In Hartshorne’s, a hyperplane $H_0$ in a projective space $\mathbb P^n$ can be defined by a global section $f\in \Gamma $($\mathbb P^n$,$\mathcal O_X(1)$), which is a linear polynomial. I have trouble understanding the following statement.
Given a closed point $x$, $x \in H_0$ if and only if $f \in m_x$, the maximal ideal of the sheaf of ring at $x$.
I guess since $x$ is a closed point, it corresponds to a maximal ideal $p$. Then if $f \notin p$, we have $x \in D(f)$, which is $\mathbb P^n - H_0$. And we get the statement by taking compliment. Am I right? Hope someone can help, thanks!