  Let $R$ be a commutative noetherian ring, $I$ an ideal of $R$, and $M$ a
finitely generated $R$-module. We consider the asymptotic injective dimensions,
projective dimensions, Bass numbers, and Betti numbers of localizations of
$M/I^n M$ at prime ideals of $R$ and prove that these invariants are stable or
have polynomial growth for large integers $n$ that do not depend on the prime
ideals.
