  Let S=K[x_1,..., x_n], let A,B be finitely generated graded S-modules, and
let m=(x_1,...,x_n). We give bounds for the Castelnuovo-Mumford regularity of
the local cohomology of Tor_i(A,B) under the assumption that the Krull
dimension of Tor_1(A,B) is at most 1. We apply the results to syzygies,
Groebner 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 (roughly) (n-1)/2 steps of the resolution of I are linear, then I^2 is a
power of m.
