# Relative Picard scheme, relative cotangent sheaf

Let $$X \to A$$ be a smooth, proper morphism of schemes, and let $$\Omega_{X/A}^1$$ denote the relative cotangent sheaf of the total space $$X$$ as a scheme over $$A$$, where we assume that $$A$$ is a one-point space. Suppose $$\mathcal{L} \subseteq \Omega_{X/A}^1$$ is a invertible subsheaf of the relative cotangent sheaf.

Is $$\mathcal{L} \in \operatorname{Pic}_{X/A}(A)$$?

In other words, is the line subbundle $$\mathcal{L}$$ an element of the $$A$$-points of the relative Picard scheme of $$X$$ as a scheme over $$A$$.

Moreover, suppose that $$\operatorname{Pic}_{X/A}(A)$$ is isomorphic to the Picard group of the fiber $$X \times_A k$$.

Would this isomiorphism imply that any line bundle of $$\operatorname{Pic}_{X/A}(A)$$ is a trivial extension of a corresponding line bundle in $$\operatorname{Pic}(X \times k)$$?