  We study the Cauchy problem for the Zakharov system in spatial dimension
$d\ge 4$ with initial datum $(u(0), n(0), \partial_t n(0)) \in
H^k(\mathbb{R}^d) \times \dot{H}^l(\mathbb{R}^d)\times
\dot{H}^{l-1}(\mathbb{R}^d)$. According to Ginibre, Tsutsumi and Velo, the
critical exponent of $(k,l)$ is $((d-3)/2,(d-4)/2)$. We prove the scattering
and the small data global well-posedness at the critical space. It seems
difficult to get the crucial bilinear estimate only by applying the $U^2,\ V^2$
type spaces introduced by Koch-Tataru. To avoid the difficulty, we use an
intersection space of $V^2$ type space and the space-time Lebesgue space
$L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.
