Algebraic Geometry Ii Cohomology Of Schemes

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Algebraic Geometry II: Cohomology of Schemes

Ulrich Görtz,Torsten Wedhorn
Algebraic Geometry II Cohomology of Schemes Book PDF

Author : Ulrich Görtz,Torsten Wedhorn
Publisher : Springer Nature
Page : 877 pages
File Size : 55,6 Mb
Release : 2023-11-22
Category : Mathematics
ISBN : 9783658430313

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Algebraic Geometry II: Cohomology of Schemes by Ulrich Görtz,Torsten Wedhorn Pdf

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.

Lectures on Algebraic Geometry II

Günter Harder
Lectures on Algebraic Geometry II Book PDF

Author : Günter Harder
Publisher : Springer Science & Business Media
Page : 376 pages
File Size : 53,9 Mb
Release : 2011-04-21
Category : Mathematics
ISBN : 9783834881595

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Lectures on Algebraic Geometry II by Günter Harder Pdf

This second volume introduces the concept of shemes, reviews some commutative algebra and introduces projective schemes. The finiteness theorem for coherent sheaves is proved, here again the techniques of homological algebra and sheaf cohomology are needed. In the last two chapters, projective curves over an arbitrary ground field are discussed, the theory of Jacobians is developed, and the existence of the Picard scheme is proved. Finally, the author gives some outlook into further developments- for instance étale cohomology- and states some fundamental theorems.

Algebraic Geometry 2

Kenji Ueno
Algebraic Geometry 2 Book PDF

Author : Kenji Ueno
Publisher : American Mathematical Soc.
Page : 196 pages
File Size : 47,6 Mb
Release : 1999
Category : Mathematics
ISBN : 0821813579

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Algebraic Geometry 2 by Kenji Ueno Pdf

Algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes was explained in Algebraic Geometry 1: From Algebraic Varieties to Schemes. In this volume, the author turns to the theory of sheaves and their cohomology. A sheaf is a way of keeping track of local information defined on a topological space, such as the local holomorphic functions on a complex manifold or the local sections of a vector bundle. To study schemes, it is useful to study the sheaves defined on them, especially the coherent and quasicoherent sheaves.

Algebraic Geometry II

I.R. Shafarevich
Algebraic Geometry II Book PDF

Author : I.R. Shafarevich
Publisher : Springer Science & Business Media
Page : 270 pages
File Size : 51,7 Mb
Release : 2013-11-22
Category : Mathematics
ISBN : 9783642609251

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Algebraic Geometry II by I.R. Shafarevich Pdf

This two-part volume contains numerous examples and insights on various topics. The authors have taken pains to present the material rigorously and coherently. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

Algebraic Geometry

Ulrich Görtz,Torsten Wedhorn
Algebraic Geometry Book PDF

Author : Ulrich Görtz,Torsten Wedhorn
Publisher : Springer Science & Business Media
Page : 615 pages
File Size : 42,5 Mb
Release : 2010-08-09
Category : Mathematics
ISBN : 9783834897220

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Algebraic Geometry by Ulrich Görtz,Torsten Wedhorn Pdf

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.

Algebraic Geometry I: Schemes

Ulrich Görtz,Torsten Wedhorn
Algebraic Geometry I Schemes Book PDF

Author : Ulrich Görtz,Torsten Wedhorn
Publisher : Springer Nature
Page : 626 pages
File Size : 53,9 Mb
Release : 2020-07-27
Category : Mathematics
ISBN : 9783658307332

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Algebraic Geometry I: Schemes by Ulrich Görtz,Torsten Wedhorn Pdf

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.

Algebraic Geometry II

David Mumford,Tadao Oda
Algebraic Geometry II Book PDF

Author : David Mumford,Tadao Oda
Publisher : Unknown
Page : 0 pages
File Size : 47,6 Mb
Release : 2015
Category : Algebraic varieties
ISBN : 9380250800

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Algebraic Geometry II by David Mumford,Tadao Oda Pdf

Several generations of students of algebraic geometry have learned the subject from David Mumford's fabled "Red Book" containing notes of his lectures at Harvard University. This book contains what Mumford had intended to be Volume II. It covers the material in the "Red Book" in more depth with several more topics added.

