Result on Betti series of the universal modules of second order derivations

Abstract

Let R be the coordinate ring of an affine irreducible curve presented by $frac{k[x,y]}{(f)}$ and m a maximal ideal of R. Assume that $R_m$, the localization of R at m, is not a regular ring. Let $Omega_2(R_m)$ be the universal module of second order derivations of $R_m$. We show that, under certain conditions, $B(Omega_2(R_m), t)$, the Betti series of$Omega_2(R_m)$, is a rational function. To conclude, we give examples related to $B(Omega_2(R_m),t)$ for various rings R.