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Torsion and cotorsion in the sheaf of Kähler differentials on some mild singularities. (English) Zbl 1285.14020

Let \(X|k\) be an algebraic variety over a field \(k\) of characteristic \(0\). The question is studied whether the sheaf of Kähler differentials \(\Omega_X\) is reflexive or torsionfree if \(X\) has “mild” singularities.
As an example, if \(X\) is a normal local complete intersection, then \(\Omega_X\) is torsionfree. It is reflexive if and only if \(X\) is non-singular in codimensin \(2\). The following theorem is proved: Let \(Z\subset \mathbb{P}^n_k\) be a smooth projective variety and \(I\) the ideal sheaf defining \(Z\). Let \(X\subseteq \mathbb{A}^{n+1}\) be the affine cone over \(Z\). If \(H^1(\mathbb{P}^n, I^2(d))=0\) for all \(d\geq 0\) then \(\Omega_X\) is torsionfree. If in addition \(Z\) is projectively normal, then \(\Omega_X\) is torsionfree if and only if also the first infinitesimal neighbourhood of \(Z\) in \(\mathbb{P}^n_k\) is projectively normal. As an application it is proved that Gorenstein terminal singularities will have in general a sheaf of Kähler differential with torsion and cotorsion.

MSC:

14F10 Differentials and other special sheaves; D-modules; Bernstein-Sato ideals and polynomials
13N10 Commutative rings of differential operators and their modules
14B05 Singularities in algebraic geometry
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