Let $k$ be a field and $I$ a homogeneous ideal of $k[x_1,...,x_n]$. Consider the graded ring $R=k[x_1,...,x_n]/I$ with unique homogeneous maximal ideal $\mathfrak m=(x_1,...,x_n)R.$ Let $J$ be a proper ideal of $R_{\mathfrak m}$ minimally generated by $t$ elements and assume $\sqrt J=\mathfrak m R_{\mathfrak m}$. Then, is it true that there exists a homogeneous ideal $L$ of $R$ such that $L_{\mathfrak m}=J$ and $L$ requires at most $t$ generators?
I think this should be true, but given $J=(f_1,...,f_t)R_{\mathfrak m}$, I'm having a hard time finding homogeneous elements $g_1,...,g_t \in R$ with $(g_1,...,g_t)_{\mathfrak m}=J$.
Please help.