On regular sequence in generating set in a homogeneous ideal in polynomial ring of maximum height Let $J$ be a homogeneous ideal in $S=k[x_1,...,x_d]$, where $k$ is an infinite field, such that $J$ has height $d$ i.e. $\dim (S/J)=0$. Then $\mu(J)\ge d$ and $\operatorname{grade}(J)=\operatorname{ht}(J)=d$. So one can choose an $R$-regular sequence of homogeneous elements $f_1,...,f_d$ in $J$. 
My question is: Can one choose an $R$-regular sequence of homogeneous elements $f_1,...,f_d$ in $J$ such that $f_1,...,f_d$ is a part of a minimal system of homogeneous generators of $J$ ?
 A: $J$ is a homogeneous ideal, and therefore it has a homogeneous (reduced) Grobner basis with respect to lexicographical ordering $x_1<\dots<x_d$. Let us denote this basis by $G$. 
Since $J$ is zero-dimensional there exist $f_1,\dots,f_d\in G$ such that $\mathrm{in}(f_i)=x_i^{t_i}$, $t_i\ge 1$. It is not hard to show that $f_1,\dots,f_d$ is a regular sequence.
We want to show that $f_1,\dots,f_d$ is a part of a minimal system of generators of $J$. Set $\mathfrak m=(x_1,\dots,x_d)$. Suppose $\sum_{i=1}^da_if_i\in\mathfrak mJ$ with $a_i\in k$. Then there are $g_1,\dots,g_d\in J$ such that $\sum_{i=1}^da_if_i=\sum_{i=1}^dx_ig_i$. This implies $\mathrm{in}(\sum_{i=1}^da_if_i)=\mathrm{in}(\sum_{i=1}^dx_ig_i)$. If $a_d\ne0$ we have $x_d^{t_d}=x_d\mathrm{in}(g_d)$, so $\mathrm{in}(g_d)=x_d^{t_d-1}$. We thus found an element $g_d\in J$ with $\mathrm{in}(g_d)=x_d^{t_d-1}$, and this contradicts the reducedness of $G$ (that is, the property that the set $\{\mathrm{in}(g):g\in G\}$ minimally generates the initial ideal of $J$). It follows that $a_d=0$.
Continuing this way we get $a_i=0$ for all $1\le i\le d$, and we are done.
