# equivariant Serre Duality.

Let $X$ be a nonsingular projective variety of dimension $n$ over a field $k$, and $\omega_X$ be its canonical sheaf. Let $G$ be a finite subgroup of the automorphism group $Aut_k(X)$, and $\mathcal{F}$ a locally free $G$-equivariant sheaf on $X$. Then $G$ acts on all the cohomology groups $H^i(X, \mathcal{F})$. Is the Serre duality $$H^i(X, \mathcal{F})\times H^{n-i}(X, \mathcal{F}^\vee\otimes \omega_X)\to H^n(X, \omega_X)=k$$ a $G$-equivariant perfect pairing? Where can I find a reference to this result?

Thank you.

• The isomorphism $\mathrm H^n(X, \omega_X) \simeq k$ is universal, that is, invariant under isomorphisms. The universal pairing is functorial under pullbacks. The result follows from this. – Angelo Mar 20 '13 at 11:14
• @Angelo: Perhaps the OP seeks a reference to justify the invariance you mention? Many references on Serre duality make the construction of the trace in a manner that is not sufficiently intrinsic to render the triviality apparent. It is equivalent to show that the natural composite map $H^n(X,\Omega^n_{X/k}) \rightarrow H^n(X,g^{\ast}(\Omega^n_{X/k})) \rightarrow H^n(X,\Omega^n_{X/k})$ is the identity (1st step pullback, 2nd step canonical at sheaf level); settling projective spaces "by bare hands" is a bit unpleasant (though easy by using the structure of the automorphism group). – user28172 Mar 20 '13 at 14:41
• @nosr, I tried to prove what you suggested and indeed it is not too difficult. However, I was trying to find a reference to include in a paper. – Jiangwei Xue Mar 26 '13 at 17:24