How to conclude the quasi-projective case of the derived McKay correspondence from the projective case?

Question

I am currently trying to understand the paper "Mukai implies McKay" from Bridgeland, King and Reid (cf. here). Let me sum up the setting we find ourselves in:

Let $M$ be a smooth quasi-projective variety over an algebraically closed field $k$ of characteristic $0$, on which a finite group $G$ acts such that $\omega_M$ is locally trivial as a $G$-sheaf. Let $Y$ be the irreducible component of the $G$-Hilbert scheme that contains the free orbits. Let $Z \subseteq Y \times M$ be the universal closed subscheme and denote by $p : Z \rightarrow Y$ and $q : Z \rightarrow M$ the projections.

Then an equivalence of bounded (equivariant) derived categories $$ \Phi = q_* p^* : D(Y) \longrightarrow D^G(M) $$ is established, by first proving the case where $M$ is projective and then deducing from this the quasi-projective case. I have understood the proof of the projective case, but I don't quite follow the generalizing step. Specifically, I am unsure about the following:

It is claimed that, when $M$ is only quasi-projective, the restricted functor $$ \Phi_c = q_* p^* : D_c(Y) \longrightarrow D_c^G(M) $$ (to objects with proper support) is an equivalence by virtue of repeating the arguments of the projective case. But the arguments involve Serre duality and spanning classes, so they do not obviously apply.

Question: What idea am I missing that allows me to repeat the projective argument? Perhaps it involves picking a smooth and projective closure $M \subseteq \overline{M}$ to which the projective argument applies — but then I don't see how to extend the action of $G$ from $M$ to $\overline{M}$.

I hope I have expressed myself in an understandable manner. Any help would be greatly appreciated!

Answer 1

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).