Let $(X_i)$ be a projective system of schemes such that $\hat{X} := \varprojlim X_i $ exist as a scheme ; let $Et(\hat{X})$(resp. $Et(X_i)$) the etale homotopy type of $\hat{X}$ (resp. of $X_i$) ; suppose that $H_n(Et(\hat{X}), \hat{\mathbb{Z}}) = 0$, do we have $\varprojlim H_n(Et(X_i), \hat{\mathbb{Z}}) = 0$ ?

  • $\begingroup$ Are they affine schemes? It so, you can see Corollary 4.4 here: arxiv.org/pdf/1905.06243.pdf $\endgroup$ Jun 15, 2020 at 6:39
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    $\begingroup$ We need the limit to be op-filtered with affine transition maps, and the $X_i$'s quasi-compact and quasi-separated. Then this is a standard result from the 1960's. $\endgroup$ Jun 15, 2020 at 8:58
  • $\begingroup$ @Denis-CharlesCisinski thank you for your comment ; pleas i need specific references $\endgroup$ Jun 15, 2020 at 14:32
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    $\begingroup$ By a Yoneda type argument, the Formula is equivalent to saying that for a locally constant sheaf A wih finite fibers, the $j$th cohomology of the limit is isomorphic to the filtered colimit of the $H^j(X_i,A)$ (with $j=0$ if $A$ is a sheaf of sets, $j=1$ if $A$ is a sheaf of groups, $j\geq 0$ if $A$ is a sheaf of abelian groups). This is established as Theorem 5.7 in Exposé VII of SGA4 in the case where $A$ is abelian. The case of non abelian coefficients is described in Remark 5.14 of loc. cit. $\endgroup$ Jun 15, 2020 at 19:47
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    $\begingroup$ By "The Formula" I meant the fact that the homology of the limit is the limit of the homologies of the $X_j$'s (where the second instance of limit means in the sense of pro-objects in the derived category of abelian groups). The vanishing result you seem to want follows then immediately from there whatever you mean by limit. $\endgroup$ Jun 15, 2020 at 19:55


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