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 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 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.