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Torsion of the differential modules and the value semigroups of one dimensional local rings. (English) Zbl 0606.13020

The author is concerned with the conjecture that for a non regular one dimensional local domain R which is analytic over a field k of characteristic zero the universally finite module of differentials over k must have non trivial torsion. He claims that this is the case if the value semigroup of the differentials contains only elements which are also values of elements of R theorem 4.1). He further gives a condition on the value semigroup of R that must hold if the conjecture was false for R (theorem 4.3). Unfortunately the paper contains some errors: Example 2.6 is not really an example for the intended situation:
The ring R given there is indeed a semigroup ring, according to a theorem of Zariski which says that for a plane algebroid curve R is a semigroup ring iff the value semigroup of the differentials brings no new values.
The proof of lemma 3.7 is incorrect: One cannot in general conclude that if (in the notation of the article) \(\partial_ ih_{j-1}\neq 0\) then \(j-1-a_ i=v(\partial_ ih_{j-1})=v(\partial_ ig_{j-1})\). - Since the main results are based on this lemma, they are still unproven. Corollary 4.2, that the conjecture holds for semigroup rings, is of course true (and well known), since these rings are quasi homogeneous.
Reviewer: R.Berger

MSC:

13N05 Modules of differentials
14H20 Singularities of curves, local rings
13H99 Local rings and semilocal rings
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