EDITED: What is the correspondence between the associated sheaves and subschemes of Weil prime divisors?

Question

I'm interested in the correspondance between the Picard group and the Weil group. Over a sufficiently well-behaved scheme $X$, like a projective curve or if $X$ is regular, these are isomorphic. The group homomorphism in general goes from sheaves to divisors and the reverse direction seems a bit opaque to me. I feel like it would really help my understanding of the topic to know what kind of sheaf a prime divisor in the Weil group corresponds to in general.

My guess (and little more than that): A prime divisor $P$ is effective and therefore corresponds to a closed subscheme of $X$ (I guess it is also defined this way too). This will have a structure sheaf $C$ and $i_*O_C$ is an $O_X$-module where $i$ is the natural embedding. It would seem very sad to me if $O(P) = i_*O_C$ did not hold. So in particular if we are working over a projective curve $O(P)$ is the skyscraper sheaf supported at $P$. $O(-P)$ is the sheaf of ideals for the subscheme $P$ which seems to support this theory.

So a trio of interdependent questions:

• Is the above guess correct and how can it be made rigorous?
• Is this true for all effective divisor? So is $O(P+Q)$ the structure sheaf of the disjoint union of two subschemes? What about $O(2P)$?

EDIT +2hrs: I've become aware of an extremely illuminating SE answer by Georges Elencwajg which provides a partial answer to this question. (I think maybe the word "pole" there should be "zero" though?) I still find it rather sad that there's no direct link to $i_*O_C$. My question becomes, I guess, what, if any, connection exists between the subsheaf of rational functions associated to a Weil divisor described in this answer and $i_*O_C$?