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Roughly, I'd like to ask how does the first terms in de Rham complex behaves for singular varieties.

Let $Y$ be a potentially singular integral scheme over a perfect field $k$ of characteristic $p$ and $F : Y \to Y$ be its Frobenius morphism. The first mapping in de Rham complex of $Y$ (after application of $F_*$) is just the differential: $$d : F_*\mathcal{O}_Y \to F_*\Omega^1_{Y/k}.$$ Its kernel clearly contains the image of the Frobenius morphism $F : \mathcal{O}_Y \to F_*\mathcal{O}_Y$. What are the sufficient assumptions for which the kernel $\textrm{Ker} (d)$ is in fact given by $F$?
For now, I can prove it for normal $Y$ with a torsion-free differential module $\Omega^1_Y$. The proof, more or less, boils down to localization at the generic point (which is injective by torsion-freeness) and then using normality ($a^p = f$ for $a \in k(Y)$ and $f \in \mathcal{O}_Y$ implies $a \in \mathcal{O}_Y$).

Is there any well-known, explicit class of varieties for which the kernel is bigger?
Maybe, there is a large class of normal singularities for which $\Omega^1_Y$ is torsion-free? Do you know any reference for this kind of problems?

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Does not hold for a cusp in char 2 (for example $F_2[t^2, t^3]$), but does hold for a node in all characteristics.

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Thanks:) I'm aware of some more general class given by $\mathrm{Spec}(\mathbb{F}_p[x,y]/(x^p - y^k))$. But I wonder if it can be described in more "systematic" way, including some higher dimension cases, for example involving some inseparable covers (like the one given by inclusion $\mathbb{F}_p[y] \to \mathbb{F}_p[x,y]/(x^p - y^k)$). –  Maciekrt Feb 5 at 14:47
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