Algebraic Geometry II: Cohomology of Schemes
With Examples and Exercises
Seiten
2023
|
1st ed. 2023
Springer Fachmedien Wiesbaden GmbH (Verlag)
978-3-658-43030-6 (ISBN)
Springer Fachmedien Wiesbaden GmbH (Verlag)
978-3-658-43030-6 (ISBN)
This book completes the comprehensive introduction to modern algebraic geometry which was started with the introductory volume Algebraic Geometry I: Schemes.
It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve to develop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously.
The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.
It begins by discussing in detail the notions of smooth, unramified and étale morphisms including the étale fundamental group. The main part is dedicated to the cohomology of quasi-coherent sheaves. The treatment is based on the formalism of derived categories which allows an efficient and conceptual treatment of the theory, which is of crucial importance in all areas of algebraic geometry. After the foundations are set up, several more advanced topics are studied, such as numerical intersection theory, an abstract version of the Theorem of Grothendieck-Riemann-Roch, the Theorem on Formal Functions, Grothendieck's algebraization results and a very general version of Grothendieck duality. The book concludes with chapters on curves and on abelian schemes, which serve to develop the basics of the theory of these two important classes of schemes on an advanced level, and at the same time to illustrate the power of the techniques introduced previously.
The text contains many exercises that allow the reader to check their comprehension of the text, present further examples or give an outlook on further results.
lt;p>Prof. Dr. Ulrich Görtz, Department of Mathematics, University of Duisburg-Essen
Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technical University of Darmstadt
Introduction.- 17 Differentials.- 18 Étale and smooth morphisms.- 19 Local complete intersections.- 20 The étale topology.- 21 Cohomology of sheaves of modules.- 22 Cohomology of quasi-coherent modules.- 23 Cohomology of projective and proper schemes.- 24 Theorem on formal functions.- 25 Duality.- 26 Curves.- 27 Abelian schemes.- F Homological algebra.- G Commutative algebra II.
| Erscheinungsdatum | 23.11.2023 |
|---|---|
| Reihe/Serie | Springer Studium Mathematik - Master |
| Zusatzinfo | VII, 869 p. 153 illus. |
| Verlagsort | Wiesbaden |
| Sprache | englisch |
| Maße | 168 x 240 mm |
| Gewicht | 1465 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Geometrie / Topologie |
| Schlagworte | Algebraic Geometry • Cohomology of Schemes • Cohomology of Sheaves of Modules • Schemes and their Cohomology • Smooth Morphisms |
| ISBN-10 | 3-658-43030-3 / 3658430303 |
| ISBN-13 | 978-3-658-43030-6 / 9783658430306 |
| Zustand | Neuware |
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