Theorem 59.68.1 (03RI): The Fundamental Exact Sequence

Theorem 59.68.1 (The Fundamental Exact Sequence). There is a short exact sequence of étale sheaves on $X$

$0 \longrightarrow \mathbf{G}_{m, X} \longrightarrow j_* \mathbf{G}_{m, \eta } \longrightarrow \bigoplus \nolimits _{x \in X_0} {i_ x}_* \underline{\mathbf{Z}} \longrightarrow 0.$

Proof. Let $\varphi : U \to X$ be an étale morphism. Then by properties of étale morphisms (Proposition 59.26.2), $U = \coprod _ i U_ i$ where each $U_ i$ is a smooth curve mapping to $X$ . The above sequence for $U$ is a product of the corresponding sequences for each $U_ i$ , so it suffices to treat the case where $U$ is connected, hence irreducible. In this case, there is a well known exact sequence

$1 \longrightarrow \Gamma (U, \mathcal{O}_ U^*) \longrightarrow k(U)^* \longrightarrow \bigoplus \nolimits _{y \in U_0} \mathbf{Z}_ y.$

This amounts to a sequence

$0 \longrightarrow \Gamma (U, \mathcal{O}_ U^*) \longrightarrow \Gamma (\eta \times _ X U, \mathcal{O}_{\eta \times _ X U}^*) \longrightarrow \bigoplus \nolimits _{x \in X_0} \Gamma (x \times _ X U, \underline{\mathbf{Z}})$

which, unfolding definitions, is nothing but a sequence

$0 \longrightarrow \mathbf{G}_ m(U) \longrightarrow j_* \mathbf{G}_{m, \eta }(U) \longrightarrow \left(\bigoplus \nolimits _{x \in X_0} {i_ x}_* \underline{\mathbf{Z}}\right)(U).$

This defines the maps in the Fundamental Exact Sequence and shows it is exact except possibly at the last step. To see surjectivity, let us recall that if $U$ is a nonsingular curve and $D$ is a divisor on $U$ , then there exists a Zariski open covering $\{ U_ j \to U\}$ of $U$ such that $D|_{U_ j} = \text{div}(f_ j)$ for some $f_ j \in k(U)^*$ . $\square$

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