Let $X\subset \mathbb{P}^n$ be a projective algebraic variety with coordinate ring $R$. $X$ is said to be arithmetically Cohen-Macaulay if $R$ is a Cohen-Macauly ring. A equivalent definition is that the natural morphism $$R\to \oplus_{k\in\mathbb{Z}}\mathcal{O}_X(k)$$ is bijictive and $H^i(X,\mathcal{O}_X(k))=0$ for all $k\in\mathbb{Z}$ and $1\leq i\leq \text{dim} X-1$. Does anyone know a proof for the equivalence? Without assuming that $X$ is Cohen-Macaulay, I can only prove that $R$ is Cohen-Macauly at $0$, the vertex of the affine cone. How to prove $R$ or the section ring is Cohen-Macauly at other points of the affine cone? Did I miss something from the idea? Or is there an complete alternative proof?