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In rational homotopy theory there is a basic construction which, given a prime number $l$ and a $CW$-complex $X$, produces a localized space $X_l$ equipped with a map $X\rightarrow X_l$ that induces an isomorphism in homology: $$H_{\bullet}(X_l,\mathbb{Z})\simeq H_{\bullet}(X,\mathbb{Z}_{(l)})$$ where $\mathbb{Z}_{(l)}$ is the localization of the ring of integers at the given prime number. On the other hand, if $X$ is a complex algebraic variety, in algebraic geometry there is a way to construct the $l$-adic cohomology groups $H^{\bullet}_{ét}(X,\mathbb{Z}_l)$, where $\mathbb{Z}_l=\lim_n(\mathbb{Z}/l^n\mathbb{Z})$ is the ring of $l$-adic numbers.

My question is: Is there any relation between this two constructions?...are $l$-adic cohomology groups isomorphic to the cohomology groups of a sort of localization of the variety?

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    $\begingroup$ By the (non-trivial!) $\ell$-adic version of the Artin comparison theorem, if $X$ is any separated $\mathbf{C}$-scheme of finite type then ${\rm{H}}^{\bullet}_{\rm{et}}(X, \mathbf{Z}_{\ell}) = {\rm{H}}^{\bullet}_{\rm{top}}(X(\mathbf{C}), \mathbf{Z}_{\ell})$. So your question ultimately doesn't involve etale cohomology at all: it is entirely a question about relating rational homotopy theory and $\ell$-adic topological cohomology for the topological space $X(\mathbf{C})$ with $X$ a separated $\mathbf{C}$-scheme of finite type. $\endgroup$
    – nfdc23
    Commented Mar 24, 2017 at 2:17
  • $\begingroup$ I don't understand what $X_l$ is or what role it plays in your motivating example. Could you elaborate on that? $\endgroup$
    – R. van Dobben de Bruyn
    Commented Mar 24, 2017 at 22:04
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    $\begingroup$ There is a notion of localizing topological spaces / homotopy types. See May-Ponto, "More Concise Algebraic Topology" or Sullivan's famous MIT notes, available here: maths.ed.ac.uk/~aar/books/gtop.pdf $\endgroup$
    – Jens Reinhold
    Commented Mar 24, 2017 at 23:23
  • $\begingroup$ @nfdc23 I see. Hence, I should probably ask if, taking into account what you're saying, the groups $H^{\bullet}_{ét}(X,\mathbb{Z})$ corresponds to the $l$-adic topological cohomology of the "homotopy type" of $X$, somehow defined without invoking the associated complex manifold $X(\mathbb{C})$. I'm wonder if this kind of homotopical constructions make any sense at the level of the étale site, or something. Thanks for coment. $\endgroup$
    – user95283
    Commented Mar 26, 2017 at 4:34
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    $\begingroup$ There is a precise relation in the spirit of @nfdc23's comment. Artin and Mazur have defined the étale homotopy type of a scheme. The homotopical version of Riemann's existence theorem then asserts that the étale homotopy type of a scheme (finite type, separated, unibranch) $X$ over $\mathbf C$ is the profinite completion of the classical homotopy type of the topological space $X(\mathbf C)$. Cf. the book of Artin-Mazur (Lecture notes in maths. 100), or their introductory paper in Proceedings of a conference on local fields (Driebergen 1966). $\endgroup$
    – ACL
    Commented Mar 26, 2017 at 9:50

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