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.
Readership
Graduate students and research mathematicians interested in
algebraic geometry.