Exodromy

Abstract

Let $X$ be a quasicompact quasiseparated scheme. Write $\mathrm{Gal}(X)$ for the category whose objects are geometric points of $X$ and whose morphisms are specializations in the étale topology. We define a natural profinite topology on the category $\mathrm{Gal}(X)$ that globalizes the topologies of the absolute Galois groups of the residue fields of the points of $X$. One of the main results of this book is that $\mathrm{Gal}(X)$ variant of MacPherson's exit-path category suitable for the étale topology: we construct an equivalence between representations of $\mathrm{Gal}(X)$ and constructible sheaves on $X$. We show that this 'exodromy equivalence' holds with nonabelian coefficients and with finite abelian coefficients. More generally, by using the pyknotic/condensed formalism, we extend this equivalence to coefficients in the category of modules over profinite rings and algebraic extensions of ${\mathbf{Q}}_{\ell }$. As an 'exit-path category', the topological category $\mathrm{Gal}(X)$ also gives rise to a new, concrete description of the étale homotopy type of $X$. We also prove a higher categorical form of Hochster Duality, which reconstructs the entire étale topos of a quasicompact and quasiseparated scheme from the topological category $\mathrm{Gal}(X)$. Appealing to Voevodsky's proof of a conjecture of Grothendieck, we prove the following reconstruction theorem for normal varieties over a finitely generated field $k$ of characteristic $0$: the functor $X\mapsto \mathrm{Gal}(X)$ from normal $k$-varieties to topological categories with an action of $G_k$ and equivariant functors that preserve minimal objects is fully faithful.

Comments very welcome. v7: 253 pages. Extensively edited and reorganized the whole document and expanded on a number of points. Added Chapter 13 which uses pyknotic/condensed mathematics to extend the Exodromy Theorem to a classification result for $\ell$-adic sheaves