Some consequences of $\textrm {dim proj}\ \Omega (A)<\infty$

Abstract:

Let $X$ be an affine variety over a field $k$ and $x$ a point on $X$. We are interested in relating the properties of $\Omega {(X)_x}$, the Kähler module of differentials of $x$, with geometric properties of $X$ at $x$. Lipman has given necessary and sufficient conditions for $\Omega {(X)_x}$ to be respectively torsion free and reflexive in the case where $X$ is locally a complete intersection at $x$. We give a generalization of these results for the case where the projective dimension (dim proj) of $\Omega {(X)_x}$ is finite.