Algebraic Geometry II
This course continues the study of algebraic geometry from the fall by
replacing algebraic varieties with the more general theory of schemes,
which makes it possible to assign geometric meaning to an arbitrary
commutative ring. One major advantage of schemes is the availability
of a well-behaved fiber product. Combined with Grothendieck's
philosophy that properties of schemes should be expressed as
properties of morphisms between schemes, fiber products make the
theory very flexible. The goal of the course is to cover the basic
definitions and properties of schemes and morphisms, and to introduce
and study the cohomology of sheaves, which provides a powerful tool
for settling geometric questions. For example, one can use
cohomological methods to give a simple proof of the classical
Riemann-Roch theorem for curves.
Prerequisites: Math 535. Familiarity with commutative algebra is an
advantage, but is not required.
Text: Hartshorne, Algebraic Geometry (Springer GTM 52).