Why do some authors define the invertible sheaf $\mathcal{L}(D)$, associated to a Cartier divisor $D = (U_{i}, f_{i})_{i\in I}$, by setting $\mathcal{L}(D)_{U_{i}} = \mathcal{O}_{X|U_{i}}\cdot f_{i}^{-1}$, instead of $\mathcal{L}(D)_{U_{i}} = \mathcal{O}_{X|U_{i}}\cdot f_{i}$. Both $\mathcal{O}_{X|U_{i}}$-modules, $\mathcal{O}_{X|U_{i}}\cdot f_{i}^{-1}$ and $\mathcal{O}_{X|U_{i}}\cdot f_{i}$ are isomorphic, but is there a deep reason to invert $f_{i}$?
Best Answer
The basic reason for this is that it fits in nicely with other places in algebraic geometry where we write something in parentheses next to a sheaf or a module.
On a projective scheme $X=\operatorname{Proj} R$ for $R$ a graded ring, we define the twisting sheaf $\mathcal{O}_X(1)$ to be the sheaf associated to the graded module $R(1)$, where this is the regular module under the usual shift operation: the degree $d$ portion of $R(1)$ is the degree $1+d$ portion of $R$. Under some mild niceness assumptions, this works out great for swapping between quasicoherent sheaves and their associated graded modules and one can get a lot done with this construction. One example in particular where we often meet these sheaves/modules for the first time is when $R=A[x_0,\cdots,x_n]$, where then $X=\Bbb P^n_A$, and the sheaves $\mathcal{O}_X(d)$ for positive $d$ have sections and those with negative $d$ have no sections.
In this same situation (perhaps plus assuming $X$ is factorial, a mild niceness assumption), if $D$ is effective, then this definition of $\mathcal{O}(D)$ has sections (the linear system of divisors equivalent to $D$), and $\mathcal{O}(-D)$ doesn't. In fact, as I originally mentioned in the comments, we get that $\mathcal{O}(-D)$ is the ideal sheaf of $D$, and we get an exact sequence $$0\to \mathcal{O}(-D)\to \mathcal{O}_X\to \mathcal{O}_D\to 0$$ which is pleasing: from our experience with $\mathcal{O}(d)$ on $\Bbb P^n_A$, we expect maps from $\mathcal{O}(d)\to\mathcal{O}(e)$ iff $d\leq e$, and sections of $\mathcal{O}(d)$ when $d\geq 0$ and no sections when $d<0$. This makes $\mathcal{O}(D)$ behave like what we're already used to, which is a nice aide to doing mathematics.