Algebraic Geometry 2

Sheaves and Cohomology

Kenji Ueno

185 pages, parution le 01/04/2001

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Résumé

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.

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.

Contents

Coherent sheaves
Proper and projective morphisms
Cohomology of coherent sheaves
Solutions to problems
Solutions to exercises
Index

L'auteur - Kenji Ueno

Kenji Ueno, Kyoto University, Japan

Caractéristiques techniques

	PAPIER
Éditeur(s)	American Mathematical Society (AMS)
Auteur(s)	Kenji Ueno
Parution	01/04/2001
Nb. de pages	185
Format	14 x 21,5
Couverture	Broché
Poids	234g
Intérieur	Noir et Blanc
EAN13	9780821813577
ISBN13	978-0-8218-1357-7