Let 𝔭 be a prime ideal in a commutative noetherian ring R. It is proved that if an R -module M satisfies ${\rm Tor}_n^R $ (k (𝔭), M) = 0 for some n ⩾ R 𝔭, where k (𝔭) is the residue field at 𝔭, then ${\rm Tor}_i^R $ (k (𝔭), M) = 0 holds for all i ⩾ n. Similar rigidity results concerning ${\rm Tor}_R^{\ast} $ (k (𝔭), M) are proved, and applications to the theory of homological dimensions are explored.