RESULTS ON BETTI SERIES OF THE UNIVERSAL MODULES OF SECOND ORDER DERIVATIONS

Let R be the coordinate ring of an affine irreducible curve presented by k[x,y] /(f) and m a maximal ideal of R. Assume that Rm, the localization of R at m, is not a regular ring. Let Ω2(Rm) be the universal module of second order derivations of Rm. We show that, under certain condi- tions, B(Ω2(Rm), t), the Betti series of Ω2(Rm), is a rational function. To conclude, we give examples related to B(Ω2(Rm), t) for various rings R.