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Let $k$ be an algebraically closed field of arbitrary characteristic, $X$ a smooth surface over $k$, and $D_i \subset X$ be an regular, effective Divisor such that $D=\sum D_i$ has normal crossings and no more than two $D_i$ meet at a point of $D$.

Then for every singular point $x \in D$ one can find an affine neighbourhood $\operatorname{spec}(A)$ of $x$ and $f,g \in A$ such that D restricted to $\operatorname{spec}(A)$ is $\operatorname{spec}(A/(fg))$. Now we know, that $\Omega^1_{A/k,x}$ is generated by $d_xf$ and $d_xg$ (e.g. by Proposition 9.1.8 of Lius book).

Now i am confused whether it is possible to choose $A$ in such a way, that already $\Omega^1_{A/k}$ is generated by $df$ and $dg$ (as an $A$-module). Am i missing a simple argument here?

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  • $\begingroup$ I am not sure I understand your confusion. If $M$ is a finitely generated module over a Noetherian ring $A$ and $a_1,\ldots,a_m$ generate $M$ localized at a maximal ideal $\mathfrak{m}$, then there is a $y\not\in\mathfrak{m}$ such that $M_y$ is generated by the $a_i$s over $A_y$. $\endgroup$
    – Mohan
    Commented Jul 16, 2012 at 22:02
  • $\begingroup$ My confusion was that I missed this really basic point. Thank you for the remainder and sorry for bothering you. $\endgroup$
    – fschueller
    Commented Jul 18, 2012 at 16:53

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