These notes bring together exercises 14 through 18 in Chapter II, Section 5 of Hartshorne’s Algebraic Geometry. The emphasis is on the relationships between the constructions: normalization and section rings, coherent extension, tensor operations, relative Spec, and vector bundles.
Background
Schemes, coherent sheaves, and the material of Hartshorne II.5.
Read the full exposition
The complete argument is available as a PDF.
Revision history ·
Revised September 22, 2026: clarified the nonempty-scheme and rank-zero conventions, independence of lifts in the associated graded, and the local arguments underlying section-ring finiteness, coherent extension, relative Spec, and bundle sections. The PDF and TeX source agree.