# Boundness of torsion in $R^i\pi_*\mathcal F$ for a smooth $\pi:X\rightarrow S$

Let $$\pi:X\rightarrow S$$ be a smooth scheme over $$S$$ and let's assume for simplicity that $$S=\mathrm{Spec} R$$ is affine, Noetherian and regular. Let $$\mathcal F$$ be a coherent sheaf on $$X$$ and let's assume as well that $$\mathcal F$$ is torsion-free. Then I can take the derived global sections $$R^\bullet\pi_* \mathcal F$$, taking further its cohomology I get some $$R$$-modules $$R^i\pi_*\mathcal F$$.

Now the question is:

Given $$R^i\pi_* \mathcal F$$ can I say anything about torsion in it (as in an $$R$$-module)? In particular is there an open dense $$U\subset S$$ such that $$R^i\pi_*\mathcal F$$ is torsion-free restricted to $$U$$?

If $$\pi$$ is proper this is automatic since then $$R^i\pi_* \mathcal F$$ is finitely generated over $$R$$. If $$\pi$$ is affine then $$R^\bullet\pi_* \mathcal F=\pi_*\mathcal F$$ and we just consider some torsion-free module $$M$$ over $$R'$$ as a module over $$R$$ under an injection $$R\subset R'$$, so the statement holds as well. For the general case I tried to use a Cech cover of $$X$$ by affine $$S$$-schemes, but what you get is a complex whose terms are infinitely generated $$R$$-modules (though torsion-free) and I'm not sure what I could use to bound the torsion in the cohomology there.

Also, not sure if this makes life any simplier, but under the assumptions above $$\mathcal F$$ is a perfect complex on $$X$$ and it would be enough to take $$\mathcal F=\mathcal O_X$$.

No, this isn't true even when $$S = \text{Spec}(\mathbf{Z})$$ because of an example of Anurag Singh. Set $$R = \mathbf{Z}[X, Y, Z, U, V, W]/(XU + YV + ZW)$$ and let $$\mathfrak a \subset R$$ be the ideal generated by $$X, Y, Z$$ in $$R$$. Set $$X = \text{Spec}(R) \setminus V(\mathfrak a)$$ Then $$H^2(X, \mathcal{O}_X) = H^3_\mathfrak a(R)$$ has a $$p$$-torsion element for infinitely many primes $$p$$ (see link given above).
• By the way, do you know if there is still a counterexamle for $S$ of characteristic 0? Aug 31 '19 at 19:01