# Is there a nice way to characterise the derived equivalence induced by a flop?

Hello, I'm interested in the case when all varieties are projective threefolds over the complex numbers.

Start with a flopping contraction $f:Y \rightarrow X$, with corresponding flop $f^+: Y^+ \rightarrow X$. It has been proved that there then exists an equivalence $\Phi : D^b(Y^+) \rightarrow D^b(Y)$.

Is there a way to understand this (or any other) equivalence explicitly?

I've heard there is a way to find an equivalence by considering a common resolution of $Y$ and $Y^+$ and then using derived pullback and pushforward, is it true?

I am mostly interested in what happens to sheaves on $Y^+$ supported on the exceptional locus.

Thanks.

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"I've heard there is a way to find an equivalence by considering a common resolution of Y and Y+ and then using derived pullback and pushforward, is it true?" True. That's all it is. –  VA. Apr 21 '10 at 3:46

As always, it depends on what you think "explicitly" means. It's a Fourier-Mukai transform; see, for example, Van den Bergh and Hille's expository article. It can also be explained in terms of so-called non-commutative crepant resolutions, see Van den Bergh.

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Thanks, this expository article seems very useful! –  babubba Apr 22 '10 at 22:48