Über den Derivationenmodul und das Jacobi-Ideal von Kurvensingularitäten
Let \(R=K[X_ 1,...,X_ n]/I\) be a reduced curve singularity over an algebraically closed field K of characteristic 0. If R is Gorenstein the derivation module \(Der_ K(R)\) contains a canonical free submodule D of rank 1, the ''module of trivial derivations''. In case R is a complete intersection, i.e. \(I=(F_ 1,...,F_{n-1}),\) the derivation \(\delta:=\sum^{n}_{k=1}(-1)^{n-k}\Delta_ k(\partial /\partial X_ k),\) where \(\Delta_ k\) denote the maximal minors of the Jacobian \(\partial(F_ 1,...,F_{n-1})/\partial(X_ 1,...,X_ n),\) is a basis element of D. The following facts are proved: Theorem 1. The length of \(Der_ K(R)/D\) is \(\leq 2\delta_ R-r_ R+1,\) where \(\delta_ R\) is the degree of singularity of R and \(r_ R\) the number of branches of R. Equality holds iff R is a quasihomogeneous singularity. Theorem 2. The minimal number of generators of \(Der_ K(R)/D\) is r(coker c\({}_ R)\), where \(c_ R:\Omega^ 1_{R/K}\to \omega^ 1_{R/K}\) is the canonical homomorphism from Kähler differentials to regular differentials and r denotes the Cohen-Macaulay type of a module. Moreover, the following conditions are equivalent: (a) \(Der_ K(R)/D\) is a cyclic R-module, (b) R is quasihomogeneous. - Theorem 3. If R is a complete intersection and J denotes the Jacobian ideal of R/K then R is quasihomogeneous iff R/J is a Gorenstein ring.
number of generators of derivation module
scientific article; zbMATH DE number 3863352