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 $\varprojlim H_n(Et(X_i), \hat{\mathbb{Z}}) = 0$ ?
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$\begingroup$ Are they affine schemes? It so, you can see Corollary 4.4 here: arxiv.org/pdf/1905.06243.pdf $\endgroup$– David CarchediJun 15, 2020 at 6:39

1$\begingroup$ We need the limit to be opfiltered with affine transition maps, and the $X_i$'s quasicompact and quasiseparated. Then this is a standard result from the 1960's. $\endgroup$– D.C. CisinskiJun 15, 2020 at 8:58

$\begingroup$ @DenisCharlesCisinski thank you for your comment ; pleas i need specific references $\endgroup$– Moutand MohammedJun 15, 2020 at 14:32

1$\begingroup$ 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. $\endgroup$– D.C. CisinskiJun 15, 2020 at 19:47

1$\begingroup$ 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 proobjects in the derived category of abelian groups). The vanishing result you seem to want follows then immediately from there whatever you mean by limit. $\endgroup$– D.C. CisinskiJun 15, 2020 at 19:55
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