  We give an application of the New Intersection Theorem and prove the
following: let $R$ be a local complete intersection ring of codimension $c$ and
let $M$ and $N$ be nonzero finitely generated $R$-modules. Assume $n$ is a
nonnegative integer and that the tensor product $M\tensor_{R}N$ is an $(n+c)$th
syzygy of some finitely generated $R$-module. If Tor$^{R}_{>0}(M,N)=0$, then
both $M$ and $N$ are $n$th syzygies of some finitely generated $R$-modules.
