A brief overview of commutative algebra: rings and ideals, Nakayama’s Lemma, localization, Krull-dimension, direct-limits, integral dependence.

Toward algebraic varieties: regular functions, algebraic sets, Hilbert’s Nullstellensatz, Zariski topology, ringed spaces, affine and projective varieties.

Toward sheaves and group schemes: functors of points, Grothendieck topologies, representable functors, group schemes, tori, Grassmannians, torsors and twisted forms, quadrics and Severi-Brauer varieties.