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

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Degeneration of Hodge-de Rham spectral sequence, exactness of a pairing and the trace morphism

First of all, Deligne-Illusie worked with relative Frobenius $F = F_{X/S}: X \to X'$, so the target of the composition should be $\Omega_{X'/S}^p$ instead of $\Omega_{X/S}^p$. Your question about why …
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