Pseudoforce and friction: confusion Consider the following situation. A block of mass $M$ is resting on a rough horizontal ground, and a frame is moving towards the right horizontally with an acceleration $a$. Suppose the coefficient of static friction on the ground on which the block is standing is $ \mu > a/g$.
Clearly the block is moving towards the left w.r.t. the frame with acceleration $a$. Suppose we try to explain this from the non-inertial frame. Then we add a pseudoforce $Ma$, towards the left, on the block. But observe that $Ma < \mu Mg$ as $\mu g > a$. So the block shouldn't move, due to static friction. 
How is this possible?
 A: The block is accelerating, but that's due to the pseudo force, not the frictional force, which is zero.

You can't just look at the horizontal forces on the block to determine whether or not static friction will hold. You have to consider what's going on with the other surface as well.
Here's a simpler situation to consider first in an inertial frame. Imagine two identical blocks stacked on top of each other. There is friction between the two blocks, but there is no friction between the lower block & the horizontal surface. Now push on the upper block while holding the lower block in place. It will slide if $F_\text{applied} > \mu M g$, as you describe. But if you instead push on each block with identical forces, they will move together and there will be no frictional force (because alone they would have identical motions). Here, the top block is accelerating even though the frictional force is zero; it's the applied force that's causing it to accelerate.
Same thing with your situation. The block and surface alone would have the same motion as each other in the non-inertial frame, so there will be no frictional force. Yes, the block is accelerating, but that's due to the pseudo force, not the frictional force, which is zero.
