Algebraic Geometry II

Cohomology of Schemes: With Examples and Exercises

Ulrich Görtz, Torsten Wedhorn

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English

Springer Spektrum
23 November 2023

Algebraic geometry

Series: Springer Studium Mathematik - Master

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.

By: Ulrich Görtz, Torsten Wedhorn
Imprint: Springer Spektrum
Country of Publication: Germany
Edition: 1st ed. 2023
Dimensions: Height: 240mm, Width: 168mm,
Weight: 1.465kg
ISBN: 9783658430306
ISBN 10: 3658430303
Series: Springer Studium Mathematik - Master
Pages: 869
Publication Date: 23 November 2023
Audience: Professional and scholarly , Undergraduate
Format: Paperback
Publisher's Status: Active

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.

Prof. Dr. Ulrich Görtz, Department of Mathematics, University of Duisburg-Essen Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technical University of Darmstadt

Reviews for Algebraic Geometry II: Cohomology of Schemes: With Examples and Exercises

“They are a great contribution to the literature for courses on basic and advanced algebraic geometry, as well as an excellent reference source.” (Cícero Carvalho, Mathematical Reviews, July, 2025)