On Betti series of the universal modules of second order derivations of k[x1, x2, ..., xs] / (f)

Let R be a coordinate ring of an affine irreducible curve represented by k[x1, x2, ..., xs] / (f) and m be a maximal ideal of R. In this article, the Betti series of Ω2(Rm) is studied. We proved that the Betti series of Ω2(Rm), where Ω2(Rm) denotes the universal module of second order derivations of Rm , is a rational function under some conditions.