Algebraic Geometry

"Hartshorne has produced a book ... of which he may be proud and for which we must be grateful. It is destined to become the standard reference for young workers in and students of algebraic geometry. They will be well served. Because algebraic geometry is important for so many fields from partial differential equations through complex analysis to number theory and algebra, this book belongs on every mathematician's shelf. We owe Hartshorne our thanks." (Bulletin of the American Mathematical Society)

This book provides an introduction to abstract algebraic geometry. The prerequisites for this approach are results from commutative algebra, which are stated as needed, and some elementary topology. There are more than 400 exercises throughout the book, offering specific examples as well as more specialized topics not treated in the main text. Three appendices present brief accounts of some areas of current research. This book can be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra.

R. Hartshorne Algebraic Geometry "Enables the reader to make the drastic transition between the basic, intuitive questions about affine and projective varieties with which the subject begins, and the elaborate general methodology of schemes and cohomology employed currently to answer these questions."MATHEMATICAL REVIEWS

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Inhalt
I Varieties.- II Schemes.- III Cohomology.- IV Curves.- V Surfaces.- Appendix A Intersection Theory.- 1 Intersection Theory.- 2 Properties of the Chow Ring.- 3 Chern Classes.- 4 The Riemann-Roch Theorem.- 5 Complements and Generalizations.- Appendix B Transcendental Methods.- 1 The Associated Complex Analytic Space.- 2 Comparison of the Algebraic and Analytic Categories.- 3 When is a Compact Complex Manifold Algebraic?.- 4 Kähler Manifolds.- 5 The Exponential Sequence.- Appendix C The Weil Conjectures.- 1 The Zeta Function and the Weil Conjectures.- 2 History of Work on the Weil Conjectures.- 3 The /-adic Cohomology.- 4 Cohomological Interpretation of the Weil Conjectures.- Results from Algebra.- Glossary of Notations.

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