Algebraic Geometry II: Schemes (Berkeley Math 256A video lectures)
by Richard E. Borcherds · UC Berkeley
Video lectures from the second half of Berkeley's Math 256A graduate course, taught by Fields medalist Richard Borcherds. Covers sheaves, Spec of a ring, the structure sheaf, affine schemes, and how general schemes are glued from them.
More resources on Affine Schemes
Stacks Project
A collaborative, open-source reference on algebraic geometry run from Columbia University, with thousands of pages building from commutative algebra to schemes and algebraic stacks. Every result has a stable tag, so readers can look up precise statements and complete proofs.
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
The Rising Sea: Foundations of Algebraic Geometry (free PDF)
Vakil's freely hosted textbook manuscript, updated October 2025. Chapters 3 and 4 develop affine schemes from the underlying set and generic points through the Zariski topology, sheaves, and the structure sheaf, with exercises embedded throughout. The most-recommended modern schemes text.
Algebraic Geometry, Chapter 12: Schemes (Gathmann)
Self-contained chapter that builds affine schemes step by step: Spec as a set of prime ideals, then the Zariski topology, then the structure sheaf, with comparisons to varieties and gluing into general schemes.
The Stacks Project — Chapter 26: Schemes
The open-source reference's chapter constructing Spec of a ring as a locally ringed space, covering the Zariski topology, the structure sheaf, quasi-coherent sheaves on affines, and the anti-equivalence between rings and affine schemes. The primary source of record for scheme theory, and nothing free or paid matches its completeness.
The Geometry of Schemes
Graduate text by Eisenbud and Harris that bridges classical varieties and Grothendieck's scheme theory through worked examples rather than heavy machinery. Readers learn to picture affine schemes, nonreduced points, and families geometrically, with many exercises; assumes basic commutative algebra.