Let $X$ be the affine cubic curve $y^2=x^3$, over a field of characteristic not equal to 2 or 3. Let $A=k[X]$, the ring of regular functions on $X$. Let $\Omega_A$ be the $A$-module of Kähler differentials on $X$, i.e. the $A$-module generated by symbols $\mathrm{d}f$, where $f\in A$, satisfying the properties $\mathrm{d}\lambda=0$, $\mathrm{d}(f+g)=\mathrm{d}f+\mathrm{d}g$ and $\mathrm{d}(fg)=f\mathrm{d}g+g\mathrm{d}f$, for all $f,g\in A$ and $\lambda\in k$. Let $\Omega[X]$ denote the set of all regular differential forms on $X$. Why is the differential form $3y\mathrm{d}x-2x\mathrm{d}y$ non-zero in $\Omega_A$, but zero in $\Omega[X]$?
Kähler differentials not the same as regular differentials on a singular curve
3
$\begingroup$
algebraic-geometry
-
0Where do these assertions come from? – 2012-05-03
-
0Dear Alan, What do you mean by "regular differential forms on $X$" when $X$ is singular? Regards, – 2012-05-04
-
0These assertions come from Chapter III, $\S5$, Exercise 9 of Shafarevich's book 'Basic Algebraic Geometry, Volume 1'. – 2012-05-05
-
0A differential form $\omega$ on a variety $X$ is regular at a point $x\in X$ if there exists an open neighbourhood $U$ of $x$ in $X$, together with functions $f_i,g_i\in k[U]$ (i=1,...,r), such that $\omega=\sum_{i=1}^{r}f_i\;\mathrm{d}g_i$ on $U$. – 2012-05-05
-
0And of course a differential form is regular on $X$ if it is regular at every point of $X$. – 2012-05-05
-
0@Alan I'm guessing it has something to do with $y^2=x^3$ implying $2ydy=3x^2 dx$. Multiplying by $x$ gives $2xydy=3x^3 dx=3y^2 dy$ and dividing by $y$ gives $2xdy=3ydx$. – 2012-05-05