Let S = K[x1, . . . , xn], let A, B be finitely generated graded S-modules, and let m = (x1, . . . , xn) ⊂ S. We give bounds for the regularity of the local cohomology of Tork (A, B) in terms of the graded Betti numbers of A and B, under the assumption that dim Tor1 (A, B) ≤ 1. We apply the results to syzygies, Gröbner bases, products and powers of ideals, and to the relationship of the Rees and symmetric algebras. For example we show that any homogeneous linearly presented m-primary ideal has some power equal to a power of m; and if the first [(n - 1)/2] steps of the resolution of I are linear, then I2 is a power of m.