Hartshorne, Algebraic Geometry, GTM 52
Math 535. Familiarity with commutative algebra is an advantage, but is not required.
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. In addition, schemes provide a natural context for introducing the theory of sheaf cohomology, which is a central tool in modern algebraic geometry. For example, one can use cohomological methods to give a simple proof of the classical Riemann-Roch theorem for curves. The goal of the course is to cover the basic definitions, properties, and applications of the above mentioned concepts.