In a manner that is reminiscent of the way a sphere may be pieced together from overlapping disk images, a scheme may be pieced together from affine schemes. An affine scheme is the geometric guise of a commutative ring with unity. Specific case: if k is a field, its n-dimensional affine space {A^{n}}_{k} is the geometric guise of the polynomial ring k [x_{1}, …, x_{n}], while its n-dimensional projective space {P^{n}}_{k} is a non-affine scheme that may be pieced together using overlapping copies of {A^{n}}_{k}.
While every scheme is locally affine, a scheme embodies global information that may not easily be discerned simply by viewing it as a union of affine schemes. The cohomology of coherent modules encodes much global geometric information. In the case of an affine scheme a coherent module is the same thing as a finitely-generated module over the ring associated with the affine scheme, and the cohomology of a coherent module is trivial.
The course is intended to complement Math 725 as offered during the fall semester of 2005.