Assume that $S_0$ is a field and that $S$ is a graded ring generated as an $S_0$-algebra by $S_1$. Let $Q\subset S$ be a prime ideal, and let $P\subset Q$ be the ideal generated by the set of homogeneous elements of $Q$. Then, it's said that $P$ is prime, and that either $P=Q$ or codim $Q/P=1$.
I have a question about the second statement. What if $Q$ is a non-homogeneous prime of codim bigger than $1$? For example, if we consider $S=k[x_1,x_2]$, with $k$ a field. Let $Q=(x_1^2+x_2+1, x_2^3+x_1+1)$, then $Q$ is a non-homogenous prime of codim $2$(if I am correct), and it's set of homogeneous elements is only the zero set, i.e. $P=0$. But then codim $Q/P$ = codim $Q\neq 1$. Hope someone could help. Thanks!
