Let $S$ be a graded ring. To understand $\operatorname{Proj}(S)$ better, I want to know under what changes of $S$ will $\operatorname{Proj}(S)$ remains unchanged. All answers from a proposition or general point of view are welcome. I will start with some examples: Fix an integer $k$.
- scaling: define a new graded ring with $S'_d = S_{dk}$. Then $\operatorname{Proj}(S)\cong \operatorname{Proj}(S')$
- replace a finite number of terms by $0$ and remain the grading. $S'_{n} = 0$ for $0\leq n \leq k$ and $S'_{n} = S_{n}$ for $n \geq k+1.$