Let F$F$ be a quasi-coherent sheaf on a scheme X$X$. To check that F$F$ vanishes it suffices to check that all the stalks of F$F$ vanish. I would like to know whether it suffices to check that all the fibers of F$F$ vanish.
(I think I am using standard terms: the tensor product over O_X$O_X$ with the local ring at x$x$ is the stalk, and the tensor product over O_X$O_X$ with the residue field at x$x$ is the fiber.)
It suffices to answer the question on an affine scheme. Let R$R$ be a commutative ring. For each prime ideal p$p$ of R$R$ let k(p)$k(p)$ be the residue field of the local ring R_p$R_p$. Let M$M$ be an R$R$-module, and suppose that Tor_i(k(p),M) = 0$\mathrm{Tor}_i(k(p),M) = 0$ for all i$i$ and all p$p$. (Tor taken in the category of R$R$-modules). Does it follow that M = 0$M = 0$?
If M$M$ is finitely generated, the answer is yes. In that case M$M$ vanishes even when Tor_0(k(p),M) = 0$\mathrm{Tor}_0(k(p),M) = 0$ for all maximal p$p$, by Nakayama's lemma. My question is whether there is a good replacement for Nakayama's lemma when M$M$ is not finitely generated.
