Dear Roy Smith: Thanks a lot for your explanation about using blowing up so that Bertini (the form given in Hartshorne) can be applied.

Dear Karl Schwede: Thanks a lot for your answer. My question about the curve is indeed a very dumb one.

The version in Hartshorne requires $X$ has at most a finite number of singular points and that $X$ projective (or equivalently, projective with a finite number of points removed). Do you have a more general form in mind? Also, your answer leads to another question (probably a dumb one that I cannot think of): curves are parametrizable, i.e. any segment on a curve is an image of a non-singular curve?

Serge Lang has an algebra book that uses the categorical way of thinking. You should give it a look.

@Alison Miller: Could you please elaborate on what you said: "I learned the material out of Cassels and Frohlich mostly, but if I had to choose a book for someone interested in taking the local-first route I'd probably suggest Neukirch's /Algebraic Number Theory/ instead."? Why do you think Neukirch is a better choice?

Thanks. I'm aware of this option, but it sounds a bit restrictive. Is there any other way?

Dear Dan: thanks for your comment. I agree that the question is somewhat vague. I am of course not asking for an isomorphism between them. I am just wondering if there is any relationship between them.

Thanks. I'm aware of this situation. I was just wondering if the two are related somehow, not exactly isomorphic.

True. But using this, we get a strong result: it seems to work for Noetherian locally ringed space of dimension 1 in general.

Yes. I'm myself doing it (undergrad). After reading Atiyah & MacDonald carefully (doing most exercises), Hartshorne is kind of a revelation of what all these commutative algebra is for. I can't believe I got -2 for this!