2

In Lectures on Étale Cohomology by J. S. Milne, Example 8.5(b) on Page 60, it is claimed that locally constant sheaves on [the étale site of] $U := \Bbb A^1_k \setminus \{0\}$, where $k$ is an algebraically closed field, correspond to modules over $\pi_1(U, \overline{u}) = \widehat{\Bbb Z}$.

But the closest statements I can find are:

  1. Locally constant sheaves on $U$ with finite stalks correspond to finite $\Bbb Z$-modules over $\pi_1(U, \overline{u})$. (ibid. Proposition 6.16; Stacks Project Tag 0DV5 (2))

  2. For a given locally constant sheaf $\mathcal{F}$ on $U$, there is a $\Bbb Z$-module $M$ and an étale covering $\{U_i \to U\}$ such that $\mathcal{F}|_{U_i} \cong \underline{M}|_{U_i}$. (ibid. Tag 09BF)

Obviously 1 cannot be applied, and for 2, in the best case scenario (the covering only has one map) we only get modules over $\pi_1(U, \overline{u}) = \widehat{\Bbb Z}$ that factor through some finite quotient.

Kenny Lau
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1 Answers1

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8.5(b) says "Let $\mathcal{F}$ be the locally constant sheaf on $U$ corresponding to a $\pi_1$-module...". It doesn't claim that all locally constant sheaves correspond to $\pi_1$-modules.

It means: "Let $\mathcal{F}$ be the locally constant sheaf on $U$ corresponding to a $\pi_{1}(U,\bar{u})$-module $F$ (as in 6.16) etc."

anon
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