Why is studying zero dimensional subschemes (for simplicity let's say closed subschemes) of $\mathbb{P}^{n}_K$ equivalent to studying saturated homogeneous ideals $I$of $K[x_0,...x_n]$ with krull dimension of $K[x_0,...x_n]/I = 1$ ?
We let $K$ be an algebraically closed field.
Any help from anyone is welcome.