Question

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.

Answer 1

If the scheme is locally noetherian, this is true and can be proved by noetherian induction. In fact, you can even replace $M$ with an object of bounded derived quasi-coherent category, if you are interested in such things.

The proof is relatively straightforward: For a complex of modules $M$ over the ring $R$, we may assume that any non-zero $f\in R$ acts by a quasi-isomorphism $f:M\to M$ (by the induction hypothesis), and then the cohomology of $M$ are defined over the field of fractions of $M$.

Answer 2

What about $R=\mathbb Z$ and $M=\mathbb Q/\mathbb Z$ ? All fibers are zero.