Proper base change for étale sheaves of spaces (Q2087378)

The main result of this paper is the nonabelian generalization of proper base change theorem in étale cohomology, which works for étale sheaves valued in spaces that satisfy certain finiteness condition named truncated coherent étale sheaves. There are various application of this theorem, including a proof that profinite étale homotopy type commutes with finite product and symmetric power of proper algebraic spaces. The contents in more detail: In Section 1 the author states the main theorem and applications as well as the strategy of the proof. In Section 2 the author gives a review of shapes and profinite completions in the infinity categorical context. Given an \(\infty\)-topos \(\mathcal{X}\), there is an essentially unique geometric morphism \(\pi_*:\mathcal{X}\longrightarrow\mathcal{S}\), the composition \(\pi_*\pi^*\) is a pro-space \(\mathrm{Sh}(\mathcal{X})\) named the shape of \(\mathcal{X}\). This is a generalization of the étale homotopy type to general topoi. In Section 3 the author studies limits of \(\infty\)-topoi and proves that being a truncated pullback guarantees that the global section in the limit topos is equivalent to the colimit of global sections in each topos. In Section 4 the author proves the proper base change theorem. The proof combines the standard technology in the abelian setting with the continuity properties of truncated objects. In Section 5 and 6 the author proves that profinite shapes functor commute with finite products and symmetric powers for proper schemes by the main theorem. The basic idea is that the main theorem allows us to directly identify \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) with \(\mathrm{Sh}(X\times Y)\) upon profinite completion, while \(\mathrm{Sh}(X)\circ\mathrm{Sh}(Y)\) is equivalent \( \mathrm{Sh}(X)\times \mathrm{Sh}(Y)\) on finite spaces by straightforward computation.