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My question is problem 2.9 of Gröbner Bases in Commutative Algebra by Ene & Herzog. If $\mathfrak m$ is the graded (or homogeneous) maximal ideal of polynomial ring $S$, the Loewy length of $S/I$ is the infimum of the numbers $k$ such that $\mathfrak m^k\subseteq I$.

The question is to prove $S/I$ and $S/\mathrm{in}_<(I)$ have the same Loewy length.

I can prove that if $\mathfrak m^k\subseteq I$ then $\mathfrak m^k\subseteq\mathrm{in}_<(I)$, but I can't show the other side. Thank you for any hint about this.

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    Hints. Suppose $m^k\subseteq in_<(I)$ and $m^k\not\subseteq I$. Consider the minimal (in the given monomial order) monomial $n\in m^k$ which is not in $I$. Pick $f_1,\dots,f_t$ a homogeneous Grobner basis of $I$. Then $in(f_i)\mid n$ for some $i$. To figure out what's going on assume $in(f_i)=n$. – 2017-01-04

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