Written at some point of time, for many purposes, but mainly so that I could understand the material better. (There will be typos! If you ever read these and find one, I would be grateful if you'd send me an email.)

In grad school:

• Flat morphisms are open
• A quick introduction to deformation theory
• Coherence of higher direct images under proper maps
(Hartshorne only does this in the projective case.)
• Universal extension for Ext groups

In undergradute:

• Irreducible representations of complex semisimple Lie algebras
• The Weyl character formula
• Quadratic reciprocity from Artin reciprocity