Projective dimension of second order symmetric derivation of Kahler modules for hypersurfaces

Abstract

$R = k[x_1,...,x_s]$ be a polynomial algebra and I be an ideal of R generated by $f \in R.$ Then $S=R/I=\frac{k[x_1,...,x_s]}{(f)} $ be an affine domain which is called hypersurfaces. $\Omega_1(S)$ denotes the module of first order derivations of K\"{a}hler modules over S. $\vee^2 (\Omega_1(S))$ denotes the module of second order derivations of symmetric algebra on $\Omega_1(S).$ In this paper, we prove that if S be an affine domain represented by $ S=\frac{k[x_1,...,x_s]}{(f)}$, then projective dimension of $\vee^2(\Omega_1(S))$ is less than or equal to 1.