TITLE: Prime spectrum of a ring, understanding geometry
QUESTION [18 upvotes]: I am doing some exercises out of Atiyah & Macdonald these days. Doing the exercises isn't the problem, but I am having trouble understanding geometric about $\text{Spec}(A)$. 
Consider $A=\mathbb{C}[X,Y]/(Y^2 - X^3 + X + 1)$ or something. Are the elements of $\text{Spec}(A)$ (prime ideals of $\mathbb{C}[X,Y]$ containing $P(X,Y)=Y^2 - X^3 + X + 1$) supposed to be "points on the curve" $0 =  Y^2 - X^3 + X + 1$ or something? I can see why $\text{Spec}(\mathbb{C}[X])$ is an affine line over $\mathbb{C}$ (plus another "generic" point), but I haven't been able to conceptualise what's going on in general. Basically, the exercises keep coming back to commutative algebra, but I am not seeing the geometic ideas behind it. 
Here's my question: could someone point me to some elementary exercises that help interpret the geometry of $\text{Spec}(A)$ for the ring $A$ above?

REPLY [5 votes]: I urge you to look at Chapter 4 of Vakil's "Foundations of Algebraic Geometry" (http://math.stanford.edu/~vakil/216blog/). It has very well though out exercises that will help you develop an intuition for what $Spec(A)$ for a commutative ring $A$ is. One of the reasons for the introduction of $Spec(A)$ was to be able to deal with nilpotent elements of a ring (which give rise to functions that vanish on all of the space but are yet to zero). Ravi discusses this in chapter 4 as well as chapter 5 (visualizing nilpotents). You should read this topic if you get a chance.
But, to see how it relates to geometry in the sense of classical affine algebraic geometry (basically the closed points of the spectrum of finitely generated algebras over algebraically closed fields), it might be useful to read other sources too such as Fulton's "Algebraic Curves" or the first chapter of Hatshorne's "Algebraic Geometry".
But, try to work through chapter 4 of Vakil's FOA systematically. It is a gem. If I remember correctly, in particular it contains exercises for the specific case you mention.