Hi,

there is Corollary III,7.12 in Hartshorne which says that:

If $X$ is a projective nonsingular variety over an algebraically closed field $k$, then the dualizing sheaf is isomorphic to the canonical sheaf.

Here the canonical sheaf is as usual $\Omega^{n}_{X}$, where $n=dim(X)$, and the dualizing sheaf is defined by some properties, see p.241.

I wonder if one also has this Corollary for an arbitrary field $k$, not necessarily alg.closed. And if not, can one still say that the dualizing sheaf is at least invertible?

And does someone know a good reference?

Thanks and greetings