# On the Fourier-Mukai transform in families

Let $X$ be an abelian variety over $\mathbb C$ and $\hat X$ its dual. Let $S$ be a scheme over $\mathbb C$, and $$0\to E\to F\to G\to 0$$ a short exact sequence of coherent sheaves on $X\times S$, all flat over $S$. In my setting $E$ and $F$ are $S$-families of torsion-free sheaves on $X$, and the quotient $G$ is finite over $S$. I wonder if it is legal to apply the Fourier-Mukai transform to the above exact sequence, and if doing so one gets an exact triangle $$\Phi(E)\to \Phi(F)\to \Phi(G)\overset{+1}{\to} \qquad\textrm{in } D^b(\hat X\times S).$$ In other words, the hope is that with the above assumptions on the families of sheaves one could use the functoriality of FM transform to construct an actual morphism between the relevant moduli spaces of sheaves/complexes on $X$ and $\hat X$, and that in addition this would give an exact triangle on $X\times S$, which is certainly the case if $S=\textrm{Spec }\mathbb C$. Does any of this make sense?

Thanks!

$\,\!$Hi Andrea. It's not just legal to apply the Fourier-Mukai transform here, it is highly encouraged. For an abelian scheme $A \to S$ the Fourier transform defines an exact equivalence of triangulated categories $D^b(A) \to D^b(\widehat A)$, where $\widehat A$ is the dual abelian scheme over $S$. This applies in particular to your case when your abelian scheme is a trivial fibration $X \times S \to S$.