Recently I am quite confused about invertible sheaves while reading Hartshorne.
In Chapter 2, exercise 5.1, it is discussed that for a locally free $O_X$-module of finite rank, denoted by $\mathcal{F}$, double dual of $\mathcal{F}$ is isomorphic to $\mathcal{F}$ itself. But since $F$ locally looks like the sheaf of modules associated to ${{(Spec\space A)}}^n$(Sorry that I can't type that notation "~"). Don't we have that dual of $\mathcal{F}$ is isomorphic to $\mathcal{F}$ itself?
Another question related to this is exercise 5.7(3), again I tried to consider locally, and it seems that I can actually take $\mathcal{G}$ arbitrarily as long as it is also invertible. But after I searched on the Internet, I found that many answer said that I have to choose $\mathcal{G}$ to be the dual of $\mathcal{F}$. I guess I have formed a wrong understanding about sheaves of modules. It would be kind if you could offer some examples to let me know where I wrong. Thanks a lot!