Sheaves of differentials.
Background on homological algebra : resolutions, derived functors, $\delta$-categories.
Triangulated categories, Derived categories of abelian categories.
Injective and flasque resolutions.
Cohomology of sheaves of abelian groups
Vanishing theorems for cohomology
Serre’s criterion for affineness
Cohomology of projective space, twisting by Serre sheaves
$Ext$ and $Tor$ for sheaves
Serre duality theorem
Schemes as functors of points, the idea of stacks