who do not share these qualities. I hope that my efforts will result in there being more mysterious, nontrivial, and counterintuitive, still highly relevant material in homological algebra for such an advantage to be realized.