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But @DavidHansen is correct that the given definition is too specific: perverse coherent sheaves can be defined in many situations other than crepant resolutions of projective threefolds...
@MikhailBorovoi Dear Professor Borovoi, yes I agree that section has many useful facts. I think that essentially what I want is an explicit version of the inverse of the map in Serre's thm 1. Given an $\overline{K}$-valued cocycle, how to construct a $K^{nr}$-valued cocycle that corresponds under inflation. Or maybe there is another way to think about it?
