Atiyah-MacDonald, as everyone in the world calls this classic book, is one of the premier texts for a serious graduate (or very gifted undergraduate) student aspiring to learn both commutative algebra for his PhD qualifying examination and much of the machinery required for algebraic geometry. These two statements require some elaboration: first, commutative algebra, as a mainstay of the algebra part of a solid PhD program, is treated wonderfully well in a number of texts, and many instructors would opt for other sources. For one thing, the chapters in Atiyah-MacDonald are cut to the bone: although the discussions overflow with elegance and abound with precision, they are not easy to use as scripts for accessible lectures. The lecturer would have to do a lot of work in order properly to motivate his presentations, at least for an average class. However, if the onus is placed on the student, i.e. if the latter is required to do combat with the text (arguably the only way really to master something in mathematics), the book is ideal, if you’ll pardon the cheap pun.
And this brings me to my second point: Atiyah-MacDonald is indeed unsurpassed as an exemplar of a text which has its pedagogical gravity located in the sets of exercises. By comparison, in classical analysis I guess the obvious counterpart is Pólya-Szegö’s Problems and Theorems in Analysis (which is of course free of text as such). Any student who works his way through the marvelously designed and orchestrated sets of problems in Atiyah-MacDonald, diligently and scrupulously, will come away from the experience knowing commutative algebra intimately: his hands will be appropriately filthy with mathematics, and he’ll know how things really work when it comes to such things as ideals and varieties, polynomial rings, chain conditions, modules, tensor products, flatness, and so on. In other words, the stage is set for the next phase, as far as, particularly, algebraic geometry is concerned: if I may, I’ll restrict myself to this angle on the subject, seeing that it is in a way the most obvious.
The salient point is that the behavior of algebraic varieties, with the business of prime, primary, maximal and radical ideals, indeed, the back-and-forth between ideal theory and the inner life of zero sets of polynomials (with the Zariski topology knocking at the door), is the heart of the subject immediately out of the gate, and a solid grounding in this aspect of commutative algebra is indispensable. There is perhaps a “standard path” to get into algebraic geometry. I don’t know, really, but I’m willing to go out on the corresponding limb: it should be Atiyah-MacDonald first, followed by Hartshorne, supplemented by Mumford’s Red Book of Varieties and Schemes, and then a selection from Grothendieck’s Séminaire de Géométrie Algébrique. Just to hedge my bets, here is a bit of a qualification, aimed primarily at the budding autodidacts in the audience: other great sources in this game are (the same) MacDonald’s Algebraic Geometry: Introduction to Schemes, covering some of the same stuff as Atiyah-MacDonald, but with a far narrower focus and with less austerity, and Shafarevich’s Basic Algebraic Geometry, which is marvelously readable. Furthermore, if you want to get a deeper look at sheaves, form an older and more “from the ground up” perspective than Harthorne’s, there’s always Serre’s elegant Faisceaux Algébriques Cohérents. In any case it’s all a wonderful set of fantastic and highly instructive books, comprising a solid education in algebraic geometry, with Atiyah-MacDonald leading the parade.
There’s no doubt about it, all this is hard work, but that is unavoidable, isn’t it? And getting off the ground with Atiyah-MacDonald is a fabulous way to start.
Michael Berg is Professor of Mathematics at Loyola Marymount University in Los Angeles, CA.