  Let $(R,\mathfrak{m})$ be a Cohen-Macaulay local ring, $I$ an
$\mathfrak{m}$-primary ideal of $R$ and $J$ its minimal reduction. We study the
depths of $F(I)$ under certain depth assumptions on $G(I)$ and length condition
on quotients of powers of $I$ and $J$, namely
$\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1}/\mathfrak{m}JI^n)$ and
$\sum_{n\geq0}\lambda(\mathfrak{m}I^{n+1} \cap J/\mathfrak{m}JI^n)$.

Commutative Algebra