This would be an elementary question and sorry if this is duplicate one - but I could not find any satisfactory answer anywhere else. :-(
I'm learning algebraic geometry not for its own but for the application to other fields, like algebraic topology. As I continue to study, I got an impression that algebraic geometry largely can be divided into, roughly speaking, something with variety and something with schemes.
I know that varieties are closely related with schemes and vice versa, and how do they relate to each other. But I got an impression that these belong to separate subfields of study. Is this really so?
May I consider these "variety approach" and "scheme approach" as separate things and safely concentrate on just one, say the scheme approach?
These two approaches seem to use somewhat different languages. Study on varieties seems to be involved with hard commutative algebra while scheme approach is not. I even could read some papers on algebraic geometry and makes sense out of it while not being familiar with commutative algebra yet. Just some exposure and experience with categorical language was enough.
Can anyone clarify these, especially the difference between "variety approach" and "scheme approach" and what these are aimed for?
(I think I know what I want from algebraic geometry and it's very scheme-oriented. But I'm not sure how much time and efforts are worth to spend on studying varieties.)