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
Best Answer
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.