Vanishing cohomology of de-Rham Witt complex

Question

Let $X$ be a smooth scheme over $\mathbb{F}_{p}$ for a prime number $p$. As far as I understand, there is a surjective morphism from $\Omega^\bullet_{W\mathcal{O}_X} \to W \Omega_{X}^\bullet$ which induces an isomorphism $\Omega^\bullet_{W\mathcal{O}_X}/(T+Fil^n \Omega^\bullet_{W\mathcal{O}_X}) \to W_{n}\Omega_X^\bullet$ where $T$ is the graded differential ideal consisting of $p-$torsion elements of $\Omega^\bullet_{W\mathcal{O}_X} $ and $Fil^n$ is the kernel of the natural projection map from $\Omega^\bullet_{W \mathcal{O}_X} \to \Omega^\bullet_{W_n\mathcal{O}_X} $. A reference for this is "Complexe de de-Rham Witt et Cohomologie Cristalline" by Luc Illusie.

The question is: Suppose there exists an integer $N$ such that $H^i(W_{n}\Omega_{X}^\bullet) $ vanish for all $i>N$. Is there any condition on $X$ or $n$ that we can impose so that $H^i(\Omega^\bullet_{W\mathcal{O}_X})$ vanish as well for $i>N$? In other words when does $H^i(T+Fil^n \Omega^\bullet_{W\mathcal{O}_X})$ vanish? Any idea/reference in this direction will be very helpful.

Answer 1

If $A$ is a finitely generated ring then so is each $W_n(A)$, and in fact one can bound the number of generators needed by a function of only $p$, $n$, and the number $d$ of generators of $A$. Write $N_p(d,n)$ for such a function. (See below.) Then $\Omega^i_{W_n(A)}$ vanishes for $i>N_p(d,n)$. In particular, this is true when $A$ is the polynomial ring in $d$ variables. Now if $X$ is a smooth scheme over dimension $d$, then it locally admits an etale map to affine $d$-space. Since $W_n$ preserves etaleness of maps, $\Omega^i_{W_n(X)}$ vanishes for $i>N_p(d,n)$. Since $W_n\Omega^i_X$ is a quotient of $\Omega^i_{W_n(X)}$, it also vanishes for $i>N_p(d,n)$. It follows that $H^i(W_n\Omega_X^\bullet)$ and $H^i(\Omega_{W_n(X)}^\bullet)$ vanish for $i>d+N_p(d,n)$. So in particular, your assumption always holds.

Now you ask about the vanishing of $H^i(\Omega^\bullet_{W\mathcal{O}_X})$. This probably involves some nasty details about inverse limits that I'd rather not think about. (For instance, it's not even clear to me what exactly you mean by ${W\mathcal{O}_X}$. Are you actually taking the limit over $n$ in some category or are you treating it as a pro-object? Etc.) But I hope what I've said helps.

[On $N_p(d,n)$: if I'm not mistaken, it can be taken to be $d+(1+p^d+\cdots+p^{nd})$. To show this, you can show that $W_n(A)$ is generated, as a module over the subring generated by Teichmueller lifts of the $d$ generators of $A$, by lifts of module generators of $W_{n-1}(A)$ and by the $V^n$ of Teichmueller lifts of monomials in the generators of $A$, where all exponents in the monomials are at most $p^n$ (thus making $p^{nd}$ additional generators). I hope I got that right. Also my index $n$ is normalized so that $W_0(A)=A$. So it is what would be $n-1$ for most people.]