  We provide optimal order pressure error estimates for the Crank-Nicolson
semidiscretization of the incompressible Navier-Stokes equations. Second order
estimates for the velocity error are long known, we prove that the pressure
error is of the same order if considered at interval midpoints, confirming
previous numerical evidence. For simplicity we first give a proof under high
regularity assumptions that include nonlocal compatibility conditions for the
initial data, then use smoothing techniques for a proof under reduced
assumptions based on standard local conditions only.
