  We study the Arnold diffusion in a priori unstable near-integrable systems in
a neighbourhood of a resonance of low order. We consider a non-autonomous
near-integrable Hamiltonian system with $n+1/2$ degrees of freedom, $n\ge 2$.
Let the Hamilton function $H$ of depend on the parameter $\varepsilon$, for
$\varepsilon=0$ the system is integrable and has a homoclinic asymptotic
manifold $\Gamma$. Our main result is that for small generic perturbation in an
$\varepsilon$-neighborhood of $\Gamma$ there exist trajectories the projections
of which on the space of actions cross the resonance. By ``generic
perturbations'' we mean an open dense set in the space of $C^r$-smooth
functions $\frac{d}{d\varepsilon}\big|_{\varepsilon=0} H$,
$r=r_0,r_0+1,\ldots,\infty,\omega$. Combination of this result with results of
\cite{DT} answers the main questions on the Arnold diffusion in a priori
unstable case: the diffusion takes place for generic perturbation, diffusion
trajectories can go along any smooth curve in the action space with average
velocity of order $\varepsilon/|\log \varepsilon|$.
