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Algebraic geometry II [electronic resource] : cohomology of schemes : with examples and exercises / Ulrich Gr̲tz, Torsten Wedhorn.

Gr̲tz, Ulrich, 1973-; Wedhorn, Torsten.
2023
QA564
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Title
Algebraic geometry II [electronic resource] : cohomology of schemes : with examples and exercises / Ulrich Gr̲tz, Torsten Wedhorn.
Author
Gr̲tz, Ulrich, 1973-
ISBN
9783658430313 (electronic bk.)
3658430311 (electronic bk.)
3658430303
9783658430306
DOI
https://doi.org/10.1007/978-3-658-43031-3
Published
Wiesbaden : Springer Spektrum, 2023.
Language
English
Description
1 online resource.
Item Number
10.1007/978-3-658-43031-3 doi
Call Number
QA564
Dewey Decimal Classification
516.3/5
Summary
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 tale morphisms including the tale 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. Contents Differentials - tale and smooth morphisms - Local complete intersections - The tale topology - Cohomology of sheaves of modules - Cohomology of quasi-coherent sheaves - Cohomology of projective and proper schemes - Theorem on formal functions - Duality - Curves - Abelian schemes - Appendix: Homological Algebra - Appendix: Commutative Algebra About the Authors Prof. Dr. Ulrich Grtz, Department of Mathematics, University of Duisburg-Essen Prof. Dr. Torsten Wedhorn, Department of Mathematics, Technical University of Darmstadt.
Bibliography, etc. Note
Includes bibliographical references and index.
Access Note
Access limited to authorized users.
Source of Description
Online resource; title from PDF title page (SpringerLink, viewed December 7, 2023).
Added Author
Wedhorn, Torsten.
Series
Springer studium mathematik. Master.
Available in Other Form
Print version: 9783658430306
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Intro
Contents
Introduction
17 Differentials
Differentials for rings and extensions of algebras
(17.1) Derivations and Kh̃ler differentials for rings.
(17.2) Extensions of algebras by modules.
Differentials for sheaves on schemes
(17.3) Conormal sheaf of an immersion.
(17.4) Derivations and Kh̃ler differentials for schemes.
(17.5) Fundamental exact sequences for Kh̃ler differentials.
(17.6) Tangent bundles.
(17.7) Differentials of Grassmannians and of projective bundles.
The de Rham complex
(17.8 )The exterior algebra.

(17.9 )Differential graded algebras.
(17.10 )The de Rham complex.
Exercises
18 tale and smooth morphisms
Formally unramified, formally smooth and formally ťale morphisms
(18.1) Definition of formally unramified, formally smooth and formally ťale morphisms.
(18.2) Formally unramified morphisms and differentials.
(18.3) Gluing local lifts.
(18.4) Formally smooth resp. formally ťale morphisms and differentials.
Unramified and ťale morphisms
(18.5) Unramified morphisms.
(18.6) tale morphisms.
(18.7) Local description of ťale morphisms.

(18.8) Characterization of ťale morphisms.
Smooth morphisms
(18.9) Geometrically regular schemes.
(18.10) Characterization of smooth morphisms.
(18.11) Characterizations of smooth morphisms in the noetherian case.
(18.12) Smooth schemes over a field.
(18.13) Smooth morphisms and differentials.
(18.14) Smooth and ťale morphisms between smooth schemes.
(18.15) Open immersions and ťale morphisms.
(18.16) Fibre criterion for smooth and ťale morphisms.
(18.17) Generic Smoothness.
Exercises
19 Local complete intersections

The Koszul complex and completely intersecting immersions
(19.1) Koszul complex.
(19.2) Regular and completely intersecting sequences.
(19.3) Regular and completely intersecting immersions.
(19.4) Regular immersions of flat and of smooth schemes.
(19.5) Blow-up of regularly immersed smooth subschemes.
Local complete intersection and syntomic morphisms
(19.6) Local complete intersection morphisms.
(19.7) Complete intersection rings.
(19.8) Local complete intersection morphisms over a field.
(19.9) Syntomic morphisms.
Exercises
20 The ťale topology

Henselian rings
(20.1) Definition of henselian rings.
(20.2) Sections of smooth morphisms.
(20.3) Sections of ťale and smooth schemes over henselian rings.
(20.4) Henselian pairs.
The ťale topology
(20.5) tale topology.
(20.6) Lifting of ťale schemes.
(20.7) Sheaves in the ťale topology.
(20.8) Points and stalks in the ťale topology.
(20.9) Stalks of the structure sheaf: (strict) henselization.
(20.10) Unibranch schemes.
(20.11) Artin approximation.
(20.12) Analytification of schemes over C.
The ťale fundamental group of a scheme

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