  We study a rotating atomic Fermi gas near a narrow s-wave Feshbach resonance
in a uniaxial harmonic trap with frequencies $\Omega_\perp$, $\Omega_z$. Our
primary prediction is the upper-critical angular velocity, $\omega_{c2}
(\delta,T)$, as a function of temperature $T$ and resonance detuning $\delta$,
ranging across the BEC-BCS crossover. The rotation-driven suppression of
superfluidity at $\omega_{c2}$ is quite distinct in the BCS and BEC regimes,
with the former controlled by Cooper-pair depairing and the latter by the
dilution of bosonic molecules. At low $T$ and $\Omega_z\ll\Omega_\perp$, in the
BCS and crossover regimes of $0 \lesssim \delta \lesssim \delta_c$,
$\omega_{c2}$ is implicitly given by $\hbar \sqrt{\omega_{c2}^2
+\Omega_\perp^2}\approx 2\Delta \sqrt{\hbar \Omega_\perp/\epsilon_F}$,
vanishing as $\omega_{c2} \sim\Omega_\perp(1-\delta/\delta_c)^{1/2}$ near
$\delta_c\approx 2\epsilon_{F} + \frac\gamma 2\epsilon_{F}
\ln(\epsilon_F/\hbar\Omega_\perp)$ (with $\Delta$ the BCS gap and $\gamma$
resonance width), and extending bulk result $\hbar\omega_{c2} \approx
2\Delta^2/\epsilon_{F}$ to a finite number of atoms in a trap. In the BEC
regime of $\delta < 0$ we find $\omega_{c2} \to\Omega^-_\perp$, where molecular
superfluidity can only be destroyed by large quantum fluctuations associated
with comparable boson and vortex densities.
