Cohomology of Vector Bundles and Syzygies

Written from a point of view of commutative algebra, no knowledge of representation theory is assumed. Several important applications are carefully considered, with numerous exercises for the reader.The central theme of this book is a detailed exposition of the geometric technique of calculating syzygies. While this is an important tool in algebraic geometry, the author has elected to write from the point of view of commutative algebra in order to avoid being tied to special cases from geometry. No prior knowledge of representation theory is assumed. Chapters on several applications are included, and numerous exercises will give the reader insight into how to apply this important method.The central theme of this book is an exposition of the geometric technique of calculating syzygies. It is written from a point of view of commutative algebra, and without assuming any knowledge of representation theory the calculation of syzygies of determinantal varieties is explained. The starting point is a definition of Schur functors, and these are discussed from both an algebraic and geometric point of view. Then a chapter on various versions of Bott's Theorem leads on to a careful explanation of the technique itself, based on a description of the direct image of a Koszul complex. Applications to determinantal varieties follow, plus there are also chapters on applications of the technique to rank varieties for symmetric and skew symmetric tensors of arbitrary degree, closures of conjugacy classes of nilpotent matrices, discriminants and resultants. Numerous exercises are included to give the reader insight into how to apply this important method....it is a useful reference, in particular for those advanced undergraduates and graduate university students who are considering the development of their knowledge in his branch of mathematics and or focusing their work for achieving a graduate degree in sich a new area of studies." Medhat H. Rahim, School and Science Mathematics' ... read this book ... instantly became the standard reference ...' Zentralblatt MATHAn exposition of the important geometric technique of calculating syzygies.