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Let $X$ be a scheme, and $F,G$ two invertible sheaves on $X$. Let $\lambda : F\rightarrow G$ be a morphism which is an isomorphism on a dense open subset of $X$ containing all depth 0 points. By locally picking bases for $F,G$, we obtain a morphism $$\mathcal{O}_X\cong F\stackrel{\lambda}{\rightarrow}G\cong \mathcal{O}_X$$ which defines an element "$s\in\mathcal{O}_X$", which in turn determines an effective Cartier divisor $D\subset X$ (using the depth 0 condition) which does not depend on the initial choice of local bases for $F$ and $G$.

By reducing to the affine case and working locally, one can explicitly come up with an isomorphism $$F\otimes\mathcal{O}_X(D)\cong G$$ but (at least in the way I've worked it out), it feels somewhat ad hoc and is not very satisfying.

Is there a better/cleaner "functorial" way to define such an isomorphism?

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  • $\begingroup$ Tensor with $G^{-1}$ to get a map $G^{-1}\otimes F\to \mathcal{O}_X$. Check that this map is a injection of sheaves, the image thus is an ideal sheaf, which by local checking can easily seen to be $\mathcal{O}_X(-D)$ for an effective (it is an ideal sheaf!) Cartier divisor $D$. Thus $G^{-1}\otimes F\cong \mathcal{O}_X(-D)$. $\endgroup$ – Mohan May 4 '17 at 22:18
  • $\begingroup$ If you can construct a canonical map in one direction or another, and then prove it's an isomorphism when restricted to affine subsets, then that map is globally an isomorphism. $\endgroup$ – Daniel Schepler May 4 '17 at 22:18

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