Let $X$ be a topological space, and $\mathcal{O}_X$ a sheaf of simplicial commutative rings on $X$ such that $(X, \pi_0\mathcal{O}_X)$ is a scheme. Is then in general for $n>0$, the presheaf $\pi_n \mathcal{O}_X$ also a sheaf on $X$? If not, do you have a counterexample?

This is what I've tried so far: if $U \subset X$ is open, and $\{U_i\}$ an open cover of $U$, then for a compatible family $\sigma_i$ in the $\pi_n\mathcal{O}_X(U_i)$, by the Dold-Kan correspondence, we can take representatives $f_i$ in the kernel of the differentials \begin{align} \partial = \sum_k (-1)^k d_k: \mathcal{O}_X(U_i)_n \to \mathcal{O}_X(U_i)_{n-1} \end{align} such that in $\mathcal{O}_X(U_{ij})_n$ it holds \begin{align} f_i |_{U_{ij}} - f_j | _{U_{ij}} = \partial(h_{ij}) \end{align} for certain $h_{ij} \in \mathcal{O}_X(U_{ij})_{n+1}$. But I don't see how I can use this to augment the $f_i$ to a compatible family in the $\mathcal{O}_X(U_i)_n$.

On the other hand, the category of shevaes of simplicial commutative rings on $X$ is equivalent to the category of simplicial objects in sheaves of rings on $X$. Since we know that there are chain complexes of sheaves of abelian groups for which the homology groups are not all sheaves, by the Dold-Kan correspondence, this suggests there might be a counterexample.

Any help is appreciated.

  • $\begingroup$ Just to be sure : by $\pi_n\mathcal{O}_X$, you mean the presheaf $U\mapsto\pi_n(\mathcal{O}_X(U))$ ? Even for the $\pi_0$ case ? Then, in general, these are not sheaves, (even $\pi_0\mathcal{O}_X$). $\endgroup$ – Roland Sep 10 '17 at 8:08
  • $\begingroup$ Yes, that's what I mean. Can you give an example of an $\mathcal{O}_X$ such that $\pi_n\mathcal{O}_X$ is not a sheaf for some $n$, preferably while $\pi_0\mathcal{O}_X$ is? $\endgroup$ – Maanroof Sep 10 '17 at 9:01
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    $\begingroup$ Well, you wrote two different arguments explaining why the presheaf $\pi_n\mathcal{O}$ might not be a sheaf. I believe these are better than a counter-example. Still I think the following example works, but I haven't check the details : take your favorite complexes of sheaves such that the homology presheaves aren't sheaves (and shift it so that it is 0 in degree 0). Then take the corresponding simplicial sheaves using Dold-Kan, and apply the free commutative ring functor (and sheafify). $\endgroup$ – Roland Sep 10 '17 at 11:21
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    $\begingroup$ I think you're right, thanks! I guess that was an $\textit{unknown known}$ from my part. $\endgroup$ – Maanroof Sep 10 '17 at 13:21

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