1 From Varieties to Schemes From varieties to schemes 2 The Spectrum Prime spectrum and Zariski topology of a commutative ring 3 Affine Scheme The affine scheme defined by the structure sheaf on a ring’s spectrum 4 Schemes Definition of a scheme as a locally affine locally ringed space 5 Projective Schemes The Proj construction from graded rings and projective spaces 6 The Topology of Schemes Generic points, Zariski topology, and irreducible components 7 Algebraic Structure of Schemes Definitions and properties of reduced and integral schemes 8 Morphisms of Schemes Four perspectives on scheme morphisms as locally ringed space morphisms 9 Properties of Scheme Morphisms Basic properties of scheme morphisms and rational maps 10 Closed Subschemes Closed subschemes and vanishing schemes defined by an ideal sheaf 11 Closed Subschemes of Projective Space Correspondence between closed subschemes of projective space and homogeneous ideals 12 Fiber Products Definition and existence of fiber products in the category of S-schemes 13 Dimension Definition of scheme dimension and relation to Krull dimension of local rings 14 Flat Morphisms Definition, geometric meaning, criteria, and examples of flat morphisms 15 Valuation Rings Valuative criteria for separated and proper morphisms 16 Quasi-Coherent Sheaves Sheaves of modules, equivalence on affine schemes, and quasi-coherence 17 Divisors and Linear Systems Cartier and Weil divisors, the sheaf O_X(D), linear systems, and ampleness 18 Sheaf Cohomology of Schemes Cohomology of quasi-coherent sheaves, Serre vanishing, the cohomological criterion for ampleness, and Hilbe... 19 Differentials and Cotangent Sheaves Kahler differentials, cotangent sheaves, tangent sheaves, Euler sequences, and canonical sheaves 20 Complete Intersections Codimension of local complete intersections, Koszul resolutions, and Hilbert polynomials 21 Smooth and Étale Morphisms Smooth, unramified, and étale morphisms; Jacobian and infinitesimal lifting criteria 22 Deformation Theory and the Cotangent Complex Square-zero extensions, first-order deformations, obstructions, and the cotangent complex 23 Functor of Points Functor of points, Yoneda embedding, representability, fiber products 24 Faithfully Flat Descent Descent data, the cocycle condition, fpqc topology, and effective descent 25 Group Schemes Group schemes, Hopf algebras, comodules, and torsors