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



Dualizing sheaf on varieties
The answer to your question is positive and follows from Theorem 6.4.32 in Qing Liu's book *Algebraic geometry and arithmetic curves*. 


Note that Liu uses Corollary 6.4.13 in the statement of his Theorem. Moreover, the base scheme is a locally Noetherian scheme, e.g., the spectrum of a field.


