Modules over unramified regular local rings

in an essential way on the fact established in 2 that for an unramified regular Tor (A, B) 0 for some R- local ring R and a torsion-free R-module A if module B, then Tor(A, B) 0 for all j >-i.In fact, if this property of Tor can be established for arbitrary regular local rings, then almost all the results of this paper extend immediately to all regular local rings.1. Some properties of Tor Before proceeding to the main results of this section we review briefly some of the basic facts concerning the codimension of a module as can be found forLet R be a local ring with maximal ideal m and A a nonzero R-module.A sequence of elements xl, xt in m is called an A-sequence if xl is not a zero-divisor in A and xi is not a zero-divisor for A/(xl,..., ,xi_)A for all i 2, t.If x,..., xt is an A-sequence, then it is easily seen that <= dim R (where dim R means the Krull dimension of R).Thus it makes sense to talk about maximal A-sequences.It can be shown that all maximal