Let $F$ be a quasi-coherent sheaf on a scheme $X$. To check that $F$ vanishes it suffices to check that all the stalks of $F$ vanish. I would like to know whether it suffices to check that all the fibers of $F$ vanish.

(I think I am using standard terms: the tensor product over $O_X$ with the local ring at $x$ is the stalk, and the tensor product over $O_X$ with the residue field at $x$ is the fiber.)

It suffices to answer the question on an affine scheme. Let $R$ be a commutative ring. For each prime ideal $p$ of $R$ let $k(p)$ be the residue field of the local ring $R_p$. Let $M$ be an $R$-module, and suppose that $\mathrm{Tor}_i(k(p),M) = 0$ for all $i$ and all $p$. (Tor taken in the category of $R$-modules). Does it follow that $M = 0$?

If $M$ is finitely generated, the answer is yes. In that case $M$ vanishes even when $\mathrm{Tor}_0(k(p),M) = 0$ for all maximal $p$, by Nakayama's lemma. My question is whether there is a good replacement for Nakayama's lemma when $M$ is not finitely generated.