Duality, Canonical Modules, and Gorenstein Rings

Abstract

The most basic geometric objects associated with a smooth manifold or a smooth variety are its tangent and cotangent bundles, and various tensors derived from them. In algebraic geometry the most easily accessible, and the most important, is the canonical bundle, the highest exterior power of the cotangent bundle. It plays a central part in duality theory. If the variety is affine, then the sections of the canonical bundle form a module over the coordinate ring of the variety called the canonical module. Because it is the module of sections of a line bundle, it is locally free of rank 1.