Dualizing sheaf on varieties

Question

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

Answer 1

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.

Answer 2

Ser Lipman's Asterisque 117 entitled "Dualizing sheaves, differentials and residues on algebraic varieties" who works over a perfect field and provides a canonical isomorphism.

Addendum: I wrote the answer in a hurry, I apologize. Let me be more explicit. In the book, working with a variety $X$ over a perfect field $k$, Lipman constructs a certain sheaf $\omega_X$ (actually a sheaf on the big Zariski site over $Spec(k)$) called the canonical sheaf, by using rational differentials and traces, together with a canonical map

$$c_X \colon \Omega^n_X \to \omega_X$$

Then he proves two things

1. The sheaf $\omega_X$ is dualizing, i.e. it represents the functor $H^d(X,-)^\vee$, where $(-)^\vee$ denotes $k$-dual and $d$ the dimension of $X$.
2. If $X$ is smooth over $k$ the map $c_X$ is an isomorphism.

The map $c_X$ is called the fundamental class and admits a big generalization using sheafified Hochschild Homology, but this is another story.