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Let $\pi:X\to S=\operatorname{Spec } O_K$ be an arithmetic surface in the sense of Arakelov geometry. Here $K$ is a number field $\pi$ is a flat map and $X$ is a projective surface. For any coherent sheaf $\mathscr F$ on $X$ we have the determinant of cohomology: $$\det R\pi_\ast\mathscr F\in \operatorname{Pic }S$$ Moreover let $\omega_{X/S}$ be the usual dualizing sheaf. Can you please explain how can I get the following "duality formula"?

$$\det R\pi_\ast\mathscr F\cong\det R\pi_\ast\mathscr (\omega_{X/S}\otimes \mathscr F^\vee)$$

(I think one should assume also the flatness of $\mathscr F$ over $\mathscr O_S$).

Does it follow from some property of the determinant of cohomology? I've found the equation in Robin De Jong PhD thesis, I'll post it below even if I think there is a typo in the main formula:

enter image description here

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Recall that Serre duality says that $H^{0}(X,F)\cong H^{1}(X, K\otimes F^{*})^{*}$. So the determinant of cohomology just follows from this. However we do need slightly more machinery than Serre duality, as the determinant of cohomology is defined in a relative setting ($X\rightarrow S$). In our case $S$ is affine, so Hartshorne Proposition 8.5 is suffice, which says if $X$ is Noetherian and $Y$ affine, then $R^{i}f_{*}(\mathcal{F})\cong \widetilde{H^{i}(X,\mathcal{F})}$ for any quasi-coherent $\mathcal{F}$ on $X$. Together with Serre duality this then gives you back what you wanted.

A good reference on determinant of cohomology is

Arbarello/Cornalba/Griffiths: "Geometry of algebraic curves II"

Chapter 8, which I learned from Robert Wilms via his answer.

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  • $\begingroup$ I have a question regarding that chapter. You know that the usual intersection number between two line bundles $L,M$ on an algebraic surface can be calculated by means of $\chi_k$ (The Poincare characteristic) applied to $L^{-1}$ and $M^{-1}$. In the aforementioned chapter there is a similar formula linking the det. of cohomology to the Deligne pariring. Actually the formulas are almost the same, the unique difference is that in the arithmetic case we apply the determinant of the cohomology to $L$ and $M$ (there is no $-1$). This sounds weird to me, why such a difference? $\endgroup$ – manifold Oct 28 '17 at 11:15
  • $\begingroup$ @manifold: I do not really know what you are talking about. I did not see any typo in the book. Maybe you can ask this in a separate question if you like. $\endgroup$ – Bombyx mori Oct 28 '17 at 14:18
  • $\begingroup$ It's not a typo, it was just a comment not worth of a complete question. I'll write a clear comment later. $\endgroup$ – manifold Oct 28 '17 at 16:13
  • $\begingroup$ $Ext^i(\mathcal F, \omega) \cong H^{n-i}(X, \mathcal F)^∗$, If $\mathcal F$ is locally free, the left side becomes $H^i(X, \mathcal F^∨ ⊗\omega_X)$ where $X$ is a smooth projective manifold. But for singular varieties, we have the failure of Serre duality. For some cases like Complete normal Cohen-Macaulay variety, we have also Serre duality $\endgroup$ – user21574 Oct 28 '17 at 18:50
  • $\begingroup$ See Theorem 1.2 for Serre duality (the relative version is due to Grothendieck )math.leidenuniv.nl/~wzomervr/2014-coco/preliminaries.pdf, we must add some weak condition also to define $f^!$ $\endgroup$ – user21574 Oct 28 '17 at 22:52

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