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.
$\begingroup$
$\endgroup$
Add a comment
|
1 Answer
$\begingroup$
$\endgroup$
The following example is from Sasha Petrov (any mistakes are mine)
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.