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Let $M$ be a finitely generated module over a local ring $R$ of
characteristic $p > 0$. If depth$(R) = s$, then the property that $M$ has finite projective
dimension can be characterized by the vanishing of the functor
$\Ext^i_R(M,^{f^n}\!\!\!R)$ for $s+1$ consecutive values $i > 0$ and for infinitely many $n$.
In addition, if $R$ is a $d$-dimensional complete intersection, then
$M$ has finite projective dimension can be characterized by the
vanishing of the functor $\Ext^i_R(M, ^{f^n}\!\!\!R)$ for some $i\geq d$
and some $n > 0$.
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