When tensoring by a vector bundle is exact?

Question

Let $D=C_1+C_2$ be two projective curves, $C_1\cap C_2=p$. Consider the following exact sequence

$0\rightarrow \mathcal{O}_{C_2}(-p)\rightarrow \mathcal{O}_D\rightarrow \mathcal{O}_{C_1}\rightarrow 0 $.

Let $F$ be a vector bundle. Is it true in general that the sequence

$0\rightarrow \mathcal{O}_{C_2}(-p)\otimes F\rightarrow \mathcal{O}_D\otimes F\rightarrow \mathcal{O}_{C_1}\otimes F\rightarrow 0 $

is also exact?

Answer 1

A vector bundle is, locally, a projective module and projective modules are flat. Now exactness can be checked locally, so...