Algebraic Geometry: Part I: Schemes. With Examples and Exercises (Advanced Lectures in Mathematics)

This comprehensive introduction to schemes is complemented by many exercises that serve to check the comprehension of the text, treat further examples and give an outlook on further results. Includes details from commutative algebra in an appendix.

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 startet, 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.

Prevarieties - Spectrum of a Ring - Schemes - Fiber products - Schemes over fields - Local properties of schemes - Quasi-coherent modules - Representable functors - Separated morphisms - Finiteness Conditions - Vector bundles - Affine and proper morphisms - Projective morphisms - Flat morphisms and dimension - One-dimensional schemes - Examples

Prof. Dr. Ulrich Görtz, Institute of Experimental Mathematics, University Duisburg-Essen
Prof. Dr. Torsten Wedhorn, Department of Mathematics, University of Paderborn

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