Let $A$ be an abelian variety and $\hat A$ be the dual abelian variety. If $P$ is the (normalized) Poincare line bundle, then Mukai defines $R\hat S:D(A)\to D (\hat A)$ via $R\hat S(?)=Rp_{\hat A,*}(Lp_A^*(?)\otimes P)$ and $RS:D(\hat A)\to D(A)$ via $R S(?)=Rp_{ A,*}(Lp_{\hat A}^*(?)\otimes P)$. He then shows that $R\hat S\circ R\hat S =(-1_A)^*[-g]$ on $D^b_{qcoh}(A)$ and $R\hat S\circ R\hat S =(-1_{\hat A})^*[-g]$ on $D^b_{qcoh}(\hat A)$. The proof uses the projection formula which can fail for non-quasi coherent sheaves and hence for $D(A)$. My question is: Is there a counter example to $R\hat S\circ R\hat S =(-1_A)^*[-g]$ in $D(A)$?
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2$\begingroup$ What do you denote by $D(A)$? $\endgroup$– SashaCommented Jun 28, 2016 at 8:01
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$\begingroup$ The derived category of $\mathcal O _A$ modules. I am not so worried about bounded vs unbounded, but I am more worried / interested about the non-quasi coherent case. $\endgroup$– HaconCommented Jun 28, 2016 at 11:50
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$\begingroup$ Dear @Hacon, if the equivalence fails for all $O_A$-modules, then Theorem 2.2 in Mukai's paper "Duality between $D(X)$ and $D(\hat{X})$ with its applications to Picard sheaves" is wrong? And Theorem 2.1 of the paper "Non-commutative tori and Fourier-Mukai duality" is similarly erroneous? $\endgroup$– Doug LiuCommented Feb 24, 2023 at 8:11
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$\begingroup$ Yes, that's right, those theorems are wrong as stated (I suspect that Mukai was thinking of quasi-coherent sheaves and simply forgot to add the hypothesis) $\endgroup$– HaconCommented Feb 25, 2023 at 14:16
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I think it is clear that $R\hat{S}$ is not an equivalence on the categories of all $O$-modules. Indeed, if it were an equivalence, it would send product to product, and also preserve quasi-coherence. Let $M_x$ be a sky-scraper sheaf at $x\in A$; its image under $R\hat{S}$ is an invertible $O$-module on $A^\vee$ (up to a shift). However, an infinite product of copies of $M_x$ is quasi-coherent, while the infinite product of invertible $O$-modules is not quasi-coherent.
