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Let $k$ be a field and $X$ a smooth $k$-scheme. We consider now the pointed (constant) simplical Nisnevich-sheaf $X_{+}$ that is represented by $X$. Let now $\nu\in U$ be a point of a smooth scheme, then the fibre of $X_{+}$ at $\nu$ is defined as $\text{colim} \\ X_{+}(V)$ where the colimit is over all Nisnevich neighborhoods $V$ of $\nu$. What is this fiber now? Is this just the constant simplicial set associated to the set of all points of $X$ with residue field isomorphic to the one of $\nu$?

And as a further question; If I have a morphism $f:X\rightarrow Y$ between two smooths $k$-schemes, when is the associated morphism of simplicial Nisnevich sheaves $f_{+}:X_{+}\rightarrow Y_{+}$ a simplicial weak equivalence? For example in the case where $Y$ is irreducible and $f$ an open immersion of a dense open subscheme $X$?

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What exactly do you mean by the second question? Specifically, what is a weak equivalence? Are you using the model category structure where $X\rightarrow Y$ is a weak equivalence if and only if it induces a weak equivalence on homotopy sheaves? –  Benjamin Antieau Nov 15 '09 at 21:55
    
Yes, exactly. Or equivalently $f:X\rightarrow Y$ is a weak equivalence if and only if $f_{\nu}:X_{\nu}\rightarrow Y_{\nu}$ is a weak homotopy equivalence of simplicial sets for all points $\nu$ –  Ted Nov 16 '09 at 0:32

2 Answers 2

Let $N:=\textrm{Neigh}\_{Nis}(U,\nu)$ denote the filtered category of Nisnevich neighborhoods of the point $\nu\in U$. Then $\textrm{colim}\_{V\in N^{op}} V\cong \textrm{spec}(\mathcal{O}^{h}\_{U,\nu})$, where $N^{op}$ denotes the opposite category of $N$ and $\mathcal{O}^{h}\_{U,\nu}$ is the henselization of the local ring $\mathcal{O}\_{U,\nu}$. My guess now would be that the fiber is the constant simplicial set associated to the set $\textrm{Hom}\_{k}(\textrm{spec}(\mathcal{O}^{h}\_{U,\nu}),X)$.

Maybe someone can clarify this as well as your second question.

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It looks like you probably need a stronger condition for the lements of the fiber. Specifically, it is not enough that residue fields should be isomorphic. There also must be a morphism on the Nisnevich local rings. Thus, I imagine that $$colim X\_+(V)$$ should correspond to the set of all points of $X$ such that the henselization of the local ring is isomorphic to the henselization of the local ring of $\nu$.

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