Deformation of quotients on a product

We consider the general problem of deforming a surjective map of modules $f : E \to F$ over a coproduct sheaf of rings $B=B_1 \otimes_A B_2$ when the domain module $E = B_1 \otimes_A E_2$ is obtained via extension of scalars from a $B_2$-module $E_2$. Assuming $B_1$ is flat over $A$, we show that the Atiyah class morphism $F \to \LL_{B/B_2} \otimes^{\bL} F[1]$ in the derived category $D(B)$ factors naturally through (the shift of) a morphism $\beta : \Ker f \to \LL_{B/B_2} \otimes^{\bL} F$. We describe the obstruction to lifting $f$ over a (square zero) extension $B_1' \to B_1$ in terms of $\beta$ and the class of the extension. As an application, we use the reduced Atiyah class to construct a perfect obstruction theory on the Quot scheme of a vector bundle on a smooth curve (and more generally).