I have taken two algebraic geometry classes, one classical and one modern. Unfortunately, I feel like I did not properly learn everything from both of them; I should have put in more time at home to make sure I was understanding all of the proofs from class.
Because of this, I am looking for a text source that I can relearn things from, and then continue on learning even more things. I am between two sources: Vakil and EGA. I was wondering if any of you can help me decide between the two or suggest a third.
Here is the state of my knowledge:
- I remember definitions fairly well. I have no trouble defining a scheme or a morphism of schemes or anything like that.
- I feel pretty comfortable with sheaf cohomology, in the sense that I now the definitions and could probably prove most things on my own. I started to struggle in class once we started applying cohomology to schemes (as opposed to working over a general topological space). Like I can define a quasicoherant sheaf, but cannot prove much about them (like the fact that their cohomology vanishes when the scheme is affine).
- I feel decent about my commutative algebra background for the most part. But a little review never hurt.