Algebraic geometry. 2. Sheaves and cohomology. Transl. from the Japanese by Goro Kato
type: Item
instance of: scholarly article
zbMATH Open document ID: 0965.14008
author: Kenji Ueno
publication date: 2001-03-28T00:00:00Z
Mathematics Subject Classification ID: 14Fxx
Mathematics Subject Classification ID: 14-01
Mathematics Subject Classification ID: 14-02
zbMATH DE Number: 1582101
zbMATH Keywords: theory of schemes
zbMATH Keywords: coherent sheaf
zbMATH Keywords: cohomology of schemes
zbMATH Keywords: projective schemes
zbMATH Keywords: algebraic morphisms
zbMATH Keywords: ample sheaves
zbMATH Keywords: projective embeddings
zbMATH Keywords: intersection theory
zbMATH Keywords: direct images of sheaves
MaRDI profile type: Publication
title: Algebraic geometry. 2. Sheaves and cohomology. Transl. from the Japanese by Goro Kato
published in: Translations of Mathematical Monographs
review text: The book under review is the second volume of \textit{K. Ueno}'s textbook. The first volume [``Algebraic geometry. 1. From algebraic varieties to schemes'', Transl., Ser. 2 (1999; Zbl 0937.14001)] provided the basic material from the theory of algebraic schemes, arranged in the first three chapters of the textbook series as a whole. Precisely, those three chapters forming volume 1 were entitled as follows:NEWLINENEWLINENEWLINEChapter 1: Algebraic Varieties:NEWLINENEWLINENEWLINEChapter 2: Schemes;NEWLINENEWLINENEWLINEChapter 3: Categories and Schemes.NEWLINENEWLINENEWLINEThe present second volume is a direct continuation of the first one and just contains the next three chapters. The focus of this second part is on sheaves and sheaf cohomology in the theory of algebraic schemes, which is, in fact, the fundamental methodical and technical toolkit for higher algebraic geometry.NEWLINENEWLINENEWLINEKeeping the notation of the first volume, the author has subdivided this second volume into three chapters entitled as follow.NEWLINENEWLINENEWLINEChapter 4: Coherent sheaves;NEWLINENEWLINENEWLINEChapter 5: Proper and projective morphisms;NEWLINENEWLINENEWLINEChapter 6: Cohomology of coherent sheaves.NEWLINENEWLINENEWLINEAccordingly, chapter 4 discusses the various constructions of sheaves on schemes out of given ones, exact sequences of sheaves, direct and inverse images of sheaves with respect to morphisms of schemes, closed subschemes and quasicoherent ideal sheaves, quasicoherent and coherent sheaves in general, as well as affine morphisms via quasicoherent algebras of sheaves.NEWLINENEWLINENEWLINEChapter 5 provides the general theory of proper morphisms and projective morphisms, notions which are crucial in algebraic geometry. This chapter, which is, quite naturally, more abstract and technically involved than the previous ones, comes with many concrete examples as motivating background, in order to facilitate the understanding of the deep theorems developed here. Apart …
reviewed by: Werner Kleinert
description: scientific article