Let $X$ be a Noetherian scheme that is not Gorenstein but possesses a dualizing complex $D$ of coherent sheaves. Then (if I understand these matters and the answer to the question [Characterization of schemes whose dualizing complex is perfect](https://mathoverflow.net/questions/107231/characterization-of-schemes-whose-dualizing-complex-is-perfect) correctly) the bounded derived category $D^b\_{coh}(X)$ of coherent sheaves on $X$ is self-dual, but this duality $D\_X$ does not send (all) perfect complexes into perfect ones (recall that an object of $D^b\_{coh}(X)$ is a perfect complex if it is locally quasi-isomorphic to a bounded complex of free sheaves). 


My question is: did anybody study the image of the triangulated subcategory of perfect complexes $D^{perf}(X)\subset D^b\_{coh}(X)$ under the coherent duality $D\_X$? Is there any name for this subcategory $D\_X(D^{perf}(X))\subset D^b\_{coh}(X)$? Is it true that $D\_X(D^{perf}(X))\cong D^{perf}(X)\otimes D$? Does it follow that $D\_X$ induces an equivalence $D^{perf}(X)^{op}\cong D^{perf}(X)$?



Which complexes of coherent sheaves are dual to perfect ones?
Since perfect complexes are dualizable, for every perfect complex $P$ and any complex $Q$ we have
$$\mathrm{hom}(P,Q)\cong \mathrm{hom}(P,1)\otimes Q\,.$$
Moreover $\mathrm{hom}(P,1)$ is perfect too (since for qcqs schemes perfect=dualizable). In particular, by taking $Q=\omega$ the dualizing sheaf this shows that the image of the duality $D(-):=\mathrm{hom}(-,\omega)$ is exactly $\mathrm{Perf}(X)\otimes\omega$.


I don't think you can imply any more than this (in particular it's unclear to me why you would expect this to give an equivalence $\mathrm{Perf}(X)^{op}\cong \mathrm{Perf}(X)$ different from the standard duality.