Algebraic Geometry

Robin Hartshorne
Algebraic Geometry Book PDF

Author : Robin Hartshorne
Publisher : Springer Science & Business Media
Page : 511 pages
File Size : 40,5 Mb
Release : 2013-06-29
Category : Mathematics
ISBN : 9781475738490

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Algebraic Geometry by Robin Hartshorne Pdf

An introduction to abstract algebraic geometry, with the only prerequisites being results from commutative algebra, which are stated as needed, and some elementary topology. More than 400 exercises distributed throughout the book offer specific examples as well as more specialised topics not treated in the main text, while three appendices present brief accounts of some areas of current research. This book can thus be used as textbook for an introductory course in algebraic geometry following a basic graduate course in algebra. Robin Hartshorne studied algebraic geometry with Oscar Zariski and David Mumford at Harvard, and with J.-P. Serre and A. Grothendieck in Paris. He is the author of "Residues and Duality", "Foundations of Projective Geometry", "Ample Subvarieties of Algebraic Varieties", and numerous research titles.

The Geometry of Schemes

David Eisenbud,Joe Harris
The Geometry of Schemes Book PDF

Author : David Eisenbud,Joe Harris
Publisher : Springer Science & Business Media
Page : 265 pages
File Size : 55,5 Mb
Release : 2006-04-06
Category : Mathematics
ISBN : 9780387226392

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The Geometry of Schemes by David Eisenbud,Joe Harris Pdf

Grothendieck’s beautiful theory of schemes permeates modern algebraic geometry and underlies its applications to number theory, physics, and applied mathematics. This simple account of that theory emphasizes and explains the universal geometric concepts behind the definitions. In the book, concepts are illustrated with fundamental examples, and explicit calculations show how the constructions of scheme theory are carried out in practice.

Lectures on Algebraic Geometry I

Günter Harder
Lectures on Algebraic Geometry I Book PDF

Author : Günter Harder
Publisher : Springer Science & Business Media
Page : 301 pages
File Size : 49,5 Mb
Release : 2008-08-01
Category : Mathematics
ISBN : 9783834895011

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Lectures on Algebraic Geometry I by Günter Harder Pdf

This book and the following second volume is an introduction into modern algebraic geometry. In the first volume the methods of homological algebra, theory of sheaves, and sheaf cohomology are developed. These methods are indispensable for modern algebraic geometry, but they are also fundamental for other branches of mathematics and of great interest in their own. In the last chapter of volume I these concepts are applied to the theory of compact Riemann surfaces. In this chapter the author makes clear how influential the ideas of Abel, Riemann and Jacobi were and that many of the modern methods have been anticipated by them.

Algebraic Geometry II

I.R. Shafarevich
Algebraic Geometry II Book PDF

Author : I.R. Shafarevich
Publisher : Springer
Page : 264 pages
File Size : 55,7 Mb
Release : 2014-10-05
Category : Mathematics
ISBN : 3642609260

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Algebraic Geometry II by I.R. Shafarevich Pdf

This two-part volume contains numerous examples and insights on various topics. The authors have taken pains to present the material rigorously and coherently. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetic algebraic geometry, complex analysis and related fields.

Algebraic Geometry Two

Игорь Ростиславович Шафаревич
Algebraic Geometry Two Book PDF

Author : Игорь Ростиславович Шафаревич
Publisher : Springer Verlag
Page : 262 pages
File Size : 45,9 Mb
Release : 1996
Category : Mathematics
ISBN : 0387546804

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Algebraic Geometry Two by Игорь Ростиславович Шафаревич Pdf

This EMS volume consists of two parts. The first part is devoted to cohomology of algebraic varieties. The second part deals with algebraic surfaces. The authors, who are well-known experts in the field, have taken pains to present the material rigorously and coherently. The book contains numerous examples and insights on various topics. This book will be immensely useful to mathematicians and graduate students working in algebraic geometry, arithmetical algebraic geometry, complex analysis and related fields.

