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In Illusie's original article, the de Rham-Witt complex is defined for a smooth scheme over a perfect characteristic $p$ base $S$, without reference to $S$. Some 25 years later, Langer and Zink defined the de Rham-Witt complex for a smooth morphism $f:X\to S$ (where $S$ is a $\mathbb{Z}_{(p)}$-scheme, though I am mainly interested in the case over $\mathbb{F}_p$).

Question. Why was it difficult to come up with a relative version? Where is the catch?

I have to admit I haven't studied the existing literature very well yet (in particular, I have only a vague understanding of the work of Langer-Zink). To Illusie the need for a relative theory must have been obvious, but as far as I know he does not give hints why the naive ways of defining it do not work. By the time the second paper (Illusie-Raynaud) came out, there were many ways of constructing the de Rham-Witt complex:

  1. As the universal de Rham $V$-pro complex,

  2. $W_n\Omega^q_X := R^q u_{X/W_n *} \mathcal{O}_{X/W_n}$ where $u: (X/W_n)_{cris} \to X_{Zar}$ is the projection (there is a way to get $R$, $d$, $F$ and $V$).

  3. If $X$ has a lift over $W$ together with a lift of Frobenius, then we can identify the de Rham-Witt complex as a suitable completion of a subcomplex of the direct limit of the de Rham complexes of the lift over the lift of Frobenius.

Why don't they easily generalize? For example #2 seems okay if we replace $/W_n$ by a (formal) lift of $S$ over $\mathbb{Z}_p$ which is possible (locally on $S$) if $S$ is smooth.

Not to mention the obvious idea of taking the quotient of $W\Omega_X$ by the pull-back of $W\Omega_S$...

Log-motivation. The need for a relative version is more apparent in the logarithmic setting: if $X$ is a proper log smooth scheme over the log point $S=Spec(\mathbb{N}\to k)$, there are two interesting de Rham complexes: $\Omega^\bullet_{X/S}$ and $\Omega^\bullet_{X/k}$ (in the latter the base has trivial log structure). We need both to define the Gauss-Manin connection, i.e. the monodromy operator on $H^\bullet(X, \Omega^\bullet_{X/S})$. If $k$ is a perfect characteristic $p>0$ field and we are interested in crystalline cohomology, we have even more choices for the base: (1) $\mathbb{N}\to W(k)$ ($1\mapsto p$), (2) $\mathbb{N}\to W(k)$ ($1\mapsto 0$) and just (3) $W(k)$ with trivial log structure. If we want to realize crystalline cohomology of $X$ over these bases by a version of the de Rham-Witt complex, we need a relative version. I know that Hyodo and Kato defined a log version of the de Rham-Witt complex and even showed an isomorphism between cohomology over (1) and (2) after $\otimes \mathbb{Q}$, but again their de Rham-Witt complex is absolute (I forgot whether it computes (1) or (2)).

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  • $\begingroup$ I guess this is connected to Grothendieck's problem of finding a good theory of crystals "over $\mathbb{Z}$". And this seems to have revealed as a very complicated task. $\endgroup$ – Leo Alonso Dec 16 '13 at 9:14
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    $\begingroup$ @LeoAlonso I'm only interested in the situation when the base has characteristic $p$. Langer and Zink indeed treat the case over $\mathbb{Z}_{(p)}$ rather than $\mathbb{F}_p$, so maybe that is where the difficulty comes from? $\endgroup$ – Piotr Achinger Dec 16 '13 at 9:24
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    $\begingroup$ It seems that leaving the "base field" case adds a lots of complexity, as in general Witt vectors vs. $p$-typical Witt vectors. Are you aware of arXiv:1311.2774? I guess, these kind of ideas would lead to a simpler construction of DRW over a field. $\endgroup$ – Leo Alonso Dec 16 '13 at 9:28
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    $\begingroup$ Perhaps of interest: Hesselholt, L., & Madsen, I. (2004, February). On the De Rham–Witt complex in mixed characteristic. (See, e.g., p.4 just after Theorem C, and the start of p.6. The version I reference is pay-walled at: sciencedirect.com/science/article/pii/S0012959304000096) $\endgroup$ – Benjamin Dickman Dec 16 '13 at 10:59
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    $\begingroup$ @LeoAlonso's reference, clickable: Cuntz and Deninger - An alternative to Witt vectors. $\endgroup$ – LSpice Jan 10 '19 at 3:34

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