I am reading the section on Weil divisors in Vakil's FOAG, where he defined a sheaf $\mathcal{O}(D)$ for a Weil divisor $D$ on a normal integral Noetherian scheme $X$ that is regular in codimension 1 via $$\mathcal{O}(D)(U):=\{t \in K(X)^× : div |_Ut + D|_U ≥ 0\} ∪ \{0\},\forall U\subset X\text{ open.}$$ Here $div |_Ut$ means take the divisor of t considered as a rational function on $U$. My question is: what is the abelian group structure of this $\mathcal{O}(D)(U)?$
My guess is the multiplication of rational functions. But then I cannot find the unit of $\mathcal{O}(D)(U),$ as the $1\in K(X)^{\times}$ may not lie in the set when $U$ intersects $\operatorname{Supp}D.$ (Does the added $0$ means an added unit?) Moreover, if $t,s\in K(X)^{\times}$ with $div |_Ut\geq -D$ and $div |_Us\geq-D,$ why can we say that $div |_U(ts)=div |_Ut+div |_Us\geq-D?$ So I get confused and hope that someone could help me. Thanks in advance.
Isn't it the addition of rational functions? if $\operatorname{div}|_U t+\mathrm{D}|_U \geq 0$ and $\operatorname{div}|_U s+\mathrm{D}|_U \geq 0$, then either $s+t = 0$ or $\operatorname{div}|_U (s+t)+\mathrm{D}|_U \geq 0$. Maybe writing the definition as
$$ \mathcal{O}(D)(U):=\{t \in K(X): t = 0 \text{ or } \operatorname{div} |_Ut + D|_U ≥ 0\},\forall U\subset X\text{ open.} $$ is cleaner? Maybe see 13.5.5 and 13.5.7 in Vakil's FOAG too? It's related to the definition of DVR.