# Dualizing sheaf on varieties

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

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Yes, this is true for a smooth projective variety over any field. It should be possible to rewrite Hartshorne's proofs to work in this case. – ulrich Aug 18 '11 at 16:30
Or you can see Residues and Duality. I should also point out that such statements should even be fine for smooth morphisms to things that aren't fields (probably Gorenstein local ring is ok, again see Residues or Brian Conrad's book) Finally, lets say that $X$ is normal (you can get away with a lot less, but lets assume this), then the reflexification/S2-ifiction of $\Omega_X^{n}$ is the dualizing sheaf. – Karl Schwede Aug 18 '11 at 16:34
Another reference is the book "Introduction to Grothendieck duality theory" by Altman and Kleiman. – ulrich Aug 18 '11 at 17:28
Very fine comment, ulrich; thanks a lot! – Descartes Aug 18 '11 at 18:37
Ah, and which book by Conrad? – Descartes Aug 18 '11 at 18:38

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.

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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.

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