Theorem 58.67.1 (The Fundamental Exact Sequence). There is a short exact sequence of étale sheaves on $X$

**Proof.**
Let $\varphi : U \to X$ be an étale morphism. Then by properties of étale morphisms (Proposition 58.26.2), $U = \coprod _ i U_ i$ 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

This amounts to a sequence

which, unfolding definitions, is nothing but a sequence

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 $f_ j \in k(U)^*$. $\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (1)

Comment #5433 by Hao on