I have a question from R. Vakil's lecture notes: chapter 14 section 3 on Effective Divisors and Invertible sheaves. It says something like given an effective divisor on a scheme $X$, we may define the ideal sheaf corresponding to $\mathscr{I}$ be $\mathcal{O}(-D)$ in the exact sequence
$$ 0\rightarrow \mathcal{O}(-D)\rightarrow \mathcal O_{X}\rightarrow \mathcal{O}_{D}\rightarrow 0$$
Then he defines the canonical section $s_{D}$ be as follows: tensoring with $\mathcal{O}(D)$ in the sequence, we get a morphism of sheaves
$$ \mathcal{O}_{X}\rightarrow \mathcal{O}(D)$$
which gives us a canonical section. I don't understand what this means, so I presume that if we take the global section, we have a mapping of $A$-modules (all rings are denoted by $A$)
$$ \mathcal{O}_{X}(X)\rightarrow\mathcal{O}(D)(X)$$
sending $1$ to some element $f\in \mathcal{O}(D)(X)=\mathcal{O}(-D)^{\vee}(X)=\text{Hom}_{\mathcal{O}_{X}}(\mathcal{O}(-D)|_{X},\mathcal{O}_{X}|_{X})$
So based on this observation, I need to show
Ex 14.3.B: show that $Z(s_{D})$ is exactly the subscheme cut out by $D$.
But an effective divisor in the sense of R. Vakil's notes is actually an effective Cartier divisor. In some other sources (See Götz-Wedhorn Algebraic Geometry I, page 305, Remark 11.31(2)) the canonical divisor $s_{D}$ is defined to be as follows: since $D$ is effective Cartier, there exists $(U_{i},f_{i})_{i}$ where $U_{i}$ is an affine covering of $X$ and $f_{i}\in \Gamma(U_{i},\mathscr{K}^{*}_{X})$
Question 1: If $D$ is effective but $X$ is NOT assumed to be reduced, can we still assume that $f_{i}\in\Gamma(U_{i},\mathcal{O}_{X})$? Because if it is not reduced, we cannot use Algebraic Hartog's lemma...
Then in Götz-Wedhorn, they define the canonical section $s_{D}$ to be exactly the data $(U_{i},f_{i})$. Or so it seems to me because it was written
Let $V$ be any open subset of $X$. Then $\Gamma(V,\mathcal{O}(D))$ consists of $s\in\Gamma(V,\mathscr{K}_{X})$ such that $sf_{i}\in\Gamma(V\cap U_{i},\mathcal{O}_{X})$. Then $s_{D}$ corresponds to $1\in \Gamma(X,\mathscr{K}_{X})$.
(Here, we assume that $X=U_{eff}$ since $D$ is an effective divisor.)
Then it is clear from here that the zeros of the canonical section $s_{D}$ is defined by $V(f_{i})$ in $U_{i}$, which is exactly the closed subscheme as by the definition of an effective Cartier divisor (in the sense of R. Vakil).
Question 2: How is this related to the $s_{D}$ in the definition in R. Vakil's notes? If they are related, then I can answer the exercise question straightaway. Otherwise, how should the definition be used to answer the question?
