Let $X$ and $Y$ be (possibly singular) projective schemes over a field $k$ (I'm also willing to assume $k$ algebraically closed or even $k = \mathbb C$). I would like to know if $$ \omega_{X \times Y} = \pi_X^* \omega_X \otimes \pi_Y^* \omega_Y,$$ where $\omega_X$ etc are the dualizing sheaves.
If both $X$ and $Y$ are smooth of dimension $r$ and $s$, then $X \times Y$ is smooth of dimension $r + s$ and so $$\omega_{X \times Y} = \Lambda^{r+s} \Omega_{X \times Y} = \Lambda^{r + s} (\pi_X^* \Omega_X \oplus \pi_Y^* \Omega_Y) = (\Lambda^r \pi_X^*\Omega_X) \otimes (\Lambda^s \pi_Y^*\Omega_Y) = \pi_X^* \omega_X \otimes \pi_Y^* \omega_Y.$$
But what about the general case?
I thought a bit about using the characterization $$\omega_X = \mathcal{Ext}^r_{\mathbb{P}^n}(\mathcal{O}_X, \omega_{\mathbb{P}^n}),$$ where $X \hookrightarrow \mathbb{P}^n$ is a closed embedding. But then I have to embed the product via the Segre embedding $$ X \times Y \hookrightarrow \mathbb{P}^n \times \mathbb{P}^m \hookrightarrow \mathbb{P}^{nm + n + m}$$ and I don't have any clue how to relate the different $\mathcal{Ext}$-sheaves in this situation.