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I am really new to this, but I am having a hard time understanding all the de Rham-Witt construction.

It seems to be really difficult to compute anything with those beasts: like I cannot find any reference with a good intuition of what $W_m\Omega^n(\mathbb{F}_p[T]/\mathbb{F}_p)$ should be: the case of $m=1$ (i.e. the usual differential forms) seems much easier to compute.

I would be happy if anyone could show me how to compute this in reasonable terms.

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This is explained in both:

Corollaire 2.15, p. 561 in Luc Illusie "Complexe de de Rham-Witt et cohomologie cristalline", Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 12 (1979) no. 4, pp. 501-661,

and:

Proposition 2.17, p. 274 in Andreas Langer and Thomas Zink "De Rham-Witt cohomology for a proper and smooth morphism", Journal of the Institute of Mathematics of Jussieu, Volume 3, Issue 2 (2004), pp. 231-314.

The result is the same, but the vocabulary is different, so you have two points of view.

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