I was reading through Katz-Mazur "Arithmetic Moduli of Elliptic Curves", Chapter 1, and ran into some small issues (which might have a lot to do actually with notation). I think most of them are due to not being comfortable enough with the algebraic geometry notions involved. I managed to read on ignoring them but they are still bugging me. I apologize if the questions seems somewhat contentless.
Here is a link to the book. Below are my questions.
$C/S$ is a smooth curve over $S$ and $D$ is an effective Cartier divisor.
(1) Does $\mathcal O_D$ mean the structure sheaf of $D$ as a scheme, or is it $\mathcal O_C/I(D)$ as a sheaf on $\mathcal O_C$? I know they are roughly the same, but I am not comfortable enough in my algebraic geometry yet to treat them as the same (if I am wrong, please point out). In the latter case, (1') would the structure sheaf of $D$ be $i^*(\mathcal O_D)$, where $i$ is the inclusion $D\hookrightarrow X$, is that correct? (1'') Otherwise, if $\mathcal O_D$ meant the structure sheaf on $D$, would $\mathscr L\otimes_{\mathcal O_C}\mathcal O_D$ mean then $\mathscr L\otimes_{\mathcal O_C}i_*(\mathcal O_D)$?
(2) On page 9, we have
$\mathscr L/\mathcal O\cong I^{-1}(D)/\mathcal O\cong I^{-1}(D)\otimes_{\mathcal O_C}\mathcal O_D\cong\mathscr L\otimes_{\mathcal O_C}\mathcal O_D=$ an invertible $\mathcal O_D$-Module. Therefore $D$ is proper over $S$ if and only if the sheaf $\mathscr L/\mathcal O$ has its support proper over $S$.
Why is this so?
(3) Here $X$ is any scheme over $S$ and $D,D'$ effective Cartier divisors. On page 13,
Globally, in terms of representatives $(\mathscr L,\ell)$ for $D$ and $(\mathscr L,\ell')$ for $D'$, the condition $D'\le D$ is precisely that the global section $\ell$ of $\mathscr L$ vanish identically in $\mathscr L|D'=\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$.
Ok, so firstly here $\mathscr L|D'=\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$ really bugs me. From what I know, restriction of a sheaf to a subscheme is via pullback of the inclusion map and it gives a sheaf on the subscheme. So then the LHS would be a sheaf on $D'$ while the right hand side is a sheaf on $X$. There is an exact sequence below involving such terms and it is definitely meant as an exact sequence of sheaves on $X$, so... what exactly does $\mathscr L|D'$ denote here?
Secondly (3') What does it mean that $\ell$ vanishes in $\mathscr L\otimes_{\mathcal O_X}\mathcal O_{D'}$ and why is this implies by $D'\le D$? I would be grateful if someone could spell this out in some detail. I don't have any issue understanding the discussion of $D'\le D$ above in terms of their ideal sheaves but somehow I can't translate this into $(\mathscr L,\ell)$ language.
Thank you for any illuminating answers/comments.