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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?


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Dear oxeimon, The two definitions are not equivalent, and the definition that Serre gives is the one which is relevant to Serre duality, which is a major source of motivation for it. Regards, – Emerton Jul 25 '12 at 3:29
Uh oh. Though, at least, it seems to me that Serre's module of differentials regular at a singular point $Q$ contains the stalks of the sheaf of differentials at $Q$? – oxeimon Jul 25 '12 at 5:54
Also, if you let $\omega_{X'}$ denote the sheaf of "regular differentials" on $X'$ in this weird sense (as opposed to $\Omega_{X'}$), is it the case that : $$\omega_{X'} = \underline{H}^{-d}(Rp^!\mathcal{O}_S)$$ Where $S = \text{Spec} k$ and $p : X'\rightarrow S$ is cohen-macauley of pure relative dimension $d$. (I don't yet know what the RHS means) – oxeimon Jul 25 '12 at 5:59
I ask because in Deligne/Rapoport's paper Les Schemas de Modules de Courbes Elliptiques (I.2), he refers to "the sheaf of regular differentials $\omega_{X}$" defined as above using the $\underline{H}^{-d}(...)$. When I first read it I thought he was just talking about $\Omega_X$, but now I guess he's actually referring to the sheaf Serre defines. Is the lower-case $\omega_X$ the usual notation for this sheaf? Can you suggest some references for where I can learn more about this sheaf and its relation to the usual sheaf of differentials $\Omega_X$? – oxeimon Jul 25 '12 at 6:04
@Emerton (the above) thanks – oxeimon Jul 25 '12 at 6:06

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