Mathematics > Algebraic Topology
[Submitted on 9 Jul 2018 (v1), last revised 23 Aug 2020 (this version, v7)]
Title:Exodromy
View PDFAbstract:Let $X$ be a quasicompact quasiseparated scheme. Write $\operatorname{Gal}(X)$ 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 $\operatorname{Gal}(X)$ 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 $\operatorname{Gal}(X)$ variant of MacPherson's exit-path category suitable for the étale topology: we construct an equivalence between representations of $\operatorname{Gal}(X)$ 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 $\operatorname{Gal}(X)$ 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 $\operatorname{Gal}(X)$. 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 $X\mapsto\operatorname{Gal}(X)$ from normal $ k $-varieties to topological categories with an action of $\operatorname{G}_{k}$ and equivariant functors that preserve minimal objects is fully faithful.
Submission history
From: Peter Haine [view email][v1] Mon, 9 Jul 2018 17:33:02 UTC (200 KB)
[v2] Mon, 6 Aug 2018 16:54:41 UTC (206 KB)
[v3] Wed, 28 Nov 2018 17:37:09 UTC (205 KB)
[v4] Tue, 16 Apr 2019 15:45:36 UTC (215 KB)
[v5] Wed, 24 Apr 2019 16:04:45 UTC (215 KB)
[v6] Wed, 10 Jul 2019 11:27:37 UTC (242 KB)
[v7] Sun, 23 Aug 2020 02:46:09 UTC (317 KB)
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