Affine varieties
 Rings, ideals, modules, Noetherianity, Hilbert basis theorem, principle ideal domains, application: proof of primary decomposition theorem and classification of finitely generated commutative groups.
 Algebraic sets, Zariski topology, Hilbert's Nulstellensatz
 Sheaves of functions, Serre's lemma
 Non-affine varieties, projective varieties
 Dimension, Noether's normalization lemma, Chevalley theorem, principal ideal theorem.
 Zariski tangent space, smooth varieties, blow up, 27 lines on a smooth cubic surface.
 Product of varieties, separated and complete varieties, Chow's
lemma, valuation criteria.