Here is the MathSciNet review by Andrei D. Halanay:
The book under review is the first of a two-volume introduction to algebraic geometry. This first volume deals mostly with prerequisites, namely homological algebra and sheaf theory. The last chapter, occupying about one third of the book, is devoted to the classical theory of Riemann surfaces and abelian varieties. In this context the methods and results of the previous chapters are applied. The exposition is very pedagogical, every notion being first motivated by an example. For instance, derived functors are introduced after the reader is presented with the functors of invariants and co-invariants for a $G$-module ($G$ is an abelian group). These functors are not right exact (respectively left exact), thus prompting the need to consider the derived functors.
The book opens with a short chapter on category theory of a more abstract character. In this chapter the notions of product and of inductive and projective limit are considered in full generality. The second chapter is dedicated to homological algebra with an emphasis on cohomology and homology of groups. The ${\rm Ext}$ and ${\rm Tor}$ functors are also discussed. About half of the book is occupied by the third and fourth chapters, which are about sheaf theory and cohomology of sheaves. The fourth chapter goes much deeper than just ``ordinary'' sheaf cohomology and can be considered as a short course in algebraic topology. Thus spectral sequences and the cohomology of manifolds, including the theorems of de Rham and Dolbeault, are considered here. The chapter ends with a treatment of Hodge theory. The last chapter treats the subject of Riemann surfaces and abelian varieties. The exposition uses the results from previous chapters, but also classical complex analytic results. In this chapter the Riemann-Roch theorem, the algebraicity of Riemann surfaces (the reconstruction of a surface from its function field) and Serre duality are proved. Abelian varieties are first introduced as Jacobians of the surfaces and then studied per se. Thus the Kodaira embedding theorem is proved in this context. Also, a first discussion about moduli spaces takes place with the introduction of the Poincaré bundle, and line bundles on abelian varieties are investigated. The chapter (and this first volume) ends with a short introduction to the algebraic methods which can be used over arbitrary ground fields, methods to be fully developed in the next volume.
The book is very well written and the author has found the right measure between proving everything and leaving some aspects as exercises. Thus the Riemann-Roch theorem, Serre duality and many other results are almost fully proved (some hard-analytic aspects being just quoted, as is to be expected) while some others are left as exercises, for instance the link between the homotopy and homology groups of manifolds or some cohomological computations. It is worth mentioning that the author introduces the derived category of the category of sheaves as the natural setting for derived functors. It is a remarkable fact that he has managed to put together in a harmonious manner very abstract notions and constructions with classical ones from the time of Abel, Jacobi and Riemann. The book can be very valuable as a course text, not only in algebraic geometry, but also in sheaf theory, modern algebraic topology or classical theory of Riemann surfaces and their Jacobians due to its pedagogical nature.
