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Kähler differentials provide an adaptation of differential forms to arbitrary commutative rings or schemes.

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Quotient of a smooth curve by a finite group and differentials

Yes, the formula holds, even when the action is not free. Here is the principle of a proof for the case of a tame action (which covers the characteristic $0$ case). Denote by $\pi:X\to Y=X/G$ the quot …
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