Let $X$ be a Noetherian, regular scheme and let $\mathcal L$ be a line bundle of $X$. Then fix a divisor $D$ on $X$. In some books I've seen the notation $\mathcal L(D)$. What is such an object?
I would say that $$\mathcal L(D)= \mathcal L\otimes\mathcal O_X(D)$$ is it correct?