An injectivity property for étale cohomology. (English) Zbl 0828.14011
In this paper, we consider the following situation: Suppose \({\mathfrak O}\) is a commutative local ring, \(M \to \text{Spec} ({\mathfrak O})\) a smooth morphism, \(\omega \in \text{Spec} ({\mathfrak O})\) the closed point, \(U \subset \text{Spec} ({\mathfrak O})\) a quasi-compact open subscheme, \(\Phi\) a complex of étale sheaves on \(U\) whose cohomology sheaves are torsion and bounded below. Denote \(Q \Phi =\) the image of \(\Phi\) in the derived category of the category of abelian sheaves on \(U_{\text{ét}}\).
\[
\begin{matrix} \overline M & @>i>> & M & @>j>> & \pi^{-1} (U) \quad & S = \text{Spec} ({\mathfrak O}) \\ \downarrow \pi_0 & & \downarrow \pi & & \downarrow \pi_U \quad & \omega = \text{Spec} (k) \\ \omega & @>i_0>> & S & @>j_0>> & U, \quad & \overline M = M \times_{\mathfrak O} \omega. \end{matrix}
\]
Let \(K = i^* Rj_*(Q (\pi^*_U \Phi)) \in D^+ (\overline M_{\text{ét}}, \mathbb{Z})\). If \(\xi @>>\varphi> M\) is a morphism where \(\xi\) is the spectrum of a field \(k (\xi)\) which is separable algebraic over \(k (\varphi (\xi))\), we can form \({\mathfrak O}^h_{M, \xi} =\) the unramified extension corresponding to \(k (\varphi (\xi)) \to k (\xi)\) of the henselization of \({\mathfrak O}_{M, \varphi (\xi)} = \Gamma (\xi, \varphi^*_{\text{ét}} {\mathfrak O}_{M_{\text{ét}}})\), and let \(M^h_\xi = \text{Spec} ({\mathfrak O}^h_{M, \xi})\).
Consider a point \(\xi \in \overline M\) and the generic point \(\eta \to \overline M^h_\xi\). Recall that \(\overline M^h_\xi\) is regular (and hence irreducible) as \({\mathfrak O}_{ \overline M, \xi}\) is. \({\mathfrak O}^h_{M, \eta}\) is the henselization of the local ring of \(M^h_\xi\) at \(\eta\), so we have a morphism \(M^h_\eta @>>\alpha> M^h_\xi\).
Theorem 1. For all \(q \in \mathbb{Z}\) the map \(H^q (M^h_\xi \times_{\mathfrak O} U\), pr\(^*_2 \Phi) @>>(\alpha \times\text{id})^*> H^q (M^h_\eta \times_{\mathfrak O} U, \text{pr}^*_2 \Phi)\) is injective.
Theorem 1 can be equivalently stated as:
Theorem 1’. The composed maps \(R^q \Gamma (\xi, K_{|\xi}) @< \sim<< R^q \Gamma (\overline M^h_\xi, K) \to R^q \Gamma (\eta, K_{|\eta})\), \(q \in \mathbb{Z}\), are injective.
Consider a point \(\xi \in \overline M\) and the generic point \(\eta \to \overline M^h_\xi\). Recall that \(\overline M^h_\xi\) is regular (and hence irreducible) as \({\mathfrak O}_{ \overline M, \xi}\) is. \({\mathfrak O}^h_{M, \eta}\) is the henselization of the local ring of \(M^h_\xi\) at \(\eta\), so we have a morphism \(M^h_\eta @>>\alpha> M^h_\xi\).
Theorem 1. For all \(q \in \mathbb{Z}\) the map \(H^q (M^h_\xi \times_{\mathfrak O} U\), pr\(^*_2 \Phi) @>>(\alpha \times\text{id})^*> H^q (M^h_\eta \times_{\mathfrak O} U, \text{pr}^*_2 \Phi)\) is injective.
Theorem 1 can be equivalently stated as:
Theorem 1’. The composed maps \(R^q \Gamma (\xi, K_{|\xi}) @< \sim<< R^q \Gamma (\overline M^h_\xi, K) \to R^q \Gamma (\eta, K_{|\eta})\), \(q \in \mathbb{Z}\), are injective.
MSC:
| 14F20 | Étale and other Grothendieck topologies and (co)homologies |
References:
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