This book introduces the reader to modern algebraic geometry. It presents Grothendieck's technically demanding language of schemes that is the basis of the most important developments in the last fifty years within this area. A systematic treatment and motivation of the theory is emphasized, using concrete examples to illustrate its usefulness. Several examples from the realm of Hilbert modular surfaces and of determinantal varieties are used methodically to discuss the covered techniques. Thus the reader experiences that the further development of the theory yields an ever better understanding of these fascinating objects. The text is complemented by many exercises that serve to check the comprehension of the text, treat further examples, or give an outlook on further results. The volume at hand is an introduction to schemes. To get started, it requires only basic knowledge in abstract algebra and topology. Essential facts from commutative algebra are assembled in an appendix. It will be complemented by a second volume on the cohomology of schemes.

For the second edition, several mistakes and many smaller errors and misprints have been corrected.

Table of Contents

1 Prevarieties

2 Spectrum of a Ring

3 Schemes

4 Fiber products

5 Schemes over felds

6 Local Properties of Schemes

7 Quasi-coherent modules

8 Representable Functors

9 Separated morphisms

10 Finiteness Conditions

11 Vector bundles

12 Affine and proper morphisms

13 Proj ective morphisms

14 Flat morphisms and dimension

15 One-dimensional schemes

16 Examples

A The language of categories

B Commutative Algebra

C Permanence for properties of morphisms of schemes

D Relations between properties of morphisms of schemes

E Constructible and open properties

About the Authors

Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen

Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technical University of Darmstadt