Course description: Modern Algebraic Geometry is based on the notion of schemes. For example, in this framework, one can generalize algebraic varieties to allow any commutative ring as base, instead of a field. Substantial parts of Complex Geometry, Number Theory and Representation Theory can be incorporated into this framework. Our first aim is to cover the basic notions of Scheme Theory, culminating in the study of the cohomology of projective schemes. If time permits, we will study further topics such as sheaves of differentials and divisors.

Topics:

1. Some topics in Commutative Algebra: tensor products, flatness, formal completion
2. General properties of schemes: spectrum of a ring, ringed topological spaces, schemes, reduced and integral schemes, dimension
3. Morphisms and base change: base change, morphisms of finite type, base extension, separated, proper and projective morphisms
4. Some local properties: normal schemes, regular schemes, flat and smooth morphisms, Zariski's Main Theorem
5. Coherent sheaves and Cech cohomology: coherent sheaves on a scheme, Cech cohomology, cohomology of projective schemes

Possible further topics:

• Sheaves of differentials
• Divisors
• Additional topics based on the class's interest