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Moutand Mohammed
Moutand Mohammed
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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), \hat{\mathbb{Z}}) = 0$$\varprojlim H_n(Et(X_i), \hat{\mathbb{Z}}) = 0$ ?

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 $\varprojlim H_n(Et(X), \hat{\mathbb{Z}}) = 0$ ?

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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Moutand Mohammed
Moutand Mohammed
  • 181
  • 5

Homology group of the étale homotopy type of a projective limit of schemes

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 $\varprojlim H_n(Et(X), \hat{\mathbb{Z}}) = 0$ ?