Course Identification

• Good knowledge of linear algebra.
• Basic knowledge of commutative algebra
• Basic knowledge of point set topology.

Chapter I Affine schemes as ring spectra

1. Equations, rings, and ideals.
2. Points of schemes and the Zariski topology
3. Hilbert's Nulstellensatz

Chapter II. General Schemes

1. Sheaves
2. Affine schemes as ringed spaces
3. General schemes
4. Products of schemes
5. Projective schemes

Chapter III. Properties of morphisms

1. Finite morphisms
2. Separated morphisms 
3. Proper morphisms and Chow's lemma 
4. Valuation criteria

If time permits: tangent spaces and tangent cones

Upon successful completion of this course students will be able to:

1. Discuss the basic notions of commutative algebra and algebraic geometry. 
2. Translate problems from algebra to geometry and vice versa.
3. Use powerful algebraic techniques in geometric problems, and to use their geometric intuition in abstract algebraic problems.
4. (Through the language of algebraic geometry) gain access to part the modern literature in the broad fields of algebra and geometry.

1. Eisenbud & Harris "The geometry of schemes"
2. Hartshorne "Algebraic Geometry"
3. Atiyah-Macdonalds "Introduction to commutative algebra"
4. Eisenbud "Commutative Algebra With a View Toward Algebraic Geometry"