To explain basic principles of modern algebraic geometry to undergraduate students with a standard education is a very difficult task. Nevertheless, Amnon Neeman has succeeded in this book in showing that it is possible. The book is based on his own teaching experience and the basic idea is simple. He has chosen to present a formulation and the proof of Serre’s famous theorem on ‘géométrie algébriques and géométrie analytique’ (GAGA) as the ultimate aim of the book. On the way to this goal, he introduces a lot of notions and tools of modern algebraic geometry (e.g. the spectrum of a ring, Zariski topology, schemes, algebraic and analytic coherent sheaves, localizations, sheaf cohomology). The whole book shows to the reader a beautiful mixture of analysis, algebra, geometry and topology, all used together in the modern language of algebraic geometry and, at the same time, nicely illustrating its power.

The book is written in a very understandable way, with a lot of details and with many remarks and comments helping to develop intuition for the field. It is an extraordinary book and it can be very useful for teachers as well as for students or non-experts from other fields.

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