# Nonzero module with vanishing derived fibers

What's an example of a nonzero $$R$$-module with vanishing derived fibers at all points of $$\mathrm{Spec}(R)$$? This was asked in When does a quasicoherent sheaf vanish? but the answer there only says there are no Noetherian examples.

Let $$S$$ be the ring $$k[x^{1/p^\infty}]$$ for some field $$k$$, and $$N$$ be the maximal ideal $$(x^{1/p^\infty})$$ in $$S$$. Then take $$R=S/x$$ and the $$R$$-module $$M=N/xN$$. We claim all its derived fibers vanish.
The only prime ideal of $$R=k[x^{1/p^\infty}]/(x)$$ is $$m=(x^{1/p^\infty})/(x)$$ with residue field $$k$$, so let's check that $$M\otimes^L_R k$$ is zero. $$N$$ is $$x$$-torsion free as a $$S$$-module so we have $$M\otimes^L_R k \cong N\otimes^L_S R \otimes^L_R k \cong N \otimes^L_S k$$ The module $$N$$ is flat over $$S$$, because it is a filtered colimit of principal ideals, so $$N \otimes^L_S k$$ is the classical tensor product which is zero because the multiplication map $$N\otimes_k N \to N$$ is surjective. Hence $$M\otimes^L_R k$$ is zero as well.