1 Spectra The prime spectrum of a commutative ring and the Zariski topology 2 Affine Schemes Affine schemes defined by the structure sheaf on the spectrum of a ring 3 Schemes The definition of a scheme as a locally affine locally ringed space 4 Topological Structure of Schemes Generic points, Zariski topology, and irreducible components 5 Algebra of Schemes Definitions and properties of reduced schemes and integral schemes 6 Morphisms of Schemes Four perspectives on scheme morphisms as morphisms of locally ringed spaces 7 Properties of Scheme Morphisms Basic properties of scheme morphisms such as affine, finite, and finite type 8 Valuation Rings Valuative criteria for separatedness and properness 9 Flat Morphisms Flat morphisms in algebraic geometry 10 Fiber Product of Schemes Definition and existence of fiber products in the category of S-schemes 11 Closed Subschemes Closed subschemes and vanishing schemes defined by ideal sheaves 12 Projective Schemes The Proj construction from a graded ring and projective schemes 13 Complete Intersections Codimension of vanishing schemes and complete intersections 14 Dimension The definition of scheme dimension and its relation to Krull dimension of local rings 15 Algebraic Groups Algebraic group action