Torsion of differentials of affine quasi-homogeneous hypersurfaces
Let \(R=K[X_1,\dots,X_N]\), \(K\) being a field of characteristic zero. \(F\in R-\{0\}\) is a quasihomogeneous polynomial (with weights \(\lambda_i\)) if it satisfies the generalised Euler equation \[ \sum \lambda_i X_i\frac{\partial F}{\partial X_i}=nF. \] Let \(A=\text{Spec }R/(F)\) be a reduced quasihomogeneous affine hypersurface with an isolated singularity at the origin. It is known that (for reduced hypersurfaces \(A\) with isolated singularities) the Kähler differentials \(\Omega^i_{A/K}\) are torsionfree for \(i\neq N-1\), \(N\) and \(\Omega^N_{A/K}\simeq A/I\), \(I=(\partial f/\partial x_1,\dots,\partial f/\partial x_N)\), \(\partial f/\partial x_i=\partial F/\partial X_i+(F)\). The main result of this paper is: Theorem. The torsion submodule \(T(\Omega^{N-1}_{A/K})\) of \(\Omega^{N-1}_{A/K}\) is a cyclic \(A\)-module generated by \[ \omega_0=\sum(-1)^{i+1}\bigl(\frac{\lambda_i} {n}\bigr)x_idx_1\wedge \dots\wedge \widehat{dx_i}\wedge\dots\wedge dx_N. \] Thus \(T(\Omega^{N-1}_{A/K})\approx \Omega^N_{A/K}\) as \(A\)-modules. -- This result is applied to deduce that for singularities corresponding to Dynkin diagrams with \(k\) vertices of type \(A_k\), \(D_k\), \(E_6\), \(E_7\) or \(E_8\) one has \(\dim_K T(\Omega^{N-1}_{A/K})=k\).
scientific article; zbMATH DE number 894681
scientific article