Cohomology and Regularity in Characteristic Zero

The goal of this chapter is to extend and prove sharper versions of the results of the preceding chapter, when working over a field of characteristic zero. Two crucial advantages are the linear reductivity of the general linear group and the Borel-Weil-Bott theorem. The class of ideals defined by shape can be enlarged to that of GL-invariant ideals, and the formulas for Castelnuovo-Mumford regularity can be significantly sharpened. As an application of the calculation of Ext modules, we explain how to describe the GL-structure for the local cohomology with support in determinantal ideals. Finally, we conclude the book with a quick survey of the important topic of free resolutions of determinantal ideals.