# Canonical bundle and diagonal morphism on varieties

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!

-
Well, $p^* \omega_X$ is locally free (and nonzero) so its support is all of $X\times X$, and on the other hand the support of $i_* \omega_X$ is contained in the diagonal $\Delta \subseteq X\times X$. –  Mattia Talpo Aug 21 '11 at 8:38
So, it is false. Thank you, Mattia! –  ernest Aug 21 '11 at 10:00
You're welcome ;) –  Mattia Talpo Aug 21 '11 at 16:23