I have been doing a project about $\operatorname{Ext}^n$ functors for my commutative algebra class. I used the approach via extensions of degree n. Basically I have shown the long exact sequence associated to a short exact sequence for $\operatorname{Hom}_{\mathcal{A}}$ where $\mathcal{A}$ is an abelian category.
To apply this construction I have proven the Chinese remainder theorem as a "toy" example. I would like to find applications in algebraic geometry that are not very technical. I feel confident with the language of sheaves but my general knowledge of algebraic geometry is very poor.
Could someone please point to some references where I can look?