This book represents Amnon Neeman’s answer to a nagging pedagogical question: how do we introduce students to algebraic geometry? His answer is brilliant, though perhaps limited in its application.
Algebraic geometry as commonly practiced today sometimes feels like at best a distant descendant of the classical study of the geometry of sets defined by polynomial equations. It is still about those same sets, but the modern scheme-theoretic approach is far more general, more powerful, and more abstract. It also has quite a long list of pre-requisites. Should we hit a beginner with all this machinery on day one? Or should we first introduce our students to the classical theory?
Some of the best introductory texts have preferred to start classical. This is true, for example, of Hartshorne’s Algebraic Geometry, Harris’s Algebraic Geometry: A First Course, Mumford’s Algebraic Geometry I, and Shafarevich’s Basic Algebraic Geometry. All of these, each in his own way, look first at varieties in projective space (often only over the complex numbers) before going off to the land of abstraction. Given the prominence of these authors, this adds up to quite a consensus.
And yet, there’s something unsatisfying about this approach. The switch to scheme-theoretic language typically comes as a wrenching change of direction. Despite all the preparation, the reader still feels asked to take it on trust that there will be good reasons for setting up all the formal equipment of sheaves, ringed spaces, schemes, cohomology, and much more. If we’re going to say, “Believe me, this is going to be useful,” why not say it from the beginning?
Neeman opts for inviting his readers to jump in at the deep end of the pool. He takes the scheme-theoretic point of view from the start, and builds his book around a hugely important theorem that makes use of all the abstract machinery: Serre’s GAGA theorem relating algebraic geometry and analytic geometry.
I really enjoyed the result. Neeman’s exposition is clear and often insightful. He takes the trouble to explain things that others take for granted, which is always a good thing. For example, one often sees the statement that schemes were developed by analogy to the theory of complex manifolds, but the latter are rarely introduced by way of sheaf theory and ringed spaces. Intead, they are defined as manifolds for which the transition maps between charts are holomorphic. Neeman outlines a proof that both approaches yield the same objects, which greatly clarifies what is going on.
Admittedly, I already know at least a little about schemes, cohomology, and so on, so I’m not the intended audience. Neeman claims to have used it in a course for upper-level undergraduates (!) who wanted to learn something about algebraic geometry, but says he does not recommend it for those who plan to do research in the subject. He lists the pre-requisites as some commutative algebra, some topology, and a little homological algebra, but admits that for maximum profit readers should probably also know something about manifolds and complex analysis.
Offering this book to American undergraduates seems like a stretch, though anyone who can find undergraduates who can handle this should certainly try it. I think it’d be most useful as supplementary reading for graduate students who have taken or are taking a more classical course in algebraic geometry. If ever there was a subject that needs to be looked at from various points of view, it is this one, so Neeman’s different take could be very useful, in particular to motivate the introduction of the scheme-theoretic machinery.
I won’t be tossing out my collection of introductory algebraic geometry texts, and I’ll probably still use the classical framework if I need to introduce someone to the subject. Nevertheless, I’m glad to have Neeman’s book as well. It will stand proudly next to the others on my shelf.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.