This page contains notes on various topics in algebraic geometry and commutative algebra that I've written for independent studies. They are aimed at undergraduates who are familiar with the basics of varieties, rings, etc.

The projective plane

The projective plane over an arbitrary field; homogeneous coordinates; affine patches; lines at infinity; homogenization and homogeneous polynomials; varieties in the projective plane; projective closure.

Modules and chain complexes

Modules; free modules; restricting scalars; module homomorphisms; submodules; kernels and images; chain complexes; exact sequences; examples.

Free resolutions

Free resolutions; examples.

Graded rings and modules

Internal direct sums; graded rings; graded modules; homogeneous ideals; graded quotient rings; graded and homogeneous linear maps; shifts; external direct sums.

Graded free resolutions

Graded free resolutions; examples; Hilbert's Syzygy Theorem; historical comments; minimal graded free resolutions; syzygies.

Hilbert functions and series

The Hilbert function, polynomial, and series of a polynomial ring; Hilbert series in general; computing Hilbert functions and series from graded free resolutions.