Paragraph 3.2 explains that duality still works for quasi-projective non-singular varieties if you restrict to the appropriate categories. Also I think there should be no particular problem with a spanning class argument?

Paragraph 3.2 explains that duality still works for quasi-projective non-singular varieties if you restrict to the appropriate categories. Also I think there should be no particular problem with a spanning class argument?

Edit : the proof that skyscraper sheaves form a spanning set for $D(X)$ only uses Serre duality (my handy reference for derived categories is "Fourier-Mukai transforms in algebraic geometry" by D. Huybrechts). This is stated without proof for $X$ only smooth in the article, but I agree it seems you need Serre duality. It gives you anyway a spanning set for $D_c(Y)$ which is what you want.

So ultimately we need to check the isomorphism $f^! (F) \cong f^*(F) \otimes f^!(\mathcal O_Y)$ if $F$ has proper support. Looking back at the original proof of Neeman I couldn't deduce quickly a proof (but maybe with other proof it's easier or maybe I don't understand it well enough). However it seems that the results of https://arxiv.org/abs/1810.06082 could apply : it gives Grothendieck duality for a non-proper morphism, but you have to modify $f^!$ by precomposing with the "proper support" functor (which is identity in our case).