Torsion of differentials of affine quasi-homogeneous hypersurfaces
type: Item
instance of: scholarly article
title: Torsion of differentials of affine quasi-homogeneous hypersurfaces
zbMATH Open document ID: 0874.13019
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-11-04T00:00:00Z
full work available at URL: contents.htm
review text: 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 \(\dim_K T(\Omega^{N-1}_{A/K})=k\).
Mathematics Subject Classification ID: 13N05
Mathematics Subject Classification ID: 14J70
Mathematics Subject Classification ID: 14F10
zbMATH DE Number: 894681
zbMATH Keywords: Kähler differentials
zbMATH Keywords: torsion submodule
zbMATH Keywords: singularities
author: Ruth I. Michler
reviewed by: Usha N. Bhosle
MaRDI profile type: Publication
OpenAlex ID: W2082365436
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DOI: 10.1216/RMJM/1181072113
description: scientific article; zbMATH DE number 894681