2013 Math 631 Daily Update

Monday December 9: We discussed the sheaf ΩX of regular differential forms on a variety X , which is sheaf of OX-modules, locally free IFF X is smooth, in which case its rank is equal to the dimension. We computed several examples: the sheaf on A is freely generated (on every open set) by dx1, . . . , dxn. On the parabola defined by y = x in A, the elements dx and dy generate on every open set, and satisfy the relation dy = 2xdx. So, dy can be used as a generator for the sheaf at every point except the origin, while dx is always a generator on every open set. On the singular curve defined by x + y + z in A, the elements dx, dy and dz always generate but satisfy the relation 3xdx + 3ydy + 3zdz = 0; in particular, this sheaf is free of rank two except at the origin—for example, on Ux, the elements dy and dz generate, since thedx = −y 2 x2 dy+ −z 2 x2 dz there. On the contrary, no two elements generate ΩX(U) in any open set U of the origin, as we expect of course since the tangent space there is three-dimensional. We defined the p-forms as the p-th exterior power of ΩX . On smooth X , the p-forms are a locally free sheaf ∧ΩX of rank ( d