Algebraic geometry. 2. Sheaves and cohomology. Transl. from the Japanese by Goro Kato
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 from the basic material on proper and projective morphisms, this chapter also contains the description of ample invertible sheaves and Segre morphisms.NEWLINENEWLINENEWLINEThe concluding chapter 6 deals with the cohomology theory of quasicoherent and coherent sheaves over a scheme. This is done in a way that provides a general theory of sheaf cohomology which is suited also for the later study of complex manifolds and functions of several complex variables. The necessary amount of methods and results from homological algebra is woven into this part of the text, in full detail, keeping thereby the self-containedness of the entire book intact. The section on the cohomology of a projective scheme, including the finiteness theorem for the cohomology of a coherent sheaf, Bezout's theorem for projective plane curves in its cohomological setting, and an ampleness criterion for invertible sheaves over a proper scheme defined over a noetherian ring, gives the first important applications of sheaf cohomology in algebraic geometry, while the final section of this chapter (and this second volume) is devoted to the (quasi-)coherence property of higher direct image sheaves with respect to certain algebraic morphisms of schemes.NEWLINENEWLINENEWLINEAs in the first volume, each chapter in this second volume comes with its own summary of main results, at the end, and with a set of carefully selected exercises, the solutions of which are explained at the end of this volume.NEWLINENEWLINENEWLINEThis second part of the whole textbook is just as masterly written as the first part of it. The author has consistently maintained his methodological strategy of providing a self-contained, comprehensive, user-friendly, nevertheless rigorous and detailed introduction to modern algebraic geometry, and the same outstanding quality may be expected for the last volume.
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