Introduction to Schemes and the

Cohomology of Coherent Modules

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.

*Algebraic Geometry*- Robin Hartshorne, Springer (Graduate Texts in Mathematics), ISBN 0387902449

The course is intended to complement Math 725 as offered during the fall semester of 2005.

**Formal Prerequisites**- The core courses in algebra, Math 520 A & B. General topology, Math 540A.
**Helpful Topics for Background**- Many things, while not formally necessary, may provide helpful motivation including commutative algebra, homological algebra, algebraic curves, and algebraic topology.