Cohomology of Vector Bundles and Syzygies
Cambridge Tracts in Mathematics Series

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.

1. Introduction; 2. Schur functions and Schur complexes; 3. Grassmannians and flag varieties; 4. Bott's theorem; 5. The geometric technique; 6. The determinantal varieties; 7. Higher rank varieties; 8. The nilpotent orbit closures; 9. Resultants and discriminants.