1
$\begingroup$

A theorem due to Artin states that for a smooth scheme $X$ of finite type over $\mathbb{C}$ and a locally constant constructible sheaf $F$ we have an isomorphism $$ H^*_{et}(X, F)\approx H^*(X(\mathbb{C}), F) $$ where the LHS is etale cohomology and the RHS is derived-functor cohomology of $F$ in analytic topology. One proof I know proceeds by establishing that there is an open covering of $X$ by $K(\pi, 1)$ spaces. However, the appearance of $K(\pi, 1)$'s feels somewhat weird to me. Is there an alternative proof that does not rely on $K(\pi, 1)$ neighbourhoods?

Improve this question
$\endgroup$
3
  • 2
    $\begingroup$ Yes, check one of the following exposes of SGA4.3. $\endgroup$
    – Piotr Achinger
    Nov 30, 2018 at 7:51
  • 1
    $\begingroup$ @PiotrAchinger do you mean SGA 4, Expose 16, theorem 4.1? $\endgroup$
    – misha
    Nov 30, 2018 at 7:57
  • 4
    $\begingroup$ Exactly. A mush stronger result (no smoothness assumptions on $X$ and $F$), proved by devissage with reduction to curves, but without elementary fibrations. $\endgroup$
    – Piotr Achinger
    Nov 30, 2018 at 8:04

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.