Hello everybody,

I'm a math student who has just got his first degree, and I am studying algebraic geometry since a few months. Something I have noticed is the (to my eyes) huge amount of commutative algebra one needs to push himself some deeper than the elementary subjects. This can be seen just counting the "A's lemmas" on Hartshorne's book or even more by looking in other books where many purely algebraic results are given with references to the proof.

The problem arises more evidently when I try to do things on my own and in particular in the exercises. Many exercises I saw required some fact about commutative algebra which, it happened, I either didn't know or I knew just "randomly" by experience or from my university class about commutative algebra (I attended the most advanced one in my university, but unfortunately it is not enough).

My question is, how should one do to feel "free" from any reasonably elementary gap about commutative algebra? Just take the references books like those by Eisenbud, Zariski-Samuel or even Bourbaki, and study them from the beginning to the end, sounds quite huge a mission. However I am actually skeptical about just looking at the results one by one, since there is the risk to never learn those background notions properly.

Hope my question is proper on this forum,

thank you in advance for your opinion!