Report on the Torsion of the Differential Module of an Algebraic Curve

There is a conjecture, that the torsionfreeness of the module of differentials in a point of an algebraic or algebroid curve should imply that the curve is non singular at that point. A report on the main results is given. Let k be a perfect field and R the local ring of a closed point of an algebraic or algebroid curve over k. There is a conjecture that R is regular if and only if the (universally finite) differential module ΩR/k is torsionfree. The nontrivial part is of course to show that for a singular point the torsion submodule τ(ΩR/k) of ΩR/k is not zero. Although a solution for the general case is not in sight there are many special cases which have been treated successfully. In all of these the conjecture has been found to be true. It is the purpose of this paper to give a survey on some of these results with hints concerning the proofs. For simplicity let us assume for the following that R is a reduced complete analytic k-algebra of dimension one with maximal ideal m and embedding dimension n, which then can be represented in the form R = k[[X1, . . . , Xn]]/I = k[[x1, . . . , xn]] , where I is a reduced ideal in the formal power series ring k[[X1, . . . , Xn]] . We will also restrict ourselves to the case char k = 0 , although many of the results are also valid for perfect ground fields. One can distinguish several cases: 1. Conditions on the number of generators of I . Let d(R) = μ(I) − (n − 1) denote the deviation of R , where μ(I) denotes the minimal number of generators of I . R is called a complete intersection if d = 0 and an almost complete intersection if d ≤ 1 . In [Be2] the cases d ≤ 1 were solved if R is a domain. This was generalized to d ≤ 3 in the reduced case by Ulrich [Ul1], [Ul2]. Denote by S the integral closure of R in its full ring of quotients K , and let D : S → SDS = ΩS/k and d : R → RdR = ΩR/k be the universally finite derivations of S and R over k respectively. Since RdR and SDS are both of rank 1 and SDS is torsionfree (even free), the kernel of the canonical homomorphism φ : RdR → SDS is T := τ(ΩR/k), so that we have an exact sequence 0 → T → RdR → SDS → SDS/RDR → 0 .