Consider the polynomial ring $S=K[X_1,\ldots,X_n]$, where $K$ is a field and let $I \subset S$ be a homogeneous ideal under standard grading. If $\mathrm{depth}(S/I) >0$, then how can I show that there exists $x \in S_1$ which is a non-zero divisor on $S/I$?
One thing I understand that Hilbert polynomial being a non-zero polynomial $I \cap S_1 \neq S_1$, in fact more than that $I \cap S_m \neq S_m$ $\forall\ m \geq 0.$ But then I can't proceed further.
Help me. Thanks.