# What to study from Eisenbud's Commutative Algebra to prepare for Hartshorne's Algebraic Geometry?

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?

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

"Is this course sufficient to prepare myself for Hartshorne's Algebraic Geometry? Or do I need to study more chapters?"
You need to to study fewer chapters, the exact number being (to first order approximation) zero.

What I mean is that Hartshorne uses a very restricted number of results in commutative algebra:
Hilbert's Nullstellensatz, Krull's principal ideal theorem, characterization of factorial rings by principality of height one primes, finiteness of integral closure and a few other theorems.
These results are very important and very hard in the sense that no student having just learned the relevant definitions could find the proof himself.
However you can then take these grandiose theorems on faith, as black boxes, or check their proofs (maybe later, at your leisure) in Eisenbud or other books on commutative algebra, like Zariski-Samuel or Matsumura.
But there is absolutely no need to read the 355 pages comprising the first 14 chapters of Eisenbud: they are very interesting but studying them seriously is not at all required and would actually prevent you from learning genuine algebraic geometry for a very long time.

I would advise you to very thoroughly study the first few chapters of Atiyah-Macdonald in order to familiarize yourself with the basic concepts, browse through the rest of the book and simultaneously start Chapter 1 in Hartshorne.
Godspeed!

• Excellent answer, Georges. I don't think I knew what a local ring was when I started learning algebraic geometry. – user64687 Jun 30 '14 at 7:30
• Dear @Asal: judging from the quality of your answers, you certainly made up for that initial lack of knowledge ! – Georges Elencwajg Jun 30 '14 at 9:19
• This is very helpful. Thank you. – PeterM Jul 8 '14 at 22:08

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.