Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question
    
There is problem with the display –  Jana Jun 18 '13 at 11:07
    
@Jana: Some kinds of LaTeX need to be enclosed in backwards quote marks. I am not sure of the exact rules but H^* and H^. usually cause problems. I changed H^. to H^\bullet for readability as well. –  Neil Strickland Jun 18 '13 at 11:21
    
@Strickland: Thanks –  Jana Jun 18 '13 at 11:28
    
@NeilStrickland, any _x needs to be _{x}. –  Fred Kline Jun 26 '13 at 10:05
add comment

1 Answer

up vote 4 down vote accepted

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.]

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.