As i understand, there are two notions of "equivalence" of projective varieties. The first is projective equivalence, under which projective varieties $X,Y$ of $\mathbb{P}^n(k)$ are mapped to each other by an invertible $(n+1) \times (n+1)$ matrix. In that case their corresponding homogeneous coordinate rings are isomorphic as $k$-algebras. The second notion is weaker and is the notion of the two varieties being isomorphic, under which a biregular morphism exists between $X,Y$.
Question: What can we say about the homogeneous coordinate rings of two isomorphic projective varieties?
Of course, the two rings will must have the same Krull dimension, but can we say anything else?
