In a nutshell, reading this book is like reading the mind of a great mathematician as he thinks about a great new idea. Anyone interested in schemes should read it. But a review needs more detail: The RED BOOK is a concise, brilliant survey of schemes, by one of the first mathematicians to learn of them from Grothendieck. He gives wonderfully intuitive pictures of schemes, especially of "arithmetic schemes" where number theory appears as geometry. The geometry shines through it all: as in differentials, and etale maps, and how unique factorization relates to non-singularity. There is a bravura discussion of Zariski's Main Theorem (the algebraic property of being "normal" implies that a variety has only one branch at each point) comparing forms of it from older algebraic geometry, topology, power series, and schemes. Mumford cites proofs of these but does not give them. In fact, this theorem was one of the first things Mumford could use, to get Zariski to respect schemes. Many accomplished algebraic geometers say this book got them started. But you probably cannot learn to work in the subject from this book alone--you either have to work with people who work with it, or use some other books besides (maybe both). The other book would probably be Hartshorne ALGEBRAIC GEOMETRY, which is far more detailed, has far more examples, goes very much farther into cohomology--and is very much longer and denser (though also clearly written). Eisenbud and Harris GEOMETRY OF SCHEMES covers a lot of the same ground as THE RED BOOK, with fewer advanced topics but many more details and examples, including classical geometric constructions like blow-ups and duals to projective plane curves. They use slightly more category theory than Mumford, more like Grothendieck. Probably none of these books will work for you unless you already know some algebraic geometry: how polynomials define a variety, the Zariski topology, what regular and birational maps are. There is more than enough in Myles Reid's humorously titled UNDERGRADUATE ALGEBRAIC GEOMETRY and UNDERGRADUATE COMMUTATIVE ALGEBRA with vividly geometric ideas in slightly scheme-theoretic language. The RED BOOK now includes the Michigan lectures, which are reputedly terrific, but I have not worked through them. |