[Math] Hilbert polynomial of product of projective varieties

Question

Let $X \subseteq \mathbb P^n, Y \subseteq \mathbb P^m$ be varieties and $h_X,h_Y$ be the hilbert polynomial. then, I know that
$$h_{X \times Y}=h_X \cdot h_Y$$
But, I can't prove.
In special case $X \times Y= \mathbb P^n \times \mathbb P^m$, I prove using the Segre embedding. But above case, I can't easily apply.

Answer 1

The homogeneous coordinate ring $k[ X \times Y ]$ is isomorphic to $\bigoplus_{d=0}^{\infty} (k[X]_{d} \otimes_{k} k[Y]_{d})$ as graded $k$-algebras. Since $\textrm{dim}_{k} (k[X]_{d} \otimes_{k} k[Y]_{d}) = \textrm{dim}_{k}k[X]_{d} \cdot \textrm{dim}_{k} k[Y]_{d}$, we conclude $h_{X \times Y} = h_{X} \times h_{Y}$.

To prove the isomorphism, try the following: let $\{ f_{i} \}_{i}$ and $\{ g_{j} \}_{j}$ be basis for the $k$-vector spaces $k[X]_{d}$ and $k[Y]_{d}$ ($d$ is fixed). The bilinear map $k[X]_{d} \times k[Y]_{d} \rightarrow k[X \times Y]_{d}, \quad (f_{i}, g_{j}) \mapsto ((x, y) \mapsto f_{i}(x)g_{j}(y))$ induces a homomorphism $k[X]_{d} \otimes_{k} k[Y]_{d} \rightarrow k[X \times Y]_{d}$. Prove this is an isomorphism.