Let $X$ be a non-empty subset of $\mathbb{P}^{n}$. Consider the affine cone over $X$, $C(X)= \theta^{-1}(X) \cup {0}$ in $\mathbb{A}^{n+1}$ Then $I(X)=I(C(X))$. Is this proof correct?
If $F\in I(X)$ then $F=f_{1}+\dots f_{d}$ with every $f_{i}\in I(X)$ homogeneous of degree $i$. Since $X$ is not empty, every $f_{i}$ is non-constant and so it vanishes on zero. Then $F$ vanishes on zero and since $F(\lambda x)=0$ for every $x\in X$ and every $\lambda$, then $F$ vanishes on the cone.
Conversely, if $F\in I(C(X))$ then $F=f_{1}+\dots f_{d}$ with every $f_{i}\in I(X)$ homogeneous of degree $i$. Since $F(0)=0$ every $f_{i}$ is non constant and $F(\lambda x)=0= \lambda^{1}f_{1}(x)+\dots+\lambda^{d}f_{d}(x)$ for every $\lambda$ then, since $k$ is algebraically closed and so infinite, every $f_{i}(x)=0$, for every point $x\in X$. Then for every $i$, $f_{i}\in I(X)$ and, since it is homogeneous, the sum $F\in I(X)$.
Thank you !