Let be a smooth variety of dimension , with sheaf of differentials . Define the to be the line bundle , whose sections are differential forms. The class of a divisor of a form is the .
In this post, we will calculate the canonical sheaf of projective space . In general, one can define the canonical sheaf for any projective scheme, in which case one usually refers to it as “dualizing”. Such calculations often reduce to the case of projective space, so the following computation is the most central one.
Let be homogeneous coordinates on . Let be the hypersurface defined by the vanishing of the coordinate , and let be its complement; note that with the coordinates , we have an identification .
On , then, the space of differential -forms is spanned by . By properties of differentials, the “formal quotient rule” holds, and we can express which implies that which has a pole of order on .
It follows that , which is the canonical class.