# Locally free sheaf tensor skyscraper sheaf

Let $X$ be a smooth projective curve over $\mathbb C$ and let $p \in X$. Let $\mathcal F$ be a locally free sheaf of rank 1 over $\mathcal O_X$. Consider the skyscraper sheaf $\mathbb C_p$ at $p$. Is it true there is a natural isomorphism of $\mathcal O_X$-modules: $$\mathcal F \otimes_{\mathcal O} \mathbb C_p \cong \mathbb C_p$$ Certainly the stalk of the tensor sheaf away from $p$ is 0 and the stalk at $p$ is $$\mathcal F_p \otimes_{\mathcal O_p} \mathbb C \cong \mathcal O_p \otimes_{\mathcal O_p} \mathbb C \cong \mathbb C,$$ but it seems to me that the first isomorphism requires a choice of local trivialization of $\mathcal F$ near $p$. And it doesn't seem like this isomorphism is independent of local trivialization near $p$.

• It does depend on a local trivilaization. – Mohan Jul 1 '16 at 19:56
• You can change the trivialization by multiplying by a nonzero complex number, so there's certainly a choice. – Hoot Jul 1 '16 at 19:57