Let $\omega$ be a differential form on a singular integral curve $X'$ over some algebraically closed field $k$ (ie, $\omega$ is an element of the stalk of the sheaf of differentials $\Omega_{X'}$ of $X'$ at the generic point), and let $Q$ is a singular point of $X'$ with local ring $\mathcal{O}'_Q$.

Then, in Serre's "Algebraic Groups and Class Fields" (chapter IV.9), he says that $\omega$ is regular at $Q$ if:

$$\sum_{P\rightarrow Q}\text{Res}_P(f\omega) = 0\qquad \text{for all }f\in\mathcal{O}'_Q$$

where the sum is taken over all points $P$ in the normalization $X$ of $X'$ that map to $Q$, and where $f\omega$ is viewed as a differential on the normalization by identifying the generic stalks of the two sheaves of differentials $\Omega_X$ and $\Omega_{X'}$.

So, I'm still relatively new to differentials, but I would have thought that you would say that a differential $\omega\in\Omega_{X',\eta}$ ($\eta$ the generic point of $X'$) is regular at $Q$ if $\omega$ actually comes from $\Omega_{X',Q}$ under the canonical inclusion $\Omega_{X',Q}\subseteq\Omega_{X',\eta}$.

Are these two definitions of regularity equivalent? and if so, is there a nice elementary way to see this?

thanks

containsthe stalks of the sheaf of differentials at $Q$? – Will Chen Jul 25 '12 at 5:54