When the kernel of a quotient of a locally free sheaf is locally free

Question

Can you please help me understand the following. Let $A$ be a locally free sheaf on a ringed space (e.g nice scheme) and $B$ some coherent sheaf with a surjection from $A$. Assuming that the homological dimension of $B$ is less or equal to $1$, (that is, for $i\geq2$ the groups $H^i(X,B)$ are equal to zero) then the kernel of the surjection $A\to B\to 0$ is also locally free.