The canonical sheaf

In algebraic geometry, there are two all important sheaves that come up, the structure sheaf and the canonical sheaf. In this video, we introduce the latter in the smooth case as essentially the top exterior product of the cotangent bundle. We hint at its importance by showing how the cotangent bundle has a natural element in its first cohomology group. We then compute the canonical sheaf in the case where we have an algebraic group and for the projective line. The canonical sheaf is an indispensable tool in the study of cohomology.