I love cohomology. And, as there is so much of it, there’s an awful lot of it to love — and to learn. It’s really remarkable that a subject that is only half a century old (modulo some foreshadowing such as Hilbert’s Satz 90 and, of course, various things from Poincaré) has caught on so dramatically: crystal clear testimony to the efficacy of the tool. (Long ago, in my mathematical childhood, I asked my de facto undergraduate advisor, V. S. Varadarajan, what cohomology was, and he replied, simply: “a tool,” and then smiled very enigmatically. Now I know why he was smiling.)
The authors of Twenty-four Hours of Cohomology , the book under review, start with the pithy phrase, “Local cohomology was invented by Grothendieck to prove Lefschetz-type theorems in algebraic geometry.” Fair enough, but more can be said. Or, rather, more can be said about cohomology, as such, as a prelude to local cohomology. To wit, we can chart the subject’s trajectory from Cartan-Eilenberg through Serre and Grothendieck to many of today’s hot themes like derived and triangulated categories, currently appearing in places as disparate as number theory and physics. (Varadarajan also told me in my student days that these two disciplines are opposite sides of the same coin: how right he was!)
The story begins in the 1950s with the Bourbaki-centered cadre of French topologists (Serre), number theorists (Weil), and algebraic geometers (Grothendieck), who set out to change the world — and did. Great theorems such as the Weil Conjectures were attacked and conquered, and wonderful new methods, indeed huge new movements, were introduced and nurtured into maturity by these men and their acolytes. It is no overstatement to say that sheaf cohomology must be placed at the very heart of this revolution.
The lectures comprising Twenty-four Hours of Cohomology reflect the evolution of the subject quite faithfully: from Cech cohomology (Serre’s approach to sheaf cohomology in the gorgeous paper Faisceaux Algébriques Cohérents) to resolutions and derived functors (one of the main foci of Homological Algebra by Cartan and Eilenberg, if not its raison d’être), while sheaf theory per se is only dealt with during the twelfth hour, as local cohomology proper needs to be discussed first (five hours before). De Rham cohomology isn’t brought up before the nineteenth hour, but this is proper, given what the authors are up to in the intervening lectures: Cohen-Macaulay rings (Lecture 10), Gorenstein rings (Lecture 11), D-modules (Lecture 17), and much more.
The septet of authors note that these lectures should be “accessible to students with a first course in commutative algebra or algebraic geometry, and in point-set topology,” but they also note later that “prior exposure to algebraic geometry and sheaf theory is helpful… [and t]he same is true for homological algebra.” Again, fair enough. But let me add that the book is very well written, indeed, and can be fruitfully used as a supplement to standard sources (I guess Weibel’s An Introduction to Homological Algebra is today’s run-away favorite), or even as a primary source. Regarding the latter option, the authors provide several possible scenarios. They note explicitly that the requisite background material occurs in Chapters 1, 2, 3, 6, 7, 8, 11, but after that it’s dealer’s choice: one can do commutative algebra, or algebraic geometry, or combinatorics (!), or even adopt a computational perspective (with Gröbner bases coming to play).
It’s all terrific stuff. I hope this book will succeed in bringing many young mathematicians to love cohomology, too, and then to go on from there.
Michael Berg is Professor of Mathematics at Loyola Marymount University in California.