Tony EZOME
Summary: I started by recalling the needed background from commutative algebra (noetherian modules and rings, finitely generated algebras, Hilbert basis theorem, integral/algebraic elements in a ring/field extension, transcendence degree of a field extension, the algebraic closure of a field) with useful exercice sessions.
Then I introduced affine algebraic sets, the Zariski topology on the affine n-space and the Hilbert’s Nullstellensatz. I specified all these notions for the case of the affine line. I also introduced the coordinate ring of an affine algebraic set, its function fields, its dimension, smooth points (Jacoby criterion as well as a criterion using the maximal ideal of the localization). I did the same study for projective algebraic varieties.
And then I introduced the decompostion into irreducible components. I also introduced homogenization and dehomogenization processes. I let student prove that any projective algebraic set is covered by a finite number of affine open subsets which are homeomorphic to closed subsets of an affine space. There were intensive exercice sessions.