Differentials of linked curve singularities
Let k be an algebraically closed field. We consider the class \({\mathcal C}\) of 1-dimensional reduced rings \(R=P/I\), where P is a formal power series ring over k. The main result of the paper is the following: If R,\(S\in {\mathcal C}\) are directly linked and have a torsion free module of differentials, then both R and S are regular. This result supports a conjecture of Berger, which says, that a singular R always has torsion differentials. - The above result and several corollaries are derived from the following more precise statement: There is a natural homomorphism, the so-called canonical class\(c_ R:\quad \Omega_{R/k}\to \omega_ R\) from the module of differentials \(\Omega_{R/k}\) into the canonical module \(\omega_ R\) of R. Since \(\omega_ R\) is torsionfree the kernel of \(c_ R\) is just the torsion \(\tau \Omega_{R/k}\) of \(\Omega_{R/k}\), and the cokernel of \(c_ R\) has finite length, i.e. \(\ell(co\ker c_ R)<\infty.\) Since R is reduced \(\omega_ R\) can be identified with an ideal \({\mathfrak k}\) of R. Set \(\sigma(R):=\ell(R/{\mathfrak k})-\ell({\mathfrak k}^{-1}/R)\). \(\sigma\) (R) does not depend on the embedding of \(\omega_ R\) into R. We have the following formula: \(\ell(co\ker c_ R)-\ell(\tau \Omega_{R/k})=\sigma(R).\) One of the consequences of this formula is the result, that \(\tau \Omega_{R/k}\neq 0\) for any singular \(R\in {\mathcal C}\) which is in the linkage class of a complete intersection.
scientific article; zbMATH DE number 3865447