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



Gabber's original proof of his purity theorem
It's different, but it also uses Weil II. See
 [Purity for intersection cohomology after Deligne-Gabber](http://www.math.purdue.edu/~dvb/preprints/delgab.pdf)
for my translation of the original.


