Torsion of the differential module of singularities of curves with maximal Hilbert function (Q1116988)

Let \(R=k[[ X_ 1,\ldots,X_ n]]/\mathfrak a\) be a one-dimensional local analytic ring over a perfect field \(k\). It is still, in the general case, an open question whether torsionfreeness of the universally finite differential module \(D(\frac{R}{k})\) implies regularity of \(R\). The author shows that this is so if \(R\) has maximal Hilbert function. In the proof he develops a formula for the drop of the length of this torsion when going from \(R\) to its first quadratic transform \(R_ 1\). From this it follows that this drop is positive if \(R\) is not regular so that the torsion of \(D(\frac{R}{k})\) must be nonzero. The main step of the proof consists in a computation of the relations in \(R_ 1\) from the relations in \(R\), where the maximality of the Hilbert function plays a decisive role.