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Here is a probably stupid question : If $F$ is a sheaf on the big Nisnevich site, then is the morphism $F(X) \to \amalg F(x)$ injective where the sum is over ALL the points of $X$ (not just the closed ones).

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    $\begingroup$ Well, a Nisnevich sheaf is also a Zariski one, so this seems to be ok.:) $\endgroup$ Sep 14, 2011 at 20:39
  • $\begingroup$ To be completely honest, my real question is about the cdh topology and $F = (KH_n)_{cdh}$, the cdh-ification of Weibel's homotopy invariant K-theory. I just thought the Nisnevich topology would be very similar but would attract more interest. The example below is not a counter example for the cdh-topology. $\endgroup$
    – name
    Sep 16, 2011 at 1:36

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It was a stupid question. And obviously not very clearly stated. By $F(x)$ I mean't $F(Spec\ k(x))$ where $k(x) = \mathcal{O}_{X,x} / \mathfrak{m}_x$ is the residue field of the point $x$ in the scheme $X$. The (étale) sheaf $N(X) = $ {$f \in \Gamma(X, \mathcal{O}_X)$ s.t $f$ is nilpotent } seems like a counter-example.

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