Suppose $R$ is a commutative Artinian local ring over an algebraically closed characteristic 0 field $k$. Suppose $f\in R$ is such that $df=0$ (in the sense that the element $df$ vanishes in the module of Kähler differentials). Is $f$ necessarily in $k$?



Kähler differentials on an Artinian local ring
I finally remembered the example (though not the reference). Take $f=x^2y^2+x^5+y^5\in R=\mathbb{C}[[x,y]]$. Then $f\_x,f\_y$ form a regular sequence in $R$ and thus $R/I$ where $I=(f\_x,f\_y)$ is an Artin local ring. One checks $f\not\in I$. Thus, $df=0\in\Omega^1\_{R/I}$, but $f\neq 0$ in $R/I$.