Etale Cohomology Theory (Revised Edition)

Lei Fu
Etale Cohomology Theory Revised Edition Book PDF

Author : Lei Fu
Publisher : World Scientific
Page : 624 pages
File Size : 49,8 Mb
Release : 2015-02-27
Category : Mathematics
ISBN : 9789814675109

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Etale Cohomology Theory (Revised Edition) by Lei Fu Pdf

Etale cohomology is an important branch in arithmetic geometry. This book covers the main materials in SGA 1, SGA 4, SGA 4 1/2 and SGA 5 on etale cohomology theory, which includes decent theory, etale fundamental groups, Galois cohomology, etale cohomology, derived categories, base change theorems, duality, and ℓ-adic cohomology. The prerequisites for reading this book are basic algebraic geometry and advanced commutative algebra.

Koszul Cohomology and Algebraic Geometry

Marian Aprodu,Jan Nagel
Koszul Cohomology and Algebraic Geometry Book PDF

Author : Marian Aprodu,Jan Nagel
Publisher : American Mathematical Soc.
Page : 138 pages
File Size : 41,9 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821849644

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Koszul Cohomology and Algebraic Geometry by Marian Aprodu,Jan Nagel Pdf

The systematic use of Koszul cohomology computations in algebraic geometry can be traced back to the foundational work of Mark Green in the 1980s. Green connected classical results concerning the ideal of a projective variety with vanishing theorems for Koszul cohomology. Green and Lazarsfeld also stated two conjectures that relate the Koszul cohomology of algebraic curves with the existence of special divisors on the curve. These conjectures became an important guideline for future research. In the intervening years, there has been a growing interaction between Koszul cohomology and algebraic geometry. Green and Voisin applied Koszul cohomology to a number of Hodge-theoretic problems, with remarkable success. More recently, Voisin achieved a breakthrough by proving Green's conjecture for general curves; soon afterwards, the Green-Lazarsfeld conjecture for general curves was proved as well. This book is primarily concerned with applications of Koszul cohomology to algebraic geometry, with an emphasis on syzygies of complex projective curves. The authors' main goal is to present Voisin's proof of the generic Green conjecture, and subsequent refinements. They discuss the geometric aspects of the theory and a number of concrete applications of Koszul cohomology to problems in algebraic geometry, including applications to Hodge theory and to the geometry of the moduli space of curves.

Algebraic Geometry and Arithmetic Curves

Qing Liu,Reinie Erne
Algebraic Geometry and Arithmetic Curves Book PDF

Author : Qing Liu,Reinie Erne
Publisher : Oxford University Press
Page : 593 pages
File Size : 53,7 Mb
Release : 2006-06-29
Category : Mathematics
ISBN : 9780191547805

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Algebraic Geometry and Arithmetic Curves by Qing Liu,Reinie Erne Pdf

This book is a general introduction to the theory of schemes, followed by applications to arithmetic surfaces and to the theory of reduction of algebraic curves. The first part introduces basic objects such as schemes, morphisms, base change, local properties (normality, regularity, Zariski's Main Theorem). This is followed by the more global aspect: coherent sheaves and a finiteness theorem for their cohomology groups. Then follows a chapter on sheaves of differentials, dualizing sheaves, and Grothendieck's duality theory. The first part ends with the theorem of Riemann-Roch and its application to the study of smooth projective curves over a field. Singular curves are treated through a detailed study of the Picard group. The second part starts with blowing-ups and desingularisation (embedded or not) of fibered surfaces over a Dedekind ring that leads on to intersection theory on arithmetic surfaces. Castelnuovo's criterion is proved and also the existence of the minimal regular model. This leads to the study of reduction of algebraic curves. The case of elliptic curves is studied in detail. The book concludes with the funadmental theorem of stable reduction of Deligne-Mumford. The book is essentially self-contained, including the necessary material on commutative algebra. The prerequisites are therefore few, and the book should suit a graduate student. It contains many examples and nearly 600 exercises.