Divisor support positive negative parts

Let $k$ be a field and let $C$ be a proper curve over $k$, so that $C$ is an integral proper $k$-scheme of dimension one and a divisor on $C$ is a finite integral sum $D={\sum }_x{n}_x[x]$ over the closed points (Degree divisor proper curve). For such a divisor define

1. the support $\mathrm{Supp}(D)=\{x:n_x\mathrm{\ne }0\}$, a finite set of closed points of $C$;
2. the positive part $D^{+}={\sum }_x\max (n_x,0)[x]$;
3. the negative part $D^{-}={\sum }_x\max (-n_x,0)[x]$, so that all coefficients of $D^{-}$ are nonnegative and $D=D^{+}-D^{-}$.

The supports of $D^{+}$ and $D^{-}$ are disjoint subsets of $\mathrm{Supp}(D)$: if $n_x>0$ then the coefficient of $x$ in $D^{-}$ is $\max (-n_x,0)=0$, and if $n_x<0$ then the coefficient of $x$ in $D^{+}$ is $\max (n_x,0)=0$. Both parts are effective divisors in the sense that all their coefficients are nonnegative, and $D$ is effective if and only if $D^{-}=0$. The same definitions apply verbatim to a Weil divisor on any integral normal locally Noetherian scheme, using prime divisors in place of closed points (Weil divisor normal noetherian scheme), and they are used on this page only for divisors on a curve, where the finite-support convention makes all three sums finite without further hypotheses.

Normality of $C$ is not required for the construction: the closed points of $C$ and the integers $n_x$ are the only data used, and the identity $D=D^{+}-D^{-}$ together with the disjointness of the two supports is a coefficientwise statement.