TITLE: What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?
QUESTION [10 upvotes]: I surveyed commutative algebra texts and found Eisenbud's "Commutative Algebra: With a View Toward Algebraic Geometry" to be the most accessible for me. The book outlines a first course in commutative algebra in the introduction. The course uses most of the material in chapters 1 to 14. Is this course sufficient to prepare my self for Hartshorne's Algebraic Geometry? Or do I need to study more chapters?
You can find the book's table of contents here.
Also, the first chapter in the book "Roots of Commutative Algebra" is a survey of a wide range of topics. Can I safely skip most of it as indicated by the chapter's intro?
Thank you

REPLY [6 votes]: I think it would be worth reading over the first few sections of Chapter 2, on localization, carefully and doing some of the more interesting exercises. Little identifications of compound localizations are going to start to pile up. Other than that, I think a lot of the difficulty in the first chapter arises from little tricks with polynomials, which is sort of orthogonal to knowing a lot of heavy theorems.
It's hard to avoid black boxing a few things -- Hartshorne uses Kähler differentials, depth, projective dimension, etc, and IIRC those aren't in Eisenbud's "first course". I don't think one can wait that long to start learning algebraic geometry.