I am currently trying to understand the paper "Mukay 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.

$\textbf{Question}:$ 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!

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!