Algebraic Geometry 2: Sheaves and Cohomology (Translations of Mathematical Monographs) (Vol 2)

Modern 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, (see Volume 185 in the same series, Translations of Mathematical Monographs). In the present book, Ueno turns to the theory of sheaves and their cohomology. Loosely speaking, 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. The primary tool in understanding sheaves is cohomology. For example, in studying ampleness, it is frequently useful to translate a property of sheaves into a statement about its cohomology. The text covers the important topics of sheaf theory, including types of sheaves and the fundamental operations on them, such as ... coherent and quasicoherent sheaves. proper and projective morphisms. direct and inverse images. Cech cohomology. For the mathematician unfamiliar with the language of schemes and sheaves, algebraic geometry can seem distant. However, Ueno makes the topic seem natural through his concise style and his insightful explanations. He explains why things are done this way and supplements his explanations with illuminating examples. As a result, he is able to make algebraic geometry very accessible to a wide audience of non-specialists. Cover Title page Contents Chapter 4. Coherent Sheaves 4.1. Exact Sequence of Sheaves (a) Sheafification of Presheaves (b) Kernels and Cokernels of Sheaf Homomorphisms (c) Exact Sequences 4.2. Quasicoherent Sheaves and Coherent Sheaves (a) \mathscr{O}_X - Modules (b) Quasicoherent Sheaves (c) Coherent Sheaves 4.3. Direct Image and Inverse Image (a) Direct Image and Inverse Image of a Sheaf under a Continuous Map (b) Direct Image and Inverse Image under a Scheme Morphism 4.4. Schemes and Quasicoherent Sheaves (a) Closed Subschemes and Ideal Sheaves (b) AfRne Morphisms and Quasicoherent \mathscr{O}_Y-Algebras Summary Exercises Chapter 5. Proper and Projective Morphisms 5.1. Proper Morphisms (a) Closed Morphisms (b) Proper Morphisms (c) Valuative Criterion 5.2. Quasicoherent Sheaves over a Projective Scheme (a) Brief Review of Projective Schemes (b) Quasicoherent Sheaves 5.3. Projective Morphisms (a) Categorical Characterization of P(E) (b) The Segre Morphism (c) Ample Invertible Sheaves Summary Exercises Chapter 6. Cohomology of Coherent Sheaves 6.1. Cohomology of Sheaves (a) Flabby Sheaves (b) Cohomology Group (c) Cohomology of Affine Schemes (d) Čech Cohomology Groups 6.2. Cohomology of a Projective Scheme (a) Cohomology of a Projective Space (b) Finiteness of Cohomology of Projective Schemes (c) Bézout's Theorem (d) Criterion for Ampleness 6.3. Higher Direct Image (a) Higher Direct Image (b) Projective Morphisms Summary Exercises Solutions to Problems Chapter 4 Chapter 5 Chapter 6 Solutions to Exercises Chapter 4 Chapter 5 Chapter 6 Index Back Cover

Modern algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes is presented in the first part of this book (Algebraic Geometry 1: From Algebraic Varieties to Schemes, AMS, 1999, Translations of Mathematical Monographs, Volume 185). In the present book, the author turns to the theory of sheaves and their cohomology. Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local algebraic functions on an algebraic manifold or the local sections of a vector bundle. Sheaf cohomology is a primary tool in understanding sheaves and using them to study properties of the corresponding manifolds. The text covers the important topics of the theory of sheaves on algebraic varieties, including types of sheaves and the fundamental operations on them, such as coherent and quasicoherent sheaves, direct and inverse images, behavior of sheaves under proper and projective morphisms, and Cech cohomology. The book contains numerous problems and exercises with solutions. It would be an excellent text for the second part of a course in algebraic geometry.

Modern algebraic geometry is built upon two fundamental notions: schemes and sheaves. The theory of schemes is presented in the first part of this book (Algebraic Geometry 1: From Algebraic Varieties to Schemes, AMS, 1999, Translations of Mathematical Monographs, Volume 185). In the present book, the author turns to the theory of sheaves and their cohomology. Loosely speaking, a sheaf is a way of keeping track of local information defined on a topological space, such as the local algebraic functions on an algebraic manifold or the local sections of a vector bundle. Sheaf cohomology is a primary tool in understanding sheaves and using them to study properties of the corresponding manifolds. The text covers the important topics of the theory of sheaves on algebraic varieties, including types of sheaves and the fundamental operations on them, such as coherent and quasicoherent sheaves, direct and inverse images, behavior of sheaves under proper and projective morphisms, and Čech cohomology. The book contains numerous problems and exercises with solutions. It would be an excellent text for the second part of a course in algebraic geometry. 1. From algebraic varieties to schemes 2. Sheaves and cohomology 3. Further study of schemes.