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
Suggested books and references:
Robin Hartshorne, Algebraic geometry, Graduate Texts in Mathematics, No. 52. Springer-Verlag, New York-Heidelberg, 1977.
Robin Hartshorne, Residues and duality, Lecture notes of a seminar on the work of A.~Grothendieck, given at Harvard 1963/64. With an appendix by P.~Deligne. Lecture Notes in Mathematics, No. 20 Springer-Verlag, Berlin-New York 1966.