Let $X\subseteq \mathbb{P}^r,Y \subseteq \mathbb{P}^s$ be two projectve varieties,what is the coordinate ring of $X\times Y$(segre embedding)?Is it true that
$$S(X\times Y)=S(X)\otimes_k S(Y)?$$
I also want to know what is the dimension of $ X\times Y$?
Edit:As Zhen Lin has indicated ,the equality $S(X\times Y)=S(X)\otimes_k S(Y)$ is not true.But what is the relationship between them,can we express $S(X\times Y)$ in terms of $S(X),S(Y)$?
It turns out that the coordinate ring for $X\times Y$ is
$$S(X\times Y) = \bigoplus_{i} S(X)_i\otimes S(Y)_i \subset S(X)\otimes S(Y)$$
where $S(X)_i$ is the degree $i$ component of $S(X)$.
Exercise 13.14 in Eisenbud's Commutative Algebra text goes through this fact.