Line bundle on projective $A$-scheme is the difference between two very ample line bundles

Question

Let $\mathcal{L}$ be an invertible sheaf on a projective $A$-scheme $X$.

Then we can always find two very ample invertible sheaves such that $$\mathcal{L}= \mathcal{M} \otimes \mathcal{N}^*$$

(here * stands for the dual).

By a projective $A$-scheme I mean a scheme X that is isomorphic to a closed subscheme of $\mathbb{P}^n_A$, for some $n.$

I have tried using the line bundle associated with the closed embedding to the projective space, which would be very ample, but I don't know how to do anything meaningful with it.

Any help, please?

Answer 1

I have a feeling for a proof: consider the line bundle associated to the projective embedding $\mathcal{O}(1)$. Then for every line bundle $\mathcal{L}$, the bundle $\mathcal{O}(n) \otimes \mathcal{L}$ will be very ample if $n$ is large enough. So now you have two very ample bundles $\mathcal{O}(n)$ and $\mathcal{O}(n) \otimes \mathcal{L}$, and you take the quotient of them to get $\mathcal{L}$.