Let $X$ be a Noetherian scheme. My question is that: for any Cartier divisor $D$, can we write it as $D_1-D_2$ where $D_1$,$D_2$ are effective? What about further assume $X$ is integral?
I can see $D$ is locally generated by fractions of sections of $O_X$. But it seems not easy to depart the denominators and numerators explicitly. Could you provide some help? Thanks.
Assume $X$ has an ample (Cartier) divisor class $H$ (equivalently, an ample invertible sheaf). Then for each Cartier divisor $D$ there exists an integer $n \gg 0$ such that the linear systems $|nH|$ and $|D + nH|$ contain effective (Cartier) divisors, say $D_1$ and $D_2$. Then $$ D_2 - D_1 = (D + nH) - nH = D. $$