In Hartshorne's book, for an scheme $X$ (noetherian integral separated and regular in codimension one) defined $Div(X)$ as the free abelian group generated by the prime divisors (closed integral subscheme of codimension one).
My doubt is if the definition in the Silverman`s book are the same for a algebraic curve.
Silverman defined $Div(X)$ as the free abelian group generated by the points of $X$.
In definition of Hartshorne not include the generic point true?
Do these groups differ?
Thanks you all.
Best Answer
If $X$ is an algebraic curve then $X$ is a Noetherian, integral, separated scheme of finite type over a field (AKA a variety) and it's of dimension 1 (curves are $1$-dimensional).
So if $X$ is $1$ dimensional, then a codimension one subvariety of $X$ is a dimension $0$ variety which is just a (closed) point.
And, in general if $X$ is a:
Etc. We want divisors to be closed subschemes if that's what you're asking. I don't know exactly what Hartshorne says but it should just be a generalization of Silverman's definition to schemes higher dimension and not-necessarily over a field.