Is Paolo Aluffi's "Algebra: Chapter 0" enough algebra preparation for a standard graduate course in algebraic geometry (using Hartshorne's text)?

Question

Does Aluffi's book have enough commutative algebra for algebraic geometry? I understand that traditional graduate algebra course using Hungerford's book or Lang's book provides enough background for such a course.

Answer 1

As someone who read Aluffi's book and then immediately took an algebraic geometry class, I can say that the book does a good job at teaching the material with an eye towards algebraic geometry. Especially notable are the homological algebra towards the end of the book and the short digression into Algebraic Geometry in chapter VII.

That said, Aluffi contains very little commutative algebra. During my Algebraic Geometry course, I had to constantly reference a commutative algebra textbook (I used Atiyah-Macdonald). Many things can be black-boxed, and the few that can't (e.g. localization) are not so bad to pick up.

So, if the question is if there's enough commmutative algebra, the answer has to be no. But if the question is how much algebra in general, I would say it is a conditional yes, as long as you know what you're getting into. First, you'll be doing a nontrivial bit of extra work with commutative algebra, and second, you'll probably want to do a more systematic study of commutative algebra and then revisit the material after.