The genericity of Arnold diffusion in the analytic category is an open problem. In this paper, we study this problem in the following a priori unstable Hamiltonian system with a time-periodic perturbation H-epsilon(p, q, I, phi, t) = h(I) + Sigma(n)(i=1) +/- (1/2 p(i)(2) + V-i(q(i))) + epsilon H-1( p, q, I, phi, t), where (p, q) epsilon R-n x T-n, (I,phi) epsilon R-d x T-d with n, d >= 1, V-i are Morse potentials, and epsilon is a small non-zero parameter. The unperturbed Hamiltonian is not necessarily convex, and the induced inner dynamics does not need to satisfy a twist condition. Using geometric methods we prove that Arnold diffusion occurs for generic analytic perturbations H-1. Indeed, the set of admissible H-1 is C-omega dense and C-3 open (a fortiori, C-omega open). Our perturbative technique for the genericity is valid in the C-k topology for all k epsilon [ 3,8) boolean OR {infinity,omega}.