Independence of \(\ell\) for the supports in the decomposition theorem (Q1667298)

The author introduced a notion of a perverse compatible system on \(\mathbb{F}_q\)-schemes, which is a variant of a compactible system for perverse \(t\)-structure. Its relations with the classical definition are investigated in Section 2. The main theorem of this paper is the following: the proper pushforward of a perverse compatible system of a direct sum of shifted semisimple perverse sheaves is also perverse compatible. The key ingredients of the proof are the existence of \(\ell\)-adic companions on smooth schemes[Theorem 2.5] and Deligne's weight theory. This main theorem gives a relative version of Gabber's result on the independence of \(\ell\) of intersection cohomology. That is, the support of proper pushforward of the \(\ell\)-adic intersection complex is independent of \(\ell\). At the end of this paper, the author remarked a generalization to \(\mathbb{F}_q\)-Artin stacks.