The book under review is a reprint of Mumford's famous Harvard lecture notes, widely used by the past few generations of algebraic geometers. Springer-Verlag has done the mathematical community a service by making these notes available once again. When the book first appeared, Grothendieck's ideas were not widely known or understood and the Red Book was the first approach to the precipitous multivolume EGA. The book covers the basic theory of schemes and varieties. Considerable emphasis is placed on the study of morphisms using down-to-earth calculations in commutative algebra. A number of topics are covered which are omitted in the now standard text of R. Hartshorne[Algebraic geometry, Springer, New York, 1977; MR0463157]. Included among these are the functor of points and specialization; the latter treatment includes some excellent illustrations. The last chapter includes treatments of coherent modules, étale morphisms, normalization, and, as dénouement, Zariski's main theorem. The informal style and frequency of examples make the book an excellent text. The introduction includes the exciting announcement that a companion volume on cohomology is to appear shortly, written in collaboration with D. Eisenbud and J. Harris.
Reviewer: Nielsen, Peter [form MathSciNet]