Let $R$ be a commutative algebra over a field $k$. Denote the $R$-module of Kahler differentials by $\Omega^1_kR$; this is the $R$-module generated by symbols of the form $da$, $a\in R$, and relations $$ d(\lambda a+b)=\lambda da+db,\;\;\; dab =adb+bda,\;\;\; d1=0, \;\;\; \forall a,b\in A, \lambda\in k$$ The $R$-module of $i$-forms is the $i$th exterior power $\Omega^i_kR:= \Lambda^i_k\Omega^1_kR$.

When $R$ has Krull dimension $n$, call $\Omega^n_kR$ the module of volume forms. When $R$ is the coordinate ring of a smooth affine variety $X$ of dimension $n$, then elements of $\Omega^nR$ determine volume forms on $X$.

For singular varieties, the behavior of $\Omega^nR$ can be weirder. For example, the simple cusp in the plane $$R=\mathbb{C}[x,y]/x^3-y^2$$ has a volume form $3ydx-2xdy$ which is non-zero, but $$y(3ydx-2xdy) = 3x^3dx-2xydy = xd(x^3)-xd(y^2)=0$$ So, the module $\Omega^1_kR$ of volume forms here has a torsion element.

My question is: Under what conditions on $R$ is $\Omega^n_kR$ torsion-free?

  • $\begingroup$ Do you mind if I ask if there is any intuition behind this particular example of a torsion element? I bumped on your question by trying to find such a torsion element as it is an exercise in Matsumura's book on Commutative Algebra. $\endgroup$ – Patrick Da Silva May 24 '15 at 4:10

This is an interesting question! When $n=1$ there is a conjecture by Berger that the module of differential is torsion-free iff the curve $X$ is non-singular. Googling ``Berger conjecture module of differentials" should give you some references. Bernd Ulrich's first paper (by Mathcsinet) was on this topic, so you can look there also. I do not know anything about higher dimensions but it is plausible that torsion-freeness of $\Omega^n_k$ imposes strong conditions on $R$, as tensor products of non-projective modules tend to have torsions.

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    $\begingroup$ If I remember correctly, there is a result of Donu Arapura which says that in addition to the above torsion-freeness, R is Cohen-Macaulay and smoothable, then R itself is smooth. $\endgroup$ – Mohan Apr 16 '13 at 22:47
  • $\begingroup$ Long... thanks for the references! Your intuition in the last sentence is the opposite of what I expected, though. The one positive example I did completely was the cone $xz=y^2$, in which $\Omega^1$ is non-projective, but $\Omega^2$ is torsion-free. Hence, I was hoping for something like 'integrally-closed' implies 'torsion-free volume forms'. $\endgroup$ – Greg Muller Apr 16 '13 at 23:13

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