Learning schemes Could someone suggest me how to learn some basic theory of schemes? I have two books from algebraic geometry, namely "Diophantine Geometry" from Hindry and Silverman and "Algebraic geometry and arithmetic curves" from Qing Liu. I have had difficulties to prove the equivalence of many definitions. For example Hindry and Silverman defines an affine variety to be an irreducible algebraic subset of $\mathbb{A}^n$ with respect to Zariski topology. On the other hand, Liu defines an affine variety to be the affine scheme associated to a finitely generated algebra over a field.

I am also currently learning about sheaves and schemes, and I'm finding Ravi Vakil's notes to be very helpful:
http://math216.wordpress.com/
