The Fourier–Mukai formulas for derived duals and adjunction maps

Let $Z$ and $X$ be separated schemes of finite type over $k$. Let $\mathcal{A}$ and $\mathcal{B}$ be DG-enhancements of $D(Z)$ and $D(X)$, and let $\bar{S}\in D(Z\times X)$ correspond to $S\in D_c(\mathcal{A}\text{-}\mathcal{B})$ that is $\mathcal{A}$- and $\mathcal{B}$-perfect. Write $\bar{L}$ and $\bar{R}$ for the Fourier–Mukai kernels corresponding to the derived left and right duals. Fourier–Mukai kernel formula conjecture. One should have

$$\bar{L}\simeq \operatorname{\mathbf R}\operatorname{\mathcal{H}om}_{Z\times X}\left(\bar{S},\pi_Z^!(\mathcal{O}_Z)\right),\qquad \bar{R}\simeq \operatorname{\mathbf R}\operatorname{\mathcal{H}om}_{Z\times X}\left(\bar{S},\pi_X^!(\mathcal{O}_X)\right).$$

Moreover, the maps in $D(Z\times Z)$ and $D(X\times X)$ corresponding to the derived trace and action maps are isomorphic to the explicit maps in the cited works that lift the adjunction counits and units of Fourier–Mukai transforms to Fourier–Mukai kernels. The conjecture gives concrete geometric formulas for abstract Morita-theoretic constructions and would make the adjunction data computable directly from the kernel; the source does not provide a resolution.