Hi,

this question concerns the canonical bundle $\omega_X$ on a smooth projective $k-$variety $X$.

I want to consider the diagonal embedding

$i:X\rightarrow X\times X$

on $X$.

Furthermore denote by $p$ the projection $X\times X \rightarrow X$ on the first factor.

Now my question:

does it hold that

$i_*\omega_X \simeq p^* \omega_X$

canonically as sheaves on $X\times X$?

I thank you very much!