In this paper, Bridgeland showed that for a projective 3-fold $X$ with Gorenstein and terminal singularity and a crepant resolution $f:Y \to X$, there exists $g:W \to X$ such that $f$ is the flop of $g$ (i.e., for a divisor $D$ on $W$ if $-D$ is $g$-nef, then $D$ is $f$-nef) and moreover there is an exact equivalence $D^bCoh(W) \simeq D^bCoh(Y)$.
On the other hand, it seems to be known (e.g. https://mathoverflow.net/a/369756/177839) that this works when $X$ is not necessarily projective but just proper (while $f$ is still a projective morphism?). So, my questions are:
- Under which condition the extension to non-projective case is true?
- Is there any reference in such a case?
Thank you in advance.