Hello everybody
Motivation of my question
Let $X$ be a scheme.
Given a morphism $\mathcal{L}\overset{\beta}\to\mathcal{O}_X$ of line bundles over $X$.
I want to understand under what conditions the image $\mathrm{Im}(\beta)$ is a line bundle. I will specify my question below. To avoid trivialities, lets assume the morphism is not surjective.
I already know the following results
- If $X$ is a smooth curve over a field, then every non-zero coherent $\mathcal{O}_X$-submodule of a line bundle is already a line bundle*. Because $\mathrm{Im}(\beta)$ is coherent and non-zero, we see that $\mathrm{Im}(\beta)$ is a line bundle in this case.
- If $X$ has the property, that every local ring $\mathcal{O}_{X,x}$ is a PID, then the submodule $\mathrm{Im}(\beta)_x=\mathrm{Im}(\beta_x)\subseteq \mathcal{O}_{X,x}$ is free of rank $1$, which implies that $\mathrm{Im}(\beta)$ is free, by finite presentedness of $\mathrm{Im}(\beta)$.
My question
What if $X$ is a smooth scheme over a base scheme $S$, say of relative dimension $1$, for example? I am mainly interested in the case of a morphism $\mathcal{L}\overset{\beta}{\to} \mathcal{O}_{X_S}$ (non-surjective), where $X_S=S\times_{\mathrm{Spec}{k}}X$ where $S$ is any scheme over a field $k$ and $X$ is a smooth irreducible curve over $k$. Under what conditions is the image $\mathrm{Im}(\beta)$ a line bundle?
Some of my ideas
Maybe one can prove that the restriction of $\mathrm{Im}(\beta)$ to the fibres $X_S\to \mathrm{Spec}(\kappa(s))$ are line bundles, by using the first result from above and then conclude, that $\mathrm{Im}(\beta)$ is a line bundle?
Alternatively one could try to understand the properties of the local rings $\mathcal{O}_{X_S,x}$, but I have doubts that they have any useful properties?
PS:
*This follows by the fact that the stalks are valuation rings and that torsion-free modules over valuation rings are flat by Tag 0539, and the fact that over finitely generated flat modules over a local ring are free by [Matsumura, Commutative Algebra, Prop. 3G, page 21], and a standard Argument about finitely presented modules.