Is $D(f)$ the smallest open set of $\operatorname{Spec}B$ such that $D_+(f)\subset D(f)$?

Question

Let $B$ be a graded ring and $\rho:\operatorname{Proj}B\to \operatorname{Spec}B$ the canonical injection, that is, $\forall \mathfrak p\in \operatorname{Proj}B$, $\rho(\mathfrak p)=\mathfrak p$. For any non-nilpotent homogeneous $f\in B_+$, is $D(f)$ the smallest open set of $\operatorname{Spec}B$ such that $D_+(f)\subset D(f)$?

Answer 1

Let $B_0=B$, then $B_+=\{0\}$, so $\operatorname{Proj}B=\varnothing$. However, it doesn't make any sense.

Can anybody give a counterexample such that $D_+(f)\neq \varnothing$

Answer 2

This is not true even if $D_+(f)$ is nonempty. Take $B=k[x]$ with $x$ in degree 1. Then $D_+(f)=\mathrm{Proj}\ B=\{(0)\}$ but $D(f)$ is the affine line minus the origin (including the generic point $\{0\}$); we can then remove any maximal ideal from $D(f)$ and get a smaller open set still containing $D_+(f)=\{0\}$.