Gluing Cohen-Macaulay modules with applications to quasihomogeneous complete intersections with isolated singularities

Abstract

This paper deals with the problem of characterizing quasihomogeneous isolated singularities. The history begins in 1971 with the beautiful result of Saito [22]: an isolated complex hypersurface singularity with defining equation f is quasihomogeneous (i.e., after a change of coordinates f can be made into a quasihomogeneous polynomial) if and only iff ~j(f), where j(f) is the ideal generated by the partial derivatives off (this ideal is also called the jacobian ideal off). In the subseqeunt years this result was extended to other fields and significantly generalized in papers by Scheja and Wiebe, see [24], [25] and [26]. Among other powerful results they showed that a complete intersection (R, m, k) with isolated singularity is quasihomogeneous if and only if there exists a k-derivation 6 of R which induces an isomorphism on the Zariski tangent space m/m 2. If dim R = 2, then the assumption that R is a complete intersection can be discarded and the requirement on the derivation 6 can be weakened: it suffices that 6 induces a nonnilpotent transformation of m/m 2. A concise account of their work can be found in Platte's paper [21]. In 1985, Wahl [29] characterized quasihomogeneous Gorenstein surface singularities in terms of certain invariants associated with the resolution of singularities. There the aforementioned criterion of Scheja and Wiebe was used. In 1984, Kunz and Waldi [15] characterized quasihomogeneous reduced Gorenstein algebroid curves over an algebraically closed field k of characteristic 0 by the condition that the cokernel R/J of the canonical homomorphism from the (universally finite) module of K~ihler differentials to the module of regular differentials of R/k is Gorenstein. If R is a complete intersection then J is the K~ihler different of R/k, i.e., the ideal generated by the maximal minors of the jacobian matrix. In 1987 the second author noticed in his thesis [16] the relevance of maximal Cohen-Macaulay modules for the problem ofquasihomogeneity. He conjectured that