Gabber's purity theorem is the statement that if $\mathscr{F}$ is a pure perverse sheaf on an open subvariety $j : U \hookrightarrow X$ then so is $j_{!*} \mathscr{F}$.

It is remarkable because it gives many objects besides (pure) local systems on smooth varieties for which purity and the Weil conjectures hold.

It is proved in $\S$5 of Faisceaux pervers by BBD(G). Here the proof is quite natural once one accepts deep theorems of Deligne: $j_{!*} \mathscr{F}$ occurs as the image of a map between an object with weights $\le m$ and an object with weights $\ge m$ and hence has weights $\{ \ge m \} \cap \{ \le m \} = \{ m \}$.

However I understand that Gabber found another proof earlier. Is this the case? Or is it similar to the one given by [BBD(G)]?

Gabber's work is often cited as:

[Ga1] O. Gabber : Pureté de la cohomologie de MacPherson-Goresky rédigé par P. Deligne, prépublication I.H.E.S., 1981


It's different, but it also uses Weil II. See Purity for intersection cohomology after Deligne-Gabber for my translation of the original.

  • 3
    $\begingroup$ Thank you very much! I wish I had more than one up vote :) $\endgroup$ – Geordie Williamson Feb 16 '16 at 11:32

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