  Given a finitely generated module $M$ over a local ring $A$ of characteristic
$p$ with $\pd M < \infty$, we study the asymptotic intersection multiplicity
$\chi_\infty(M, A/\underline{x})$, where $\underline{x} = (x_1, \ldots, x_r)$
is a system of parameters for $M$. We show that there exists a system of
parameters such that $\chi_\infty$ is positive if and only if $\dim
\Ext^{d-r}(M, A) = r$, where $d = \dim A$ and $r = \dim M$. We use this to
prove several results relating to the Grade Conjecture, which states that
$\grade M + \dim M = \dim A$ for any module $M$ with $\pd M < \infty$.
