In this paper we study the regularity and the projective dimension of the Stanley-Reisner ring of a $k$-decomposable simplicial complex and explain these invariants with a recursive formula. To this aim, the graded Betti numbers of decomposable monomial ideals which is the dual concept for $k$-decomposable simplicial complexes are studied and an inductive formula for the Betti numbers is given. As a corollary, for a shellable simplicial complex $\Delta$, a formula for the regularity of the Stanley-Reisner ring of $\Delta$ is presented. Finally, for a chordal clutter $\mathcal{H}$, an upper bound for $\T{reg}(I(\mathcal{H}))$ is given in terms of the regularities of edge ideals of some chordal clutters which are minors of $\mathcal{H}$.
